By Zak Toews with assistance of AI

4th Year Software Engineering Student and Hobbyist

Data Collected

All data is anonymous. No personal data is recorded.

This is subject to change as the analytics evolve.

Notes about Data

The AnalyticsEvent table has two different id/createdAt formats mixed together:

• main-site events (raw SQL insert) always used gen_random_uuid() for id and Postgres’s NOW() for createdAt — a UUID and microsecond-precision timestamp.
• trip-planner events (via Prisma) used Prisma’s client-side cuid() and Date.now() defaults instead of the database — a cuid and only millisecond-precision timestamp.

This got fixed at the schema level (trip-planner/prisma/schema.prisma now uses dbgenerated("gen_random_uuid()") / dbgenerated("now()") for both fields, so new trip-planner rows match main-site’s format exactly). But a schema default only changes future inserts — rows written before the fix keep their original cuid/millisecond format forever. So any notebook reading this table needs to tolerate both formats existing side by side in the same column.

Tools and Techniques

Everything below builds on a small set of Python libraries and a handful of core data-mining concepts. This section is a quick reference for what each one does and why it’s useful - skip ahead if you’re already familiar, or come back here whenever a technique further down feels unfamiliar.

Importing the Data

We load the raw AnalyticsEvent export, normalize the two timestamp/id formats described above into one consistent shape, then narrow down to main-site events only and treat event as a categorical column so the rest of the notebook can group and count by event type cleanly.

%matplotlib inline

import json

import matplotlib.pyplot as plt
import numpy as np
import pandas as pd
from sklearn.cluster import KMeans
from sklearn.decomposition import PCA
from sklearn.preprocessing import StandardScaler

# Read the csv as a Pandas table
# (the data-dump pipeline writes CSVs under notes/notebooks/data/ - see CLAUDE.md -
# shared by both draft and published notebooks)
events = pd.read_csv("data/analytics_event.csv")
# Show the table as it is read in
events

# Normalize the different time formats to meet  ISO8601
events["createdAt"] = pd.to_datetime(events["createdAt"], format="ISO8601", utc=True)
# parse the json properties
events["properties"] = events["properties"].apply(json.loads)

# Print the total amounts of events an service
print(f"{len(events)} events, {events['service'].nunique()} services")

# show the amounts of each type of service... We are only concerned with mainsite
events["service"].value_counts()

2316 events, 5 services

service
main-site       1178
trip-planner     476
quantum          246
sky              232
notes            184
Name: count, dtype: int64

# Confirm both id/timestamp eras are actually present, and that parsing didn't drop anything.
# Not every render has both eras - a fresh/sparse local dev database may only have one
# (or neither), so this reports what's actually present instead of assuming a full
# production-shaped history.
old_era = events[events["id"].str.startswith("cm")]   # trip-planner cuid era
new_era = events[~events["id"].str.startswith("cm")]  # gen_random_uuid era

if len(old_era) > 0:
    sample_old = old_era.iloc[0]
    print("cuid era     :", sample_old["id"], sample_old["createdAt"])
else:
    print("cuid era     : none present in this dataset")

if len(new_era) > 0:
    sample_new = new_era.iloc[0]
    print("uuid/NOW era :", sample_new["id"], sample_new["createdAt"])
else:
    print("uuid/NOW era : none present in this dataset")

assert events["createdAt"].isna().sum() == 0, "some timestamps failed to parse"

cuid era     : cmqqq040x001jqgjo2poi6ym1 2026-06-23 14:09:06.603000+00:00
uuid/NOW era : 4e55128c-2284-49a9-b8cb-7104c20870c2 2026-06-23 18:37:34.118451+00:00

# Now we can strip out the trip planner data
main_site = events[events["service"] == "main-site"].copy()

# Show the top 5 results from the table
main_site.head()

# confirm column data types
main_site.dtypes

id                            str
event                         str
properties                 object
sessionId                     str
createdAt     datetime64[us, UTC]
service                       str
dtype: object

# change event type to categorical to associate an integer with the names
main_site.event = main_site.event.astype('category')

main_site.dtypes

id                            str
event                    category
properties                 object
sessionId                     str
createdAt     datetime64[us, UTC]
service                       str
dtype: object

Visualizing the Data

A first look at how people actually use the site: how long sessions last, how many actions they trigger, and how far they scroll - each shown as a box-and-whisker plot with the underlying numbers spelled out in prose below it.

# 1. Find the start and end time for each session
session_times = main_site.groupby('sessionId')['createdAt'].agg(['min', 'max'])

# 2. Calculate duration in minutes
session_times['duration_minutes'] = (session_times['max'] - session_times['min']).dt.total_seconds() / 60.0

# 3. Cap duration for these visualizations only - a single session with an idle open
# tab can run for hours and swamps the scale of a box plot / scatter matrix. 100
# minutes is a generous cap for someone actually reading the site. This only affects
# the charts below - the classifier later in this notebook still trains on the real,
# uncapped duration.
DURATION_CAP_MINUTES = 100
session_times['duration_minutes'] = session_times['duration_minutes'].clip(upper=DURATION_CAP_MINUTES)

# 4. Create the boxplot
session_times.boxplot(column='duration_minutes')

Reading the box plot: the box spans the 1st to 3rd quartile, the green line marks the median, and the whiskers show the normal range of the data - circled points beyond them are outliers. Duration here is capped at 100 minutes (see the code above) so a rare session where someone left the tab open for hours doesn’t flatten the whole chart.

# 1. Calculate the components dynamically
stats = session_times['duration_minutes'].describe()

median_val = stats['50%']
q1_val = stats['25%']
q3_val = stats['75%']
max_val = stats['max']

# 2. Format the dynamic print statement
print(
    f"Therefore, we can conclude that a typical user session duration is {median_val:.2f} minutes (the median).\n"
    f"The middle 50% of our users stay between {q1_val:.2f} and {q3_val:.2f} minutes (the box range).\n"
    f"The chart above caps duration at {DURATION_CAP_MINUTES} minutes, so the {max_val:.2f} minute maximum shown\n"
    f"here is the capped value, not the true longest session - a handful of real sessions run far longer\n"
    f"than that (people who leave the tab open), which is exactly why the cap exists.\n\n"
    f"We most also take into consideration that the duration is calculated by the first and last event per session.\n"
    f"This means that a user could stay on the page for some time after without triggering another event."
)

Therefore, we can conclude that a typical user session duration is 0.00 minutes (the median).
The middle 50% of our users stay between 0.00 and 0.02 minutes (the box range).
The chart above caps duration at 100 minutes, so the 100.00 minute maximum shown
here is the capped value, not the true longest session - a handful of real sessions run far longer
than that (people who leave the tab open), which is exactly why the cap exists.

We most also take into consideration that the duration is calculated by the first and last event per session.
This means that a user could stay on the page for some time after without triggering another event.

# 1. Count how many events happen in each session
session_counts = main_site.groupby('sessionId').size().reset_index(name='event_count')

# 2. Boxplot the distribution of activity
session_counts.boxplot(column='event_count')

# 1. Calculate the components dynamically
stats = session_counts['event_count'].describe()

median_val = stats['50%']
q1_val = stats['25%']
q3_val = stats['75%']
max_val = stats['max']

# 2. Format the dynamic print statement
print(
    f"Therefore, we can conclude that a typical user session consists of {median_val:.2f} events (the median).\n"
    f"The middle 50% of our users trigger between {q1_val:.2f} and {q3_val:.2f} events (the box range).\n"
    f"While the absolute maximum recorded activity was {max_val:.2f} events, the whiskers show the normal\n"
    f"range of data, meaning extreme high values are likely just click-happy users."
)

Therefore, we can conclude that a typical user session consists of 1.00 events (the median).
The middle 50% of our users trigger between 1.00 and 2.00 events (the box range).
While the absolute maximum recorded activity was 19.00 events, the whiskers show the normal
range of data, meaning extreme high values are likely just click-happy users.

# 1. Filter for scroll events
scroll_data = main_site[main_site['event'] == 'scroll_depth'].copy()

# 2. Extract the number from the JSON string safely
def extract_percent(prop_str):
    try:
        # If it's already a dict, grab it; if it's a string, parse it as JSON
        if isinstance(prop_str, dict):
            return prop_str.get('percent')
        
        data = json.loads(prop_str)
        return data.get('percent')
    except (json.JSONDecodeError, TypeError, AttributeError):
        return None

scroll_data['depth'] = scroll_data['properties'].apply(extract_percent)

# 3. Ensure it is numeric
scroll_data['depth'] = pd.to_numeric(scroll_data['depth'], errors='coerce')

# 4. Create the plot! 
# (We drop 'by=sessionId' here to avoid the ugly X-axis problem)
plt.figure(figsize=(6, 6))
scroll_data.boxplot(column='depth')

plt.title("Overall User Scroll Depth Distribution")
plt.ylabel("Scroll Percentage (%)")
plt.show()

# 1. Calculate the components dynamically
stats = scroll_data['depth'].describe()

median_val = stats['50%']
q1_val = stats['25%']
q3_val = stats['75%']
max_val = stats['max']

# 2. Format the dynamic print statement
print(
    f"Therefore, we can conclude that a typical user scrolls to {median_val:.1f} % (the median) of the main page.\n"
    f"The middle 50% of our users scroll between {q1_val:.1f} and {q3_val:.1f} events (the box range)."
)

Therefore, we can conclude that a typical user scrolls to 50.0 % (the median) of the main page.
The middle 50% of our users scroll between 25.0 and 75.0 events (the box range).

# 1. Create a clean metrics DataFrame per session
# Total events per session
session_metrics = main_site.groupby('sessionId').size().reset_index(name='total_actions')

# Session duration in minutes
times = main_site.groupby('sessionId')['createdAt'].agg(['min', 'max'])
session_metrics['duration_minutes'] = ((times['max'] - times['min']).dt.total_seconds() / 60.0).values

# Count how many links were clicked in each session
link_clicks = main_site[main_site['event'] == 'link_clicked'].groupby('sessionId').size()
session_metrics['links_clicked'] = session_metrics['sessionId'].map(link_clicks).fillna(0)

# 2. Drop the sessionId column (since it's text) just for the plot, and cap duration the
# same way as the box plot above. This clips a copy used only for the chart -
# session_metrics itself stays uncapped, since the classifier later in this notebook
# trains on the real duration.
plot_data = session_metrics[['total_actions', 'duration_minutes', 'links_clicked']].copy()
plot_data['duration_minutes'] = plot_data['duration_minutes'].clip(upper=DURATION_CAP_MINUTES)

# 3. Generate the Scatter Matrix!
# Needs at least a couple of sessions with values in every column, or pandas' internal
# min/max range calculation blows up on an empty array - a fresh/sparse local dev
# database (few or no main-site sessions yet) can hit that, so this adapts the same way
# the classifier evaluation cell further down does.
# (The '_ =' syntax just suppresses messy text output in Jupyter notebooks)
if len(plot_data.dropna()) >= 2:
    _ = pd.plotting.scatter_matrix(
        plot_data,
        diagonal='kde',
        figsize=(10, 10),
        alpha=0.6,    # Makes dots slightly transparent to see overlaps
        density_kwds={'color': 'red'} # Color the KDE lines red
    )
    plt.show()
else:
    print(f"Only {len(plot_data)} session(s) available - too few to plot a scatter matrix.")

A correlation matrix, visualized as a heatmap - warmer colors mean two metrics move together more strongly. The prose below walks through what these numbers mean for links_clicked and total_actions specifically.

corr_matrix_viz = plot_data.corr()

plt.figure(figsize=(6, 5))
plt.imshow(corr_matrix_viz, cmap="coolwarm", vmin=-1, vmax=1)
plt.xticks(range(len(corr_matrix_viz.columns)), corr_matrix_viz.columns, rotation=45, ha="right")
plt.yticks(range(len(corr_matrix_viz.columns)), corr_matrix_viz.columns)
plt.colorbar(label="Correlation")
plt.title("Correlation between session metrics")
plt.tight_layout()
plt.show()

# 1. Calculate correlations and statistics dynamically
corr_matrix = plot_data.corr()
stats = plot_data.describe()

# Extract key summary metrics
median_duration = stats.loc['50%', 'duration_minutes']
median_actions = stats.loc['50%', 'total_actions']
max_duration = stats.loc['max', 'duration_minutes']

# Pull individual correlation coefficients (ranges from -1 to 1)
click_duration_corr = corr_matrix.loc['links_clicked', 'duration_minutes']
actions_duration_corr = corr_matrix.loc['total_actions', 'duration_minutes']

# Dynamically evaluate the relationship between link clicks and time on site
if click_duration_corr > 0.5:
    feature_value_desc = "strong positive relationship. This proves that clicking links directly drives user retention and keeps people on the site longer"
elif click_duration_corr > 0.1:
    feature_value_desc = "slight positive relationship, showing some connection between link engagement and session length"
else:
    feature_value_desc = "weak or non-existent relationship. This suggests that link clicks don't impact how long someone stays, meaning they might be clicking an external link and leaving immediately"

# Dynamically analyze user personas based on action vs duration correlation
if actions_duration_corr > 0.6:
    persona_desc = "highly consistent. Users are steadily interacting with the site the entire time they are here (Classic Explorers)."
else:
    persona_desc = "fragmented. We have a mix of rapid-fire clickers alongside users who generate very few actions over long stretches of time (Window Shoppers vs. Idle Tabs)."

# 2. Format the dynamic print statement
print(
    f"--- EXECUTIVE SUMMARY OF USER BEHAVIOR ---\n\n"
    f"Based on our session metrics, a typical user experience lasts for {median_duration:.4f} minutes "
    f"and generates {median_actions:.2f} total actions.\n\n"
    f"1. IDENTITY USER PERSONAS\n"
    f"The relationship between total activity and duration tells us our audience personas are {persona_desc}\n\n"
    f"2. DISCOVER FEATURE VALUE\n"
    f"When looking at link engagement, there is a {feature_value_desc}.\n\n"
    f"3. SEE YOUR 'AVERAGE' USER EXPERIENCE\n"
    f"Looking at the distribution curves, the shape of our traffic shows how varied our audience is. "
    f"While our median session length sits comfortably at {median_duration:.4f} minutes, the longest session "
    f"shown here reached {max_duration:.4f} minutes (this chart is capped at {DURATION_CAP_MINUTES} minutes - "
    f"see the code above). This spread reveals the true 'shape' of our user base—separating "
    f"the quick bounce traffic from the deeply engaged power users who exhaustively browse the site."
)

--- EXECUTIVE SUMMARY OF USER BEHAVIOR ---

Based on our session metrics, a typical user experience lasts for 0.0000 minutes and generates 1.00 total actions.

1. IDENTITY USER PERSONAS
The relationship between total activity and duration tells us our audience personas are fragmented. We have a mix of rapid-fire clickers alongside users who generate very few actions over long stretches of time (Window Shoppers vs. Idle Tabs).

2. DISCOVER FEATURE VALUE
When looking at link engagement, there is a slight positive relationship, showing some connection between link engagement and session length.

3. SEE YOUR 'AVERAGE' USER EXPERIENCE
Looking at the distribution curves, the shape of our traffic shows how varied our audience is. While our median session length sits comfortably at 0.0000 minutes, the longest session shown here reached 100.0000 minutes (this chart is capped at 100 minutes - see the code above). This spread reveals the true 'shape' of our user base—separating the quick bounce traffic from the deeply engaged power users who exhaustively browse the site.

Classifying Contact Interest

session_metrics above already looks like a supervised learning setup: numeric features per session (total_actions, duration_minutes, links_clicked) with an obvious target missing. Nothing in this schema hands us a class to predict — so let’s engineer one instead of only clustering.

Label: did the session ever show interest in the Contact section — i.e. a section_viewed or nav_clicked event with section: "contact"? That’s a reasonable proxy for “this visitor was interested enough to consider reaching out,” and it turns the same session-level features into a decision tree classification problem.

# 1. Add max scroll depth reached per session as another behavioral feature
max_scroll = scroll_data.groupby('sessionId')['depth'].max()
session_metrics['max_scroll_depth'] = session_metrics['sessionId'].map(max_scroll).fillna(0)

# 2. Label each session: did it ever show interest in the Contact section?
contact_events = main_site[main_site['event'].isin(['section_viewed', 'nav_clicked'])]
# .apply() on an empty Series returns a result that doesn't line up as a boolean mask
# (indexing with it collapses to a 0-column frame), so a sparse/empty local dev dataset
# needs this handled explicitly rather than falling through to the apply/filter below.
if len(contact_events) > 0:
    is_contact = contact_events['properties'].apply(lambda p: p.get('section') == 'contact')
    contact_sessions = set(contact_events[is_contact]['sessionId'])
else:
    contact_sessions = set()
session_metrics['contact_interest'] = session_metrics['sessionId'].isin(contact_sessions)

session_metrics[['sessionId', 'total_actions', 'duration_minutes', 'links_clicked', 'max_scroll_depth', 'contact_interest']]

Each row above is one session with its behavioral features (total_actions, duration_minutes, links_clicked, max_scroll_depth) alongside the engineered contact_interest label - this is the exact table the classifier below is trained on.

import sklearn.tree

# 1. Pick the behavioral features and the engineered label
feature_cols = ['total_actions', 'duration_minutes', 'links_clicked', 'max_scroll_depth']
X = session_metrics[feature_cols].values
y = session_metrics['contact_interest'].astype(int).values

# 2. Fit a shallow decision tree (kept shallow so it stays readable while the dataset is small).
# Needs at least one main-site session to fit at all - a fresh/empty local dev database can
# have zero, so this (and the two cells below that depend on contact_classifier) adapt the
# same way the rest of this notebook does.
if len(session_metrics) > 0:
    contact_classifier = sklearn.tree.DecisionTreeClassifier(max_depth=3)
    contact_classifier.fit(X, y)
    # 3. Explain the tree as text
    print(sklearn.tree.export_text(contact_classifier, feature_names=feature_cols))
else:
    contact_classifier = None
    print("No main-site sessions available - skipping the decision tree.")

|--- max_scroll_depth <= 62.50
|   |--- links_clicked <= 0.50
|   |   |--- duration_minutes <= 0.00
|   |   |   |--- class: 0
|   |   |--- duration_minutes >  0.00
|   |   |   |--- class: 0
|   |--- links_clicked >  0.50
|   |   |--- duration_minutes <= 0.07
|   |   |   |--- class: 0
|   |   |--- duration_minutes >  0.07
|   |   |   |--- class: 0
|--- max_scroll_depth >  62.50
|   |--- max_scroll_depth <= 87.50
|   |   |--- duration_minutes <= 0.67
|   |   |   |--- class: 1
|   |   |--- duration_minutes >  0.67
|   |   |   |--- class: 0
|   |--- max_scroll_depth >  87.50
|   |   |--- class: 1

# Visualize the same tree
if contact_classifier is not None:
    plt.figure(figsize=(12, 8))
    _ = sklearn.tree.plot_tree(
        contact_classifier,
        feature_names=feature_cols,
        class_names=['no_contact_interest', 'contact_interest'],
        filled=True
    )
else:
    print("No classifier to visualize.")

Reading the tree: each split asks a yes/no question about one feature - true goes left, false goes right. class: 1 (contact_interest) leaves show in a warmer shade in the plot above, class: 0 in a cooler one. The text dump above and this plot describe the exact same tree; the plot’s easier to scan, the text is easier to copy or quote.

# How good is the classifier? This adapts to however much data actually exists at render
# time below a per-class threshold a held-out split isn't meaningful, so we fall back to training accuracy.
from sklearn.model_selection import train_test_split

if contact_classifier is None:
    print("No classifier to evaluate.")
else:
    class_counts = session_metrics['contact_interest'].value_counts()
    min_class_count = class_counts.min()
    min_per_class_for_split = 10

    if min_class_count >= min_per_class_for_split:
        X_train, X_test, y_train, y_test = train_test_split(
            X, y, test_size=0.3, stratify=y, random_state=42
        )
        eval_classifier = sklearn.tree.DecisionTreeClassifier(max_depth=3)
        eval_classifier.fit(X_train, y_train)
        accuracy = eval_classifier.score(X_test, y_test)
        from sklearn.metrics import ConfusionMatrixDisplay

        ConfusionMatrixDisplay.from_estimator(
            eval_classifier, X_test, y_test,
            display_labels=['no_contact_interest', 'contact_interest']
        )
        plt.show()
        print(
            f"{len(session_metrics)} sessions available ({class_counts.to_dict()} by class) - "
            f"enough per class to hold out a test set.\n"
            f"Trained on {len(X_train)}, tested on {len(X_test)} held-out sessions.\n"
            f"Held-out accuracy: {accuracy:.2f}"
        )
    else:
        accuracy = contact_classifier.score(X, y)
        print(
            f"Only {len(session_metrics)} sessions available ({class_counts.to_dict()} by class) - "
            f"too few per class to hold out a meaningful test set.\n"
            f"Training accuracy (read skeptically - not a generalization estimate): {accuracy:.2f}"
        )

506 sessions available ({False: 461, True: 45} by class) - enough per class to hold out a test set.
Trained on 354, tested on 152 held-out sessions.
Held-out accuracy: 0.96

from sklearn.metrics import precision_score, recall_score, f1_score

if contact_classifier is not None and min_class_count >= min_per_class_for_split:
    predicted = eval_classifier.predict(X_test)
    precision = precision_score(y_test, predicted, zero_division=0)
    recall = recall_score(y_test, predicted, zero_division=0)
    f1 = f1_score(y_test, predicted, zero_division=0)
    print(f"Precision: {precision:.2f}, Recall: {recall:.2f}, F1: {f1:.2f}")
else:
    print(f"Only {len(session_metrics)} sessions available - too few per class for a held-out precision/recall/F1 evaluation yet.")

Precision: 1.00, Recall: 0.57, F1: 0.73

from sklearn.neighbors import KNeighborsClassifier
from sklearn.model_selection import train_test_split

if contact_classifier is not None and min_class_count >= min_per_class_for_split:
    n_neighbors_range = [1, 3, 5, 7, 9]
    for seed in [1, 42]:
        Xtr, Xte, ytr, yte = train_test_split(X, y, test_size=0.3, stratify=y, random_state=seed)
        scores = {k: KNeighborsClassifier(n_neighbors=k).fit(Xtr, ytr).score(Xte, yte) for k in n_neighbors_range}
        print(f"random_state={seed}: best n_neighbors={max(scores, key=scores.get)}  scores={scores}")
else:
    print("Not enough sessions per class to demonstrate split instability yet.")

random_state=1: best n_neighbors=3  scores={1: 0.9736842105263158, 3: 0.9802631578947368, 5: 0.9736842105263158, 7: 0.9671052631578947, 9: 0.9671052631578947}
random_state=42: best n_neighbors=1  scores={1: 0.9736842105263158, 3: 0.9736842105263158, 5: 0.9671052631578947, 7: 0.9605263157894737, 9: 0.9605263157894737}

from sklearn.model_selection import cross_val_score

cv_folds = 5
if contact_classifier is not None and min_class_count >= cv_folds:
    cv_scores = cross_val_score(sklearn.tree.DecisionTreeClassifier(max_depth=3), X, y, cv=cv_folds)
    plt.boxplot(cv_scores)
    plt.ylabel("Accuracy")
    plt.title(f"{cv_folds}-fold CV Accuracy")
    plt.show()
    print(f"Mean: {cv_scores.mean():.2f}, Std-dev: {cv_scores.std():.2f}")
else:
    print(f"Need at least {cv_folds} sessions in the smaller class for {cv_folds}-fold CV.")

Mean: 0.96, Std-dev: 0.02

if min_class_count >= min_per_class_for_split:
    depths = list(range(1, 8))
    rows = []
    for d in depths:
        tree = sklearn.tree.DecisionTreeClassifier(max_depth=d).fit(X_train, y_train)
        rows.append({"max_depth": d, "train": tree.score(X_train, y_train), "test": tree.score(X_test, y_test)})
    depth_accuracy = pd.DataFrame(rows)

    plt.plot(depth_accuracy["max_depth"], depth_accuracy["train"], "ro-", label="Train")
    plt.plot(depth_accuracy["max_depth"], depth_accuracy["test"], "bv--", label="Test")
    plt.xlabel("max_depth"); plt.ylabel("Accuracy"); plt.legend()
    plt.title("Decision Tree: train vs. test accuracy by depth")
    plt.show()
else:
    print("Not enough sessions per class for a train/test overfitting curve yet.")

import sklearn.neighbors
import sklearn.ensemble
import sklearn.naive_bayes
from sklearn.dummy import DummyClassifier

if contact_classifier is not None and min_class_count >= cv_folds:
    n_neighbors = min(5, min_class_count - 1)  # can't exceed the smallest class size
    models = {
        "Decision Tree": sklearn.tree.DecisionTreeClassifier(max_depth=3),
        "K-Nearest Neighbors": sklearn.neighbors.KNeighborsClassifier(n_neighbors=n_neighbors),
        "Random Forest": sklearn.ensemble.RandomForestClassifier(n_estimators=100, max_depth=3, random_state=42),
        "Bagging": sklearn.ensemble.BaggingClassifier(
            estimator=sklearn.tree.DecisionTreeClassifier(max_depth=3), n_estimators=50, random_state=42
        ),
        "Gaussian Naive Bayes": sklearn.naive_bayes.GaussianNB(),
        "Dummy (baseline)": DummyClassifier(strategy="most_frequent"),
    }
    comparison = pd.DataFrame([
        {"model": name, "mean_accuracy": cross_val_score(model, X, y, cv=cv_folds).mean()}
        for name, model in models.items()
    ]).sort_values("mean_accuracy", ascending=False)

    comparison.plot.bar(x="model", y="mean_accuracy", legend=False)
    plt.ylabel("Mean CV Accuracy")
    plt.xticks(rotation=20)
    plt.show()
    comparison
else:
    print(f"Need at least {cv_folds} sessions in the smaller class to compare classifiers.")

from sklearn.metrics import roc_curve

if min_class_count >= min_per_class_for_split:
    plt.figure(figsize=(6, 6))
    for name, model in models.items():
        model.fit(X_train, y_train)
        probs = model.predict_proba(X_test)[:, 1]
        fpr, tpr, _ = roc_curve(y_test, probs)
        plt.plot(fpr, tpr, label=name)
    plt.plot([0, 1], [0, 1], 'k--', label="Random guess")
    plt.xlabel("False Positive Rate")
    plt.ylabel("True Positive Rate")
    plt.title("ROC curves: contact_interest classifiers")
    plt.legend(fontsize=8)
    plt.show()
else:
    print("Not enough sessions per class for a held-out ROC comparison yet.")

if contact_classifier is not None and min_class_count >= cv_folds:
    n_neighbors = min(5, min_class_count - 1)
    metric_scores = {
        metric: cross_val_score(
            sklearn.neighbors.KNeighborsClassifier(n_neighbors=n_neighbors, metric=metric), X, y, cv=cv_folds
        ).mean()
        for metric in ["euclidean", "manhattan", "cosine"]
    }
    print(metric_scores)
else:
    print(f"Need at least {cv_folds} sessions in the smaller class to compare KNN metrics.")

{'euclidean': np.float64(0.9604154533100369), 'manhattan': np.float64(0.9604154533100369), 'cosine': np.float64(0.9228499320520287)}

from sklearn.preprocessing import StandardScaler

if contact_classifier is not None and min_class_count >= cv_folds:
    X_scaled = StandardScaler().fit_transform(X)
    n_neighbors = min(5, min_class_count - 1)
    unscaled_score = cross_val_score(sklearn.neighbors.KNeighborsClassifier(n_neighbors=n_neighbors), X, y, cv=cv_folds).mean()
    scaled_score = cross_val_score(sklearn.neighbors.KNeighborsClassifier(n_neighbors=n_neighbors), X_scaled, y, cv=cv_folds).mean()
    print(f"KNN mean CV accuracy — unscaled: {unscaled_score:.2f}, scaled: {scaled_score:.2f}")
else:
    print(f"Need at least {cv_folds} sessions in the smaller class to compare scaled vs. unscaled features.")

KNN mean CV accuracy — unscaled: 0.96, scaled: 0.97

from sklearn.model_selection import GridSearchCV

if contact_classifier is not None and min_class_count >= cv_folds:
    tree_grid = {"criterion": ["gini", "entropy"], "max_depth": [2, 3, 4, 5], "min_samples_split": [2, 3, 5]}
    knn_grid = {
        "n_neighbors": [k for k in [1, 3, 5, 7] if k < min_class_count],
        "metric": ["euclidean", "manhattan", "cosine"],
        "weights": ["uniform", "distance"],
    }

    for label, feats in [("unscaled", X), ("scaled", X_scaled)]:
        tree_search = GridSearchCV(sklearn.tree.DecisionTreeClassifier(), tree_grid, cv=cv_folds).fit(feats, y)
        knn_search = GridSearchCV(sklearn.neighbors.KNeighborsClassifier(), knn_grid, cv=cv_folds).fit(feats, y)
        print(f"[{label}] best tree: {tree_search.best_params_} -> {tree_search.best_score_:.2f}")
        print(f"[{label}] best knn:  {knn_search.best_params_} -> {knn_search.best_score_:.2f}")
else:
    print(f"Need at least {cv_folds} sessions in the smaller class to run GridSearchCV.")

[unscaled] best tree: {'criterion': 'gini', 'max_depth': 2, 'min_samples_split': 2} -> 0.97
[unscaled] best knn:  {'metric': 'euclidean', 'n_neighbors': 5, 'weights': 'distance'} -> 0.97
[scaled] best tree: {'criterion': 'gini', 'max_depth': 2, 'min_samples_split': 2} -> 0.97
[scaled] best knn:  {'metric': 'euclidean', 'n_neighbors': 5, 'weights': 'uniform'} -> 0.97

if contact_classifier is not None and len(session_metrics) >= 2:
    X_scaled_for_pca = StandardScaler().fit_transform(X)
    pca_2d = PCA(n_components=2, random_state=1).fit_transform(X_scaled_for_pca)

    plt.figure(figsize=(6, 6))
    plt.scatter(pca_2d[:, 0], pca_2d[:, 1], c=y, cmap="coolwarm", edgecolor="k")
    plt.xlabel("PC1")
    plt.ylabel("PC2")
    plt.title("Sessions in PCA space, colored by contact_interest")
    plt.show()
else:
    print("Not enough sessions to plot a PCA scatter yet.")

Clustering Sessions into Behavioral Groups

Everything above predicts a label we engineered ourselves (contact_interest). This section instead asks an unsupervised question: without any labels at all, do sessions naturally fall into distinct behavioral groups? Visualizing the Data’s executive summary speculated about “user personas” from a single correlation coefficient - this section checks that claim with an actual model instead.

As with KNN above, these algorithms are distance-based, so features are scaled first:

from sklearn.preprocessing import StandardScaler

cluster_features = StandardScaler().fit_transform(X)
n_sessions = len(session_metrics)

from sklearn.cluster import KMeans
from sklearn.metrics import silhouette_score

if n_sessions >= 4:
    k_range = range(2, min(8, n_sessions))
    inertia = [KMeans(n_clusters=k, random_state=42).fit(cluster_features).inertia_ for k in k_range]
    silhouette = [
        silhouette_score(cluster_features, KMeans(n_clusters=k, random_state=42).fit_predict(cluster_features))
        for k in k_range
    ]

    fig, axes = plt.subplots(1, 2, figsize=(10, 4))
    axes[0].plot(list(k_range), inertia, 'bx-')
    axes[0].set_xlabel("k"); axes[0].set_ylabel("Inertia"); axes[0].set_title("Elbow method")
    axes[1].plot(list(k_range), silhouette, 'bx-')
    axes[1].set_xlabel("k"); axes[1].set_ylabel("Silhouette score"); axes[1].set_title("Silhouette method")
    plt.tight_layout()
    plt.show()

    best_k = list(k_range)[int(np.argmax(silhouette))]
    print(f"Best k by silhouette: {best_k}")
else:
    best_k = None
    print(f"Only {n_sessions} sessions - too few to sweep k meaningfully.")

Best k by silhouette: 4

# Fit at the chosen k and profile each cluster - this is what turns the "user personas"
# language above into an actual finding instead of an eyeballed guess.
if best_k is not None:
    kmeans = KMeans(n_clusters=best_k, random_state=42).fit(cluster_features)
    session_metrics['cluster'] = kmeans.labels_
    session_metrics.groupby('cluster')[feature_cols + ['contact_interest']].mean()
else:
    print("Skipping cluster profiling - not enough sessions.")

import scipy.cluster.hierarchy as hierarchy

if n_sessions >= 3:
    linked = hierarchy.linkage(cluster_features, method='ward')

    plt.figure(figsize=(8, 5))
    hierarchy.dendrogram(linked, labels=session_metrics['sessionId'].str[:8].tolist())
    plt.xlabel("Session (truncated ID)")
    plt.ylabel("Distance")
    plt.title("Session clustering dendrogram")
    plt.xticks(rotation=90)
    plt.tight_layout()
    plt.show()
else:
    print(f"Only {n_sessions} sessions - too few for a meaningful dendrogram.")

from sklearn.cluster import DBSCAN
from sklearn.neighbors import NearestNeighbors

if n_sessions >= 5:
    n_neighbors_for_eps = min(len(feature_cols) * 2, n_sessions - 1)  # lab 8's own rule of thumb: 2x dimensionality

    neighbors = NearestNeighbors(n_neighbors=n_neighbors_for_eps).fit(cluster_features)
    distances, _ = neighbors.kneighbors(cluster_features)
    distances = np.sort(distances[:, -1])

    plt.figure(figsize=(6, 4))
    plt.plot(distances)
    plt.xlabel("Session, sorted by distance")
    plt.ylabel(f"Distance to {n_neighbors_for_eps}th neighbor")
    plt.title("DBSCAN eps estimation")
    plt.show()
else:
    print(f"Only {n_sessions} sessions - too few to estimate a stable DBSCAN eps.")

eps = 1.5  # replace with whatever value the elbow above actually shows

if n_sessions >= 5:
    dbscan = DBSCAN(eps=eps, min_samples=n_neighbors_for_eps).fit(cluster_features)
    session_metrics['dbscan_cluster'] = dbscan.labels_
    session_metrics['dbscan_cluster'].value_counts()

from sklearn.covariance import EllipticEnvelope
from sklearn.svm import OneClassSVM
from sklearn.neighbors import LocalOutlierFactor

if n_sessions >= 10:
    outliers_fraction = 0.1
    n_neighbors_lof = min(20, n_sessions - 1)

    anomaly_algorithms = {
        "Robust Covariance": EllipticEnvelope(contamination=outliers_fraction, random_state=42),
        "One-Class SVM": OneClassSVM(nu=outliers_fraction, kernel="rbf", gamma="scale"),
        "Local Outlier Factor": LocalOutlierFactor(n_neighbors=n_neighbors_lof, contamination=outliers_fraction),
    }

    results = pd.DataFrame({"sessionId": session_metrics["sessionId"]})
    for name, algorithm in anomaly_algorithms.items():
        try:
            if name == "Local Outlier Factor":
                # LOF only supports fit_predict in this mode - no separate .fit() then .predict()
                labels = algorithm.fit_predict(cluster_features)
            else:
                labels = algorithm.fit(cluster_features).predict(cluster_features)
            results[name] = labels  # -1 = outlier, 1 = inlier
        except ValueError as e:
            # Robust Covariance needs real spread within its "cleanest" data subset to
            # estimate a covariance matrix at all - real traffic with many near-identical
            # bounce sessions (single pageview, no scroll, no clicks) can make that subset
            # exactly degenerate. Skip just this algorithm rather than failing the cell.
            print(f"Skipped {name}: {e}")

    results
else:
    print(f"Only {n_sessions} sessions - too few to fit anomaly detectors reliably (need >= 10).")

Skipped Robust Covariance: The covariance matrix of the support data is equal to 0, try to increase support_fraction

/usr/local/lib/python3.11/site-packages/sklearn/neighbors/_lof.py:327: UserWarning: Duplicate values are leading to incorrect results. Increase the number of neighbors for more accurate results.
  warnings.warn(

if n_sessions >= 10:
    algo_cols = [name for name in anomaly_algorithms if name in results.columns]
    results["flagged_by"] = (results[algo_cols] == -1).sum(axis=1)
    results.sort_values("flagged_by", ascending=False)

if n_sessions >= 10:
    flagged = results[results["flagged_by"] >= 2]["sessionId"]
    session_metrics[session_metrics["sessionId"].isin(flagged)][feature_cols + ["contact_interest"]]

Association Analysis: What Site Behaviors Go Together

A third kind of question, distinct from both prediction and clustering: which site behaviors tend to co-occur within the same session? Each session becomes a “basket” of the distinct events it triggered (e.g. section_viewed:contact, scroll_depth:100), borrowing the same asymmetric-binary-attribute idea used for classic market-basket analysis (which products get bought together).

def event_to_item(row):
    event = row['event']
    props = row['properties']
    if isinstance(props, dict) and len(props) > 0:
        key = next(iter(props))
        return f"{event}:{props[key]}"
    return event

main_site['item'] = main_site.apply(event_to_item, axis=1)

# one basket per session: the *set* of distinct items it triggered (order/repeats don't matter)
transactions = main_site.groupby('sessionId')['item'].apply(lambda items: set(items)).tolist()
all_items = sorted(set(item for t in transactions for item in t))

print(f"{len(transactions)} sessions (transactions), {len(all_items)} distinct items")
all_items

506 sessions (transactions), 27 distinct items

['link_clicked:ayuda',
 'link_clicked:email',
 'link_clicked:exact',
 'link_clicked:github',
 'link_clicked:notes',
 'link_clicked:old-site',
 'link_clicked:planner',
 'link_clicked:quantum',
 'link_clicked:sky',
 'link_clicked:youtube',
 'link_clicked:zakacat',
 'nav_clicked:about',
 'nav_clicked:contact',
 'nav_clicked:experience',
 'nav_clicked:home',
 'nav_clicked:research',
 'page_load',
 'page_load:atlas',
 'scroll_depth:100',
 'scroll_depth:25',
 'scroll_depth:50',
 'scroll_depth:75',
 'section_viewed:about',
 'section_viewed:contact',
 'section_viewed:experience',
 'section_viewed:research',
 'video_played:atlas_demo_full']

import itertools

min_support_count = max(2, len(transactions) // 5)  # tune this - see note below

def itemset_support_count(itemset, transactions):
    return sum(1 for t in transactions if set(itemset).issubset(t))

frequent = []
for size in [1, 2, 3]:
    for combo in itertools.combinations(all_items, size):
        count = itemset_support_count(combo, transactions)
        if count >= min_support_count:
            frequent.append({"itemset": combo, "support_count": count, "support": count / len(transactions)})

frequent_df = pd.DataFrame(frequent).sort_values("support", ascending=False)
frequent_df

support_lookup = {frozenset(row['itemset']): row['support'] for _, row in frequent_df.iterrows()}

def proper_subsets(s):
    s = list(s)
    return itertools.chain.from_iterable(itertools.combinations(s, r) for r in range(1, len(s)))

min_confidence = 0.6
rules = []
for _, row in frequent_df[frequent_df['itemset'].apply(len) > 1].iterrows():
    itemset = row['itemset']
    for left in proper_subsets(itemset):
        left_key = frozenset(left)
        if left_key not in support_lookup:
            continue
        right = frozenset(itemset) - left_key
        confidence = row['support'] / support_lookup[left_key]
        if confidence >= min_confidence:
            rules.append({"left": left, "right": tuple(right), "support": row['support'], "confidence": confidence})

rules_df = pd.DataFrame(rules).sort_values("confidence", ascending=False)
rules_df

AI assisted study guide built from https://mathshistory.st-andrews.ac.uk/Biographies/

1. Biographical Essentials

• Origin: Swiss mathematician born in Basel.
• Education: Entered the University of Basel at age 13; studied under Johann Bernoulli.
• Career Path: * Originally intended for the ministry (theology), but Bernoulli convinced his father of his mathematical genius.
  • Spent most of his career in St. Petersburg (Russia) and Berlin (Prussia).
• Resilience: He lost sight in his right eye in 1735 and became totally blind in 1766. Remarkably, he produced nearly half of his total work while blind by dictating to scribes.

2. Key Contributions to Mathematical Notation

Euler standardized the language of modern mathematics. If you see these on an exam, Euler is the reason: * Function Notation: * The Base of Natural Logarithms: (approx. 2.718) * Imaginary Unit: * Summation Symbol: * Constants: Popularized (ratio of circumference to diameter). * Trigonometry: Defined as functions/ratios rather than just lengths of lines.

3. Famous Formulas & Theorems

4. Landmark Problems & Fields

• Graph Theory: Solved the Seven Bridges of Königsberg problem (1736), proving it was impossible to cross all seven bridges exactly once. This is considered the origin of graph theory and topology.
• Calculus of Variations: Developed the Euler-Lagrange equation, fundamental to optimization and physics.
• Number Theory: Proved the Basel Problem (sum of reciprocal squares):
• Fluid Dynamics: Developed the Euler Equations for the motion of inviscid (frictionless) fluids.

5. Fun Facts

• Fact: Euler was the most prolific mathematician in history (over 850 publications).
• Fact: He introduced the use of for triangle sides and for opposite angles.
• Fact: He was the first to treat logarithms as functions and define them using the number .
• Fact: He stayed at the St. Petersburg Academy twice, fleeing Berlin after falling out of favor with Frederick the Great.

1. Biographical & Professional Context

• Origin: German theoretical physicist, born in Kiel.
• Education: Studied at Munich and Berlin under giants like Helmholtz and Kirchhoff.
• Career: Spent most of his career at the University of Berlin.
• Nobel Prize: Awarded the 1918 Nobel Prize in Physics for the discovery of energy quanta.
• Historical Note: He was a “reluctant revolutionary.” Deeply conservative and trained in classical physics, he only proposed the quantum hypothesis as an “act of desperation” to fix a specific problem in thermodynamics.

2. The Core Discovery: Black-Body Radiation

Before Planck, classical physics (the Rayleigh-Jeans Law) predicted the “Ultraviolet Catastrophe”—the impossible idea that an ideal radiator would emit infinite energy at short wavelengths (ultraviolet).

3. Key Scientific Contributions

• Quanta: Introduced the concept that energy is quantized. This ended the era of “Classical Physics” and began “Quantum Physics.”
• Thermodynamics: His early work focused on the Second Law of Thermodynamics and the concept of Entropy (). He eventually linked entropy to probability using Boltzmann’s constant ().
• Planck Units: He proposed a system of natural units based only on fundamental constants (), known as Planck length, Planck time, etc.
• Support for Einstein: Planck was one of the first major scientists to recognize and champion Albert Einstein’s Special Theory of Relativity (1905).

4. Philosophical Stance & Later Life

• Scientific Realism: Unlike later quantum physicists (like Bohr or Heisenberg), Planck struggled with the idea of “indeterminacy.” He believed in an objective, causal reality.
• The “Planck Principle”: He famously remarked that science doesn’t triumph by convincing its opponents, but rather because its opponents eventually die and a new generation grows up familiar with the new ideas.
• Personal Tragedy: His life was marked by sorrow; he lost his first wife and all four of his children from that marriage (including a son executed for a plot to assassinate Hitler in 1944).

5. Fun Facts

• Q: What specific problem did Planck solve?
• A: The Black-Body Radiation problem / Ultraviolet Catastrophe.
• Q: What is the “Quantum of Action”?
• A: Another name for Planck’s Constant ().
• Q: How did Planck view the nature of energy?
• A: As discrete “elements” or “quanta” rather than a continuous stream.
• Q: Which institution is named after him?
• A: The Max Planck Society (formerly the Kaiser Wilhelm Society), Germany’s premier research organization.

1. Biographical Overview

• Identity: A French mathematician often described as one of the last “universalists” because he contributed to almost every branch of mathematics.
• Education: Ranked 1st in the entrance exams for both the École Polytechnique and the École Normale Supérieure (chose the latter).
• Career: Held prestigious chairs at the Collège de France and École Polytechnique.
• Personal Context: He was deeply affected by the Dreyfus Affair (Alfred Dreyfus was a relative by marriage), which turned him into a lifelong human rights activist. He lived to the age of 97, remaining mathematically active into his 90s.

2. Number Theory: The Prime Number Theorem (1896)

This is arguably his most famous result. He proved the Prime Number Theorem (PNT) independently of Charles-Jean de la Vallée Poussin in the same year. * The Theorem: It describes the asymptotic distribution of prime numbers. * The Formula: If is the prime-counting function (the number of primes less than or equal to ), then: * Significance: He used complex analysis (specifically the Riemann zeta function ) to prove that has no zeros on the line .

3. Mathematical Physics & PDEs

Hadamard laid the groundwork for how we solve physical problems using math. * Well-Posed Problems: He introduced the definition of a “well-posed” problem. A problem is well-posed if: 1. A solution exists. 2. The solution is unique. 3. The solution’s behavior changes continuously with the initial conditions (stability). * Method of Descent: A technical method he created for solving the wave equation in lower dimensions by “descending” from higher dimensions. * Cauchy Problem: He produced definitive work on the Cauchy problem for linear hyperbolic partial differential equations.

4. Other Key Contributions

• Hadamard Matrix: A square matrix whose entries are either or and whose rows are mutually orthogonal. These are vital today in error-correcting codes and signal processing.
• Hadamard Inequality: A result regarding the maximum volume of a “box” in -dimensions, or specifically, an upper bound on the determinant of a matrix:
• Functional Analysis: He was a pioneer in this field and actually coined the term “functional” to describe functions that take other functions as arguments.
• Psychology of Invention: Wrote a famous book, The Psychology of Invention in the Mathematical Field, where he argued that mathematical thought is often wordless and relies on mental images and the unconscious mind.

5. Fun Facts

• Major Achievement: Proving the Prime Number Theorem ().
• Term Creator: He named the field/concept of a “functional.”
• Concept: The “Well-Posed Problem” (existence, uniqueness, stability).
• Constraint: His proof of the PNT required showing the Riemann Zeta function has no zeros where the real part is 1.

1. Biographical Context

• Origin: German-born theoretical physicist (Ulm, Germany).
• Early Career: Famously worked as a technical assistant at the Swiss Patent Office in Bern, where he developed many of his greatest ideas during his “spare time.”
• Global Impact: Moved to the Institute for Advanced Study in Princeton, NJ, in 1933 after fleeing Nazi Germany.
• Nobel Prize (1921): Awarded for his discovery of the law of the photoelectric effect, not for Relativity (which was still considered controversial by the Nobel committee at the time).

2. The “Annus Mirabilis” (1905)

While still a patent clerk, Einstein published four papers in Annalen der Physik that revolutionized science: 1. Photoelectric Effect: Proposed that light consists of discrete “quanta” (photons). This provided the first solid evidence for quantum theory. 2. Brownian Motion: Provided empirical evidence for the existence of atoms by explaining the random motion of particles in a fluid. 3. Special Relativity: Introduced the idea that the laws of physics are the same for all non-accelerating observers and that the speed of light is constant. 4. Mass-Energy Equivalence: Derived the most famous equation in history:

3. General Relativity (1915)

Einstein expanded his theory to include acceleration and gravity. * The Concept: Gravity is not a “force” (as Newton thought) but a curvature of spacetime caused by mass and energy. * The Field Equations: * Experimental Proof: Confirmed in 1919 by Arthur Eddington during a solar eclipse, showing that gravity bends starlight passing near the sun.

4. Key Concepts & Terms

• Postulate of Special Relativity: The speed of light () in a vacuum is the same for all observers, regardless of their motion.
• Time Dilation: Time moves slower for an object in motion relative to a stationary observer.
• Length Contraction: Objects in motion appear shorter in the direction of travel to a stationary observer.
• Equivalence Principle: The idea that the local effects of gravity are indistinguishable from the effects of acceleration.
• Cosmological Constant (): Originally added to his equations to allow for a static universe; he later called it his “biggest blunder” after Hubble discovered the universe is expanding.

5. Fun Facts

• Q: Why did Einstein win the Nobel Prize?
• A: For the Photoelectric Effect (proving light behaves as a particle).
• Q: What is the significance of ?
• A: It shows that mass and energy are interchangeable; a small amount of mass can be converted into a huge amount of energy.
• Q: What is the “EPR Paradox”?
• A: A paper written with Podolsky and Rosen challenging the “spooky action at a distance” (quantum entanglement) in Copenhagen-style quantum mechanics.
• Q: Did Einstein believe in a deterministic universe?
• A: Yes. He famously said, “God does not play dice with the universe,” expressing his skepticism of the probabilistic nature of quantum mechanics.

1. Biographical & Academic Context

• Origin: German physicist and mathematician born in Breslau (now Poland).
• Education: Studied at several universities, but his time at Göttingen under Hilbert and Minkowski was most influential.
• Academic Hub: He made the University of Göttingen the world’s premier center for theoretical physics in the 1920s.
• Exile: Being of Jewish descent, he was forced to leave Germany in 1933, eventually becoming the Tait Professor of Natural Philosophy at the University of Edinburgh.
• Nobel Prize (1954): Awarded “for his fundamental research in quantum mechanics, especially for his statistical interpretation of the wavefunction.”

2. The Statistical Interpretation (The Born Rule)

This is his most critical contribution to science and a common exam topic. * The Problem: Schrödinger’s wave equation described a wave (), but it wasn’t clear what that wave physically represented. * The Solution: Born proposed that the square of the magnitude of the wavefunction gives the probability density. * The Formula: The probability of finding a particle at a point is: * Significance: This introduced probability into the heart of physics, replacing the absolute certainty (determinism) of Newtonian mechanics.

3. Matrix Mechanics

• Collaboration: Worked closely with his assistant Werner Heisenberg and student Pascual Jordan.
• Contribution: When Heisenberg developed a strange new symbolic logic for transitions in atoms, Born recognized it as Matrix Algebra (which was then obscure to most physicists).
• Result: They published the “Three-Man Paper” (Dreimännerarbeit), which provided the first complete mathematical formulation of quantum mechanics.
• Commutation Relation: He helped derive the fundamental link between position () and momentum ():

4. Solid State Physics & Optics

• Born-Oppenheimer Approximation: A foundational technique in molecular physics and quantum chemistry that allows for the separation of nuclear and electronic motion.
• Crystal Dynamics: Developed the theory of lattice dynamics, explaining how atoms in a solid vibrate.
• Principles of Optics: Wrote one of the most famous textbooks on the subject (Optik), which is still a standard reference.

5. Fun Facts

• Q: What is the “Born Rule”?
• A: The rule stating that represents the probability density of finding a particle.
• Q: Who were some of his famous students?
• A: Heisenberg, Oppenheimer, Fermi, Pauli, and Maria Goeppert-Mayer.
• Q: How did he differ from Einstein on Quantum Mechanics?
• A: Born championed the probabilistic nature of the universe; Einstein famously disagreed, leading to their lifelong “God does not play dice” debate (despite remaining close friends).
• Q: What was his stance on nuclear weapons?
• A: Like many of his peers, he was deeply concerned with the social responsibility of scientists and was a signer of the Russell-Einstein Manifesto.

1. Biographical & Professional Context

• Origin: Danish physicist born in Copenhagen.
• Academic Hub: Founded the Institute of Theoretical Physics in Copenhagen (now the Niels Bohr Institute), which became the epicenter for quantum research in the 1920s and 30s.
• Nobel Prize (1922): Awarded for his investigation of the structure of atoms and the radiation emanating from them.
• World War II: Bohr assisted in the escape of Jewish scientists from Nazi Germany and later fled to Sweden, the UK, and the US (joining the Manhattan Project under the pseudonym “Nicholas Baker”).

2. The Bohr Model of the Atom (1913)

Bohr revolutionized the planetary model of the atom by introducing quantization. * Key Postulates: 1. Electrons orbit the nucleus in specific stationary states (orbits) without radiating energy. 2. Electrons can only exist in orbits where their angular momentum () is an integer multiple of : 3. Radiation is emitted or absorbed only when an electron “jumps” from one orbit to another. * Energy Transition Formula: The energy of the emitted photon () corresponds to the difference between energy levels:

3. The Correspondence Principle

• The Concept: Bohr argued that quantum mechanics must transition into classical physics when dealing with large systems or high quantum numbers ().
• Significance: This served as a vital bridge for physicists to develop quantum theory while ensuring it remained consistent with established classical laws in the macroscopic limit.

4. The Principle of Complementarity

This is the philosophical heart of the Copenhagen Interpretation. * The Idea: Items can have “complementary” properties that cannot be observed or measured simultaneously (like the wave-particle duality of light/electrons). * Bohr’s View: To get a full understanding of a physical object, both perspectives are necessary, even though they are mutually exclusive in a single experiment. * Legacy: This led to a famous lifelong debate with Albert Einstein, who remained skeptical of the probabilistic nature of this interpretation.

5. Fun Facts

• Q: What was the main flaw Bohr fixed in the Rutherford model?
• A: He explained why electrons don’t spiral into the nucleus by quantizing their orbits.
• Q: What is the “Copenhagen Interpretation”?
• A: The standard view of quantum mechanics (developed by Bohr and Heisenberg) asserting that physical systems don’t have definite properties until they are measured.
• Q: What element is named after him?
• A: Bohrium (element 107).
• Q: What was his famous motto/coat of arms symbol?
• A: The Yin and Yang symbol, representing his Principle of Complementarity (“Contraria sunt complementa”).

1. Biographical & Historical Context

• Origin: Austrian theoretical physicist born in Vienna.
• Academic Path: Held the prestigious chair of theoretical physics at the University of Berlin (succeeding Max Planck) before leaving Germany in 1933 due to his opposition to Nazism.
• Nobel Prize (1933): Shared with Paul Dirac for the discovery of new productive forms of atomic theory.
• Later Life: Spent many years at the Dublin Institute for Advanced Studies, where he wrote on physics, biology, and philosophy.

2. The Schrödinger Equation (1926)

This is his most monumental contribution. It describes how the quantum state of a physical system changes with time. * The Wave Function (): Unlike Heisenberg’s “Matrix Mechanics,” Schrödinger used a wave equation approach, which was more familiar to physicists trained in classical acoustics and optics. * The Time-Dependent Equation: * The Hamiltonian (): This operator represents the total energy of the system (kinetic + potential). * Significance: This equation is to quantum mechanics what is to classical mechanics.

3. Wave Mechanics vs. Matrix Mechanics

• Unity: Initially, Heisenberg’s and Schrödinger’s theories seemed like competing rivals.
• Equivalence: Schrödinger proved that his “Wave Mechanics” and Heisenberg’s “Matrix Mechanics” were mathematically equivalent—they were just two different “languages” describing the same underlying reality.

4. Famous Thought Experiments & Concepts

• Schrödinger’s Cat: A paradox intended to critique the Copenhagen Interpretation. It illustrates the problem of superposition (the cat being “dead and alive”) and how quantum effects scale up to the macroscopic world.
• Quantum Tunneling: His equations allowed for the possibility of particles “tunneling” through energy barriers that they classically shouldn’t be able to cross.
• “What is Life?” (1944): He wrote an influential book exploring the physical basis of genetics. He proposed the idea of an “aperiodic crystal” that stored genetic information, which directly inspired Watson and Crick in their discovery of DNA.

5. Fun Facts

• Q: What was Schrödinger’s primary tool for describing the atom?
• A: Differential equations (Wave Mechanics).
• Q: Did Schrödinger like the probabilistic “Born Rule” interpretation?
• A: Not initially. Like Einstein, he was uncomfortable with the idea of “probability waves” and preferred a more literal, continuous wave interpretation.
• Q: What is a “stationary state” in his theory?
• A: An energy level where the probability density does not change over time.
• Q: What was his contribution to biology?
• A: His book What is Life? suggested that genetic material must be a complex molecule, paving the way for molecular biology.

1. Biographical & Professional Context

• Origin: French physicist born in Dieppe into a noble family (he later became the 7th Duc de Broglie).
• Education: Originally studied History at the Sorbonne before switching to Physics after serving as a radio operator in World War I (at the Eiffel Tower).
• The “Thesis”: His most famous work was his 1924 doctoral thesis, Recherches sur la théorie des quanta. It was so revolutionary that his examiners consulted Einstein, who immediately recognized its brilliance.
• Nobel Prize (1929): Awarded for his discovery of the wave nature of electrons.

2. Wave-Particle Duality of Matter

Before de Broglie, light was known to have both wave and particle properties (Photoelectric Effect). De Broglie proposed the inverse: that matter (particles) must also have wave properties.

3. Explaining Bohr’s Atom

One of de Broglie’s greatest achievements was providing a physical reason for Bohr’s quantized orbits. * Standing Waves: He proposed that an electron can only exist in an orbit where its wave doesn’t cancel itself out. * Quantization Condition: The circumference of the electron’s orbit must be an integer multiple () of its wavelength: * Significance: This turned Bohr’s “ad hoc” rule for angular momentum into a logical consequence of wave mechanics.

4. Experimental Confirmation

• Davisson-Germer Experiment (1927): Electrons were fired at a nickel crystal and produced a diffraction pattern. Since diffraction is a wave phenomenon, this proved de Broglie’s hypothesis that particles behave like waves.
• Legacy: This discovery was the direct inspiration for Schrödinger to develop his wave equation ().

5. Later Concepts: Pilot Wave Theory

• Non-Copenhagen View: De Broglie initially proposed the Pilot Wave Theory (or de Broglie–Bohm theory).
• The Idea: Particles are “real” and are “guided” by a physical wave (a “pilot wave”).
• Outcome: He eventually abandoned this due to criticism from the Copenhagen school (Bohr/Heisenberg), but it remains a significant alternative interpretation of quantum mechanics today.

6. Fun Facts

• Q: What was de Broglie’s central hypothesis?
• A: That all matter exhibits wave-particle duality (Matter Waves).
• Q: How does wavelength relate to mass?
• A: Inversely. As mass () increases, the wavelength () becomes so small it is undetectable (which is why humans don’t “diffract” through doors).
• Q: Who confirmed his theory experimentally?
• A: Davisson and Germer (via electron diffraction).
• Q: Which famous equation did his work inspire?
• A: The Schrödinger Equation.

1. Biographical & Intellectual Context

• Origin: Austrian theoretical physicist born in Vienna.
• Prodigy: At age 21, he wrote a 200-page review of General Relativity that even Einstein praised for its depth and clarity.
• Reputation: Known as the “Whip of God” or the “Conscience of Physics” because of his devastatingly sharp critiques of incorrect theories (famously coining the phrase “Not even wrong”).
• Nobel Prize (1945): Nominated by Einstein and awarded for the discovery of the Exclusion Principle.

2. The Pauli Exclusion Principle (1925)

This is his most fundamental contribution to chemistry and physics. It explains the structure of the periodic table and the stability of matter. * The Principle: No two fermions (electrons, protons, neutrons) in a system can occupy the identical quantum state simultaneously. * Quantum Numbers: An electron in an atom is defined by four quantum numbers: 1. (Principal) 2. (Angular momentum) 3. (Magnetic) 4. (Spin) — Pauli realized a fourth degree of freedom was needed. * Significance: This principle prevents atoms from collapsing and explains why electrons fill shells in a specific order.

3. The Prediction of the Neutrino (1930)

Pauli solved a crisis in nuclear physics regarding Beta Decay. * The Problem: Experiments showed that energy and momentum appeared to be “lost” during beta decay, threatening the Law of Conservation of Energy. * The Solution: Pauli proposed a “desperate remedy”—a neutral, nearly massless particle that carried away the missing energy. * The Particle: Later named the Neutrino () by Enrico Fermi. It wasn’t experimentally detected until 1956, 26 years after Pauli’s prediction.

4. Pauli Spin Matrices

Pauli developed the mathematical framework for describing the spin of particles with half-integer angular momentum. * The Matrices (): * Quantum Computing Link: These matrices are the basis for the X, Y, and Z gates used in quantum circuits today.

5. The “Pauli Effect” (Anecdotal)

• The Myth: It was jokingly said that sensitive experimental equipment would break or explode simply by Pauli entering the room.
• The Legend: Fellow physicists (like Otto Stern) famously banned Pauli from their laboratories to protect their experiments.

6. Fun Facts

• Q: What particle did Pauli predict to save the Law of Conservation of Energy?
• A: The Neutrino.
• Q: What class of particles obeys the Exclusion Principle?
• A: Fermions (particles with half-integer spin).
• Q: What is the “Spin-Statistics Theorem”?
• A: A fundamental result (refined by Pauli) linking a particle’s spin to the type of quantum statistics it obeys (Fermi-Dirac vs. Bose-Einstein).
• Q: Which quantum number did Pauli effectively introduce?
• A: The fourth quantum number, representing electron spin ().

1. Biographical & Professional Context

• Origin: German theoretical physicist born in Würzburg.
• Education: Studied under Arnold Sommerfeld in Munich and worked as an assistant to Max Born in Göttingen and Niels Bohr in Copenhagen.
• Nobel Prize (1932): Awarded for the creation of quantum mechanics, the application of which has, inter alia, led to the discovery of the allotropic forms of hydrogen.
• WWII Role: Led the German nuclear energy project (the Uranium Club). His role remains a subject of intense historical debate regarding whether he intentionally stalled the project or simply lacked the resources.

2. Matrix Mechanics (1925)

Heisenberg rejected the idea of “orbits” because they couldn’t be observed. He argued physics should only deal with observables (like the frequencies of light emitted by atoms). * Innovation: He developed a system of non-commutative algebra to describe these observables. * The Collaboration: Max Born recognized this algebra as Matrix Mechanics. * The Fundamental Commutation Relation: * Significance: This showed that in quantum mechanics, the order in which you measure things (like position and momentum ) matters.

3. The Uncertainty Principle (1927)

This is his most famous contribution. It sets a fundamental limit on how precisely we can know certain pairs of physical properties. * The Principle: The more precisely the position of a particle is determined, the less precisely its momentum can be known, and vice versa. * The Formula: * Variables: * : Uncertainty in position. * : Uncertainty in momentum. * Philosophical Impact: It destroyed the “Clockwork Universe” (Determinism). If we cannot know the present state of a particle perfectly, we cannot predict its future perfectly.

4. The Copenhagen Interpretation

Along with Niels Bohr, Heisenberg formulated the standard way of understanding quantum mechanics. * Key Idea: A quantum system does not have definite properties until it is measured. * Wavefunction Collapse: The act of measurement “forces” the system into a specific state.

5. Later Scientific Work

• Isospin: Introduced the concept of isospin to explain the symmetry between protons and neutrons in the nucleus.
• Ferromagnetism: Provided a quantum mechanical explanation for why certain materials become magnetic (the exchange interaction).
• S-matrix Theory: Attempted to describe particle interactions without needing a detailed underlying field theory.

6. Fun Facts

• Q: What was Heisenberg’s “observables only” philosophy?
• A: He believed physics should only model things that can be measured (spectra), not unobservable things like “electron paths.”
• Q: How did his theory differ from Schrödinger’s?
• A: Heisenberg used discrete matrices (Matrix Mechanics); Schrödinger used continuous waves (Wave Mechanics).
• Q: What is the “gamma-ray microscope” thought experiment?
• A: A mental exercise Heisenberg used to illustrate the Uncertainty Principle: to “see” an electron, you must hit it with a high-energy photon, which inevitably changes the electron’s momentum.
• Q: What is the significance of the “hbar” ()?
• A: It is the reduced Planck constant (), the fundamental scale of the quantum world.

1. Biographical & Professional Context

• Origin: British theoretical physicist born in Bristol.
• Education: Originally trained as an Electrical Engineer, which influenced his “mathematical beauty” approach to physics.
• Character: Famous for his extreme silence and literal-mindedness (colleagues jokingly defined a “Dirac” as the unit of one word per hour).
• Nobel Prize (1933): Shared with Erwin Schrödinger for the discovery of new productive forms of atomic theory.
• Legacy: Held the Lucasian Chair of Mathematics at Cambridge (the same chair held by Newton and later Hawking).

2. The Dirac Equation (1928)

This is his most monumental achievement. He sought an equation for the electron that was consistent with Special Relativity. * The Equation: * Key Components: * : A four-component spinor (rather than a simple wave function). * : The Dirac Matrices (4x4 matrices). * Major Successes: 1. It naturally predicted the electron spin () as a mathematical necessity of relativity. 2. It gave the correct magnetic moment of the electron.

3. Prediction of Antimatter

The Dirac Equation had a “problem”: it allowed for negative energy states. * The Interpretation: Rather than ignoring these states, Dirac eventually proposed they represented “holes” in an infinite sea of electrons (the Dirac Sea). * The Positron: He predicted a particle with the same mass as the electron but an opposite (positive) charge. * Discovery: In 1932, Carl Anderson experimentally discovered the positron, confirming Dirac’s theory and proving the existence of antimatter.

4. Notation & Formalism (The Language of Quantum Computing)

Dirac created the standard “language” used in modern quantum mechanics and quantum computing. * Bra-Ket Notation: A shorthand for vectors and inner products in Hilbert space. * Ket: (a state vector). * Bra: (the conjugate transpose). * Bracket: (an inner product/probability amplitude). * Delta Function: The Dirac Delta (), a “generalized function” used to model point-like densities. * Poisson Brackets: He was the first to realize the deep mathematical link between classical Poisson brackets and the quantum commutators .

5. Quantum Electrodynamics (QED)

• Foundations: Dirac is considered one of the founders of QED, the first theory to successfully quantize the electromagnetic field.
• Monopoles: He theoretically showed that if even a single magnetic monopole existed in the universe, it would explain why electric charge is quantized.

6. Fun Facts

• Q: What two theories did the Dirac Equation unify?
• A: Quantum Mechanics and Special Relativity.
• Q: What did his equation predict that was later discovered by Carl Anderson?
• A: The Positron (Antimatter).
• Q: What is the significance of in your coding work (Qiskit/PennyLane)?
• A: It is Dirac’s Ket notation, representing the quantum state vector.
• Q: What was Dirac’s “Mathematical Beauty” principle?
• A: He believed that physical laws should have mathematical beauty and that a beautiful equation was more likely to be “right” than an ugly one that fit data.

1. Biographical & Professional Context

• Origin: Hungarian-American mathematician, physicist, and computer scientist born in Budapest.
• Prodigy: Known for his photographic memory and lightning-fast mental calculations (reportedly divided 8-digit numbers in his head by age six).
• The “Martians”: Part of a group of brilliant Hungarian scientists (along with Szilard and Teller) nicknamed “The Martians” for their seemingly superhuman intelligence.
• Career: A founding member of the Institute for Advanced Study (IAS) in Princeton, working alongside Einstein and Gödel.
• War Effort: Played a critical role in the Manhattan Project, specifically the design of the explosive lenses needed for the implosion-type atomic bomb.

2. Foundations of Quantum Mechanics (1932)

Before von Neumann, quantum mechanics was a collection of brilliant but loosely connected ideas (Schrödinger’s waves vs. Heisenberg’s matrices). * The Mathematical Synthesis: In his book Mathematical Foundations of Quantum Mechanics, he proved that Wave Mechanics and Matrix Mechanics were mathematically equivalent. * Hilbert Space: He introduced the rigorous framework of Hilbert Space () as the setting for all quantum states. * Operators: Defined physical observables as Hermitian operators acting on that space. * Density Matrix: Introduced the Density Matrix (), which is essential for describing “mixed states” and is a core concept in modern quantum information theory and your work with PennyLane/Qiskit.

3. Computer Science & The Von Neumann Architecture

He is the father of modern computing as we know it today. * Von Neumann Architecture: Proposed the design where the instruction data and the program data are stored in the same memory. * Components: Defined the standard structure: 1. A processing unit (ALU and registers). 2. A control unit (instruction register and program counter). 3. Memory. 4. External mass storage. 5. Input/Output mechanisms. * Stochastic Computing: Explored how to build reliable computers from unreliable components, a precursor to error-correction theories.

4. Game Theory & Economics

• Minimax Theorem (1928): Proved that in zero-sum games with perfect information, there is always a strategy that minimizes the maximum possible loss for both players.
• Theory of Games and Economic Behavior: Co-authored with Oskar Morgenstern, this founded the entire field of Game Theory, revolutionizing economics and social sciences.

5. Cellular Automata & Self-Replication

• Self-Replicating Machines: He designed a theoretical “Universal Constructor” that could create a copy of itself, proving that machine reproduction was mathematically possible.
• Cellular Automata: Created the first cellular automata models (later popularized by Conway’s “Game of Life”) to study complex systems.

6. Fun Facts

• Q: What was von Neumann’s main contribution to the QM debate?
• A: He provided the rigorous mathematical proof (using Hilbert Space) that unified the different versions of quantum mechanics.
• Q: What is the “Von Neumann Bottleneck”?
• A: The limited throughput between the CPU and memory in his standard architecture, which restricts processing speed.
• Q: What mathematical object did he introduce to describe mixed quantum states?
• A: The Density Matrix ().
• Q: What was his role in the Cold War?
• A: He was a key strategist for the U.S. government, applying Game Theory to nuclear deterrence and the “Mutually Assured Destruction” (MAD) doctrine.

1. Biographical & Academic Context

• Origin: Swiss physicist born in Zürich.
• Academic Pedigree: He was the first graduate student of Werner Heisenberg at the University of Leipzig.
• Global Career: Fled Nazi Germany in 1933 and became the first professor of theoretical physics at Stanford University.
• Nobel Prize (1952): Shared with Edward Purcell for the development of new methods for nuclear magnetic precision measurements (Nuclear Magnetic Resonance or NMR).
• CERN: He served as the first Director-General of CERN in Geneva (1954–1955).

2. Solid State Physics: Bloch Waves

Before Bloch, it was a mystery how electrons could move through a solid metal without being scattered by every single atom. * Bloch’s Theorem: He applied quantum mechanics to crystal lattices. He proved that electrons in a periodic potential (like a crystal) move as “waves” modulated by the lattice. * The Formula (Bloch Function): * Significance: This is the basis of the Energy Band Theory (conductors, insulators, and semiconductors). Your modern computer and the SENG hardware you study rely entirely on this principle.

3. Nuclear Magnetic Resonance (NMR)

Bloch discovered a way to measure the magnetic moment of atomic nuclei in liquids and solids. * The Concept: When placed in a strong magnetic field, nuclei align with the field. By applying a radio-frequency (RF) field, the nuclei “flip” or precess. * Bloch Equations: A set of macroscopic equations that describe the nuclear magnetization as a function of time. * Legacy: This discovery led directly to the invention of MRI (Magnetic Resonance Imaging) in medicine.

4. The Bloch Sphere (Quantum Computing Link)

While the “Bloch Sphere” is a geometric representation named in his honor, it is derived from his work on magnetic resonance and the spin of particles. * Definition: A geometrical representation of the pure state space of a two-level quantum mechanical system (a qubit). * Coordinates: Any qubit state can be represented as a point on the surface of the sphere: * Application: In PennyLane or Qiskit, the Bloch Sphere is the standard way to visualize gate rotations (X, Y, Z gates).

5. Magnetism & Neutrons

• Spin Waves: He developed the theory of “magnons” (spin waves) to explain how magnetization changes with temperature.
• Neutron Moment: He performed the first precise measurement of the magnetic moment of the neutron.

6. Fun Facts

• Q: What is a Bloch Wave?
• A: A quantum mechanical wave function for a particle (usually an electron) moving in a periodic potential/crystal lattice.
• Q: What did his Nobel-winning work lead to?
• A: Nuclear Magnetic Resonance (NMR) and MRI technology.
• Q: How does his work relate to semiconductors?
• A: His Band Theory explains why some materials conduct electricity and others don’t, based on the behavior of electrons in crystals.
• Q: What is the significance of the Bloch Sphere in SENG/Quantum labs?
• A: It is the primary tool for visualizing the state and rotation of a single qubit.

1. Biographical & Professional Context

• Origin: American mathematician and electrical engineer born in Michigan.
• Education: Dual degrees in Mathematics and Electrical Engineering from the University of Michigan; PhD from MIT.
• The “Most Important Master’s Thesis”: At age 21, he wrote A Symbolic Analysis of Relay and Switching Circuits, which changed the world by linking logic to electronics.
• Career: Spent much of his career at Bell Labs and MIT.
• Personality: Known for his playful genius; he famously invented a flame-throwing trumpet, a motorized pogo stick, and “The Ultimate Machine” (a box that turns itself off).

2. Digital Circuit Theory (1937)

Before Shannon, circuit design was an ad-hoc art. He proved that it was a science based on Boolean Algebra. * The Discovery: He showed that the “on/off” states of electrical switches (relays) could represent the “True/False” values of Boolean logic. * Significance: This is the foundation of all modern digital computers. Every SENG course you take on architecture or logic gates traces back to this specific realization.

3. Information Theory (1948)

Shannon founded the entire field of Information Theory with his landmark paper, A Mathematical Theory of Communication. * The Bit: He popularized the term “bit” (binary digit) as the fundamental unit of information. * Source Coding Theorem: Proved that there is a statistical limit to how much a message can be compressed without losing information (the origin of ZIP files and JPEG compression). * Noisy-Channel Coding Theorem: Proved that data can be transmitted with zero errors over a “noisy” channel, provided the transmission rate is below the Shannon Capacity ().

4. Entropy in Information

Shannon borrowed the concept of entropy from thermodynamics (specifically from Boltzmann) to measure the “uncertainty” or “information content” of a message. * The Formula (Shannon Entropy): * Interpretation: * If a result is certain (), entropy is 0. * If a result is a coin flip (), entropy is maximized (1 bit of information).

5. Cryptography & AI

• Communication Theory of Secrecy Systems: Shannon proved that the One-Time Pad is the only unbreakable cipher (provided the key is random and never reused).
• Artificial Intelligence: He created Theseus, a mechanical mouse that could learn to navigate a maze using a memory of relay switches—one of the earliest examples of machine learning and AI.
• Computer Chess: Wrote the first significant paper on how a computer could be programmed to play chess (Programming a Computer for Playing Chess).

6. Fun Facts

• Q: What did Shannon bridge in his Master’s thesis?
• A: Electrical engineering (switching circuits) and Philosophy/Logic (Boolean Algebra).
• Q: What is the “Shannon Limit”?
• A: The maximum rate at which information can be transmitted over a communication channel with a specific noise level.
• Q: How does Shannon Entropy relate to Data Mining (SENG 474)?
• A: Entropy is used in Decision Trees (like ID3 or C4.5) to calculate Information Gain when splitting data.
• Q: What was “Theseus”?
• A: A mechanical mouse that demonstrated the first practical application of “electronic” learning/memory.

1. Biographical & Professional Context

• Origin: American theoretical physicist born in Queens, New York.
• Manhattan Project: Recruited as a young prodigy to work at Los Alamos, where he oversaw the human “computer” groups calculating implosion rates.
• The “Great Explainer”: Known for his ability to explain complex concepts in simple terms (The Feynman Lectures on Physics).
• Nobel Prize (1965): Shared with Tomonaga and Schwinger for fundamental work in Quantum Electrodynamics (QED).
• Challenger Disaster: In 1986, he served on the Rogers Commission and famously demonstrated the failure of O-rings using a glass of ice water.

2. Quantum Electrodynamics (QED) & Feynman Diagrams

Feynman revolutionized how physicists calculate particle interactions. * Feynman Diagrams: Instead of massive, complex equations, he introduced a visual bookkeeping system for the interactions of subatomic particles. * Components: * Straight lines represent fermions (like electrons). * Wavy/wiggly lines represent bosons (like photons). * Vertices represent interactions. * Significance: These diagrams are actually shorthand for complex mathematical integrals used to calculate “scattering amplitudes.”

[Image of a Feynman diagram showing electron-positron annihilation]

3. The Path Integral Formulation

Feynman provided a third way to look at quantum mechanics (distinct from Schrödinger’s waves and Heisenberg’s matrices). * The Concept: A particle doesn’t just take one path from point A to point B; it takes every possible path simultaneously. * The Math: The probability amplitude is found by summing the phases of all possible paths: * S: The Action of the path. * Significance: This formulation is the backbone of modern Quantum Field Theory (QFT).

4. The Father of Quantum Computing (1981)

In a famous keynote titled “Simulating Physics with Computers”, Feynman noted that classical computers could not efficiently simulate quantum systems because of the exponential complexity. * The Proposal: “Nature isn’t classical, dammit, and if you want to make a simulation of nature, you’d better make it quantum mechanical.” * The Goal: He proposed using a computer governed by quantum laws to simulate quantum physics—the birth of the field you are studying in PennyLane and Qiskit.

5. Nanotechnology: “There’s Plenty of Room at the Bottom”

In a 1959 talk, Feynman predicted the field of nanotechnology. * The Vision: He suggested that individual atoms could be manipulated and that entire encyclopedias could be written on the head of a pin. * Impact: This inspired the development of scanning tunneling microscopes and the modern field of molecular engineering.

6. Fun Facts

• Q: What is the “Feynman Technique” for learning?
• A: Explain a concept to a child (or someone with no background); identify your gaps in understanding; go back to the source material; simplify and create an analogy.
• Q: What problem did Feynman Diagrams solve?
• A: They made the incredibly difficult calculations of Quantum Electrodynamics (QED) manageable and intuitive.
• Q: How did Feynman contribute to the Challenger investigation?
• A: He proved that the O-ring seals lost elasticity at freezing temperatures, leading to the shuttle’s explosion.
• Q: What was his stance on the simulation of physics?
• A: He argued that only a quantum computer could accurately and efficiently simulate the quantum world.

1. Biographical & Professional Context

• Origin: Northern Irish physicist born in Belfast.
• Career: Spent most of his career at CERN in Geneva as a theoretical particle physicist, though his most famous work was done as a “hobby” in foundations of quantum mechanics.
• Legacy: Often described as the man who proved “Einstein was wrong” about local realism, though he deeply respected Einstein’s desire for clarity.
• The “Bell’s Theorem” Paper (1964): Titled On the Einstein-Podolsky-Rosen Paradox, it is one of the most cited and profound papers in the history of science.

2. The EPR Paradox & Local Realism

Before Bell, the physics community mostly ignored the EPR (Einstein-Podolsky-Rosen) Paradox. * Einstein’s View: Quantum mechanics must be “incomplete.” There must be Hidden Variables that determine a particle’s state before we measure it. * Local Realism: The belief that: 1. Realism: Objects have definite properties even when not observed. 2. Locality: No influence can travel faster than the speed of light.

3. Bell’s Theorem (Bell’s Inequality)

Bell proved that no theory based on Local Hidden Variables could ever reproduce all the predictions of quantum mechanics. * The Concept: He created a mathematical “limit” (an inequality) for the correlation between measurements of two entangled particles. * The Result: If quantum mechanics is correct, the correlation between particles will violate this inequality. * The Inequality (CHSH version): * Significance: Quantum mechanics predicts a value of , which is greater than 2. This means nature is non-local.

4. Experimental Confirmation

Bell’s work was purely theoretical until others built experiments to test it. * Alain Aspect (1982): Conducted the first definitive experiment using entangled photons, proving that Bell’s Inequality was violated. * 2022 Nobel Prize: Awarded to Aspect, Clauser, and Zeilinger for these experiments, directly validating Bell’s 1964 theorem. * Impact: This confirmed that Entanglement is a “spooky” but real connection that defies classical local logic.

5. “Speakable and Unspeakable in Quantum Mechanics”

• The Book: A collection of Bell’s essays where he argues for precision in language (the “Speakable”).
• Criticism of “Measurement”: He disliked the word “measurement” because it implies a human observer is necessary; he preferred the term “beables” for things that actually exist in the physical world.
• The “Bertlmann’s Socks” Analogy: A famous analogy he used to explain the difference between classical correlation (wearing matching socks) and quantum entanglement.

6. Fun Facts

• Q: What did Bell’s Theorem prove?
• A: That no local hidden variable theory can replicate the predictions of quantum mechanics (i.e., the universe is non-local).
• Q: What is a “Bell State” in Quantum Computing?
• A: One of the four specific maximally entangled states of two qubits (e.g., ).
• Q: How does this relate to SENG/Quantum labs (Qiskit/PennyLane)?
• A: When you use a CNOT gate and a Hadamard gate to entangle two qubits, you are creating a system that violates Bell’s Inequality.
• Q: What is the “Superdeterminism” loophole?
• A: A theoretical way to save locality by suggesting that the universe is completely predetermined, including the choices of the experimenters.

By Zak Toews

4th Year Software Engineering Student

Part 1: Documenting my Quantum Journey

import matplotlib.pyplot as plt
from qiskit.visualization import plot_bloch_vector

# Coordinates for the 6 basis states [x, y, z]
states = {
    "|0>": [0, 0, 1],
    "|1>": [0, 0, -1],
    "|+>": [1, 0, 0],
    "|->": [-1, 0, 0],
    "|i>": [0, 1, 0],
    "|-i>": [0, -1, 0]
}

# Create subplots with 3D projection
fig = plt.figure(figsize=(20, 5))

for i, (label, coord) in enumerate(states.items()):
    # Add a 3D subplot for each basis vector
    ax = fig.add_subplot(1, 6, i + 1, projection='3d')
    plot_bloch_vector(coord, title=label, ax=ax)

plt.show()

Matrix Symbols and Multiplication Properties

1. Definitions * Complex Conjugate: or * Transpose: * Inverse: * Conjugate Transpose (Adjoint):

2. Multiplication Properties

3. Properties of Unitary Matrices () A matrix is unitary if its conjugate transpose is also its inverse.

4. Properties of Tensor Products ()

5. Dirac Notation & Computational Basis The computational basis for a single qubit consists of the states and .

• Column Vectors (Kets):

• Row Vectors (Bras):

• Linear Combination of Unit Kets: A general quantum state is a superposition: where and .

Inner Product and Orthnormality The inner product of two vectors and is written as . For the computational basis:

Measurement and the Born Rule When measuring a state in the computational basis:

• Probability of outcome :

• Probability of outcome :

• General Born Rule: For a state , the probability of measuring an eigenvalue associated with eigenvector is:

6. Quantum Logic Gates (Single-Qubit)

7. Common Gate Identities Note: Identities are often given “up to a global phase” ().

8. Tensor Product of Vectors (-qubits) If we have individual qubits in states , the combined state is the tensor product:

For the computational basis, the state where is a binary string : This results in a column vector of size .

9. Tensor Product of Matrices (Kronecker Product) For two matrices (size ) and (size ), the tensor product is a matrix of size :

10. 2-Qubit Computational Basis The basis for a 2-qubit system is formed by the tensor product of two 1-qubit basis vectors. There are possible states:

General Rule for n-qubits For an -qubit state where is the decimal representation of the binary string: * The vector has a length of . * The vector has a at the -th index (starting from 0) and elsewhere.

Mixed Tensor Products (Superposition) If and , then:

Basis Definitions

• Hadamard Basis (-basis):

Tensor Products of Basis Vectors

Mixed Basis States

Measurement in the Hadamard Basis To measure a state in the Hadamard basis using the Born Rule, you project onto the or states:

11. 2-Qubit Gate Tensor Products The following matrices represent the operations when gates are applied in parallel to two separate qubits.

12. 3-Qubit Hadamard Tensor Product () Applying a Hadamard gate to three qubits simultaneously creates a uniform superposition of all 8 possible basis states.

13. General n-qubit Identity and Parity * : An identity matrix of size . * : A diagonal matrix where the entry is if the basis state has an even number of s and if it has an odd number of s.

14. Entanglement: The 4 Bell States

The Bell states (or EPR pairs) are the four maximally entangled two-qubit states. They form an orthonormal basis for the 4-dimensional Hilbert space of two qubits.

Generating Bell States via Circuit The general recipe to create a Bell state from a basis state is:

Matrix of the CNOT Gate The CNOT gate flips the target (second) qubit if the control (first) qubit is :

The Four Bell States

Initial State	Bell State Notation	Vector / Formula

*Of most of which was generated by Gemini [3]

It is also possible to visualize the equations in LaTeX using the draw method of Statevector object in Qiskit. For example:

Show code

from qiskit.quantum_info import Statevector
from qiskit import QuantumCircuit

qc = QuantumCircuit(2)
qc.x(1)
qc.h(1)
qc.z(1)
psi = Statevector(qc)
psi.draw("latex")

import pennylane as qml

# We need 3 wires for teleportation
dev = qml.device("default.qubit", wires=3)

@qml.qnode(dev)
def teleportation_circuit():
    
    # 1. PREPARE THE STATE TO TELEPORT (on wire 0)
    # Let's just create a random state using a rotation
    qml.RX(1.23, wires=0)
    qml.RY(0.45, wires=0)

    # 2. CREATE ENTANGLEMENT (between Alice/wire 1 and Bob/wire 2)
    qml.Hadamard(wires=1)
    qml.CNOT(wires=[1, 2])

    # 3. ALICE'S OPERATIONS (Bell State Measurement)
    qml.CNOT(wires=[0, 1])
    qml.Hadamard(wires=0)

    # 4. MEASURE AND CONDITIONALLY APPLY CORRECTIONS
    # In PennyLane, we use m_0 and m_1 to represent Alice's measurement results
    m_0 = qml.measure(0)
    m_1 = qml.measure(1)

    # Bob applies corrections based on Alice's classical bits
    # If m_1 is 1, apply PauliX; if m_0 is 1, apply PauliZ
    qml.cond(m_1, qml.PauliX)(wires=2)
    qml.cond(m_0, qml.PauliZ)(wires=2)

    # Return the probabilites in the state
    return qml.probs(wires=2)

# Execute
result = teleportation_circuit()
print(result)

[0.6504817 0.3495183]

Part 1 Bibliography

[1] GeeksforGeeks, “Matrix Operations,” GeeksforGeeks, Nov. 23, 2020. https://www.geeksforgeeks.org/maths/matrix-operations/
[2] “Matrix Calculator - Reshish,” matrix.reshish.com. https://matrix.reshish.com/
[3] Google, “Gemini,” gemini.google.com, 2025. https://gemini.google.com/app
[4] “Map | PennyLane Codebook,” Pennylane.ai, 2025. https://pennylane.ai/codebook/learning-paths
[5] “IBM Quantum Computing | Qiskit,” Ibm.com, 2026. https://www.ibm.com/quantum/qiskit#tutorials (accessed Feb. 12, 2026).
[6] “Fertilizer and other quantum computer chemistry,” Quantum Flagship. https://qt.eu/applications/fertilizer-and-other-quantum-computer-chemistry
[7] G. Dunn, V. Sinha, Laurent-Pierre Baculard, S. Ali, and W. Chang, “Quantum Computing Moves from Theoretical to Inevitable,” Bain, Sep. 23, 2025. https://www.bain.com/insights/quantum-computing-moves-from-theoretical-to-inevitable-technology-report-2025/
[8] L. Botelho, “Grover’s Algorithm,” PennyLane Demos, Jul. 03, 2023. https://pennylane.ai/qml/demos/tutorial_grovers_algorithm

Part 2: Documenting my Quantum Journey

Part 2 Bibliography

[1] https://www.quera.com/glossary/quantum-fourier-transform [2] https://www.quera.com/glossary/quantum-phase-estimation [3] https://gemini.google.com/app [4] Relevant, grad-student made presentations from the SEng 457 course BrightSpace page.

SENG 440 Embedded Systems — University of Victoria, Electrical and Computer Engineering

Summer 2025 · Instructor: Mihai Sima · Students: Robert Bell and Zak Toews · Group 18

Using register Key Word

By using the register keyword before a variable of 32 bits or less, the compiler can be motivated to use registers as the variables instead of putting them on the stack and fetching and storing them in memory. Store and Load operations are relatively costly operations at 2+ cycles a call. These Load and Store operations could ideally be avoided all together, saving many clock cycles.

Loop Unrolling

Reasons for loop unrolling include reduced loop control overhead and increased efficiency with pipelining.

The Cortex-M4 CPU utilizes a 3-stage pipelining strategy of Fetch, Decode, and Execute. This means that in optimal code, 3 instructions can be executed in parallel.

Figure 4. From the STM32 manual, we get a brief visual of the Cortex-M4 pipeline with the 3 pipeline operations in an optimised state.

By unrolling in certain circumstances, we can remove overhead of the loop AND reduce waiting or NOP usage resulting in less overall clock cycles.

CMSIS-DSP Intrinsics

The hardware for this project allowed for the use of the ARM CMSIS, a standardized layer between the user specific C code and the ARM Cortex-M hardware, that emphasizes portability, readability, performance, and faster development. CMSIS also provides developers access to a whole suite of debugging tools such as the DWT (Data Watchpoint and Trace unit) and SWO (Serial Wire Output), which provided for higher visibility on the project, concerning both debugging and performance evaluations. The DWT->CYCCNT provided access to the number of cycles which was used to quantify the performance of the code during the application of optimization techniques.

CMSIS-DSP is a library of optimized math functions used for digital signal processing. These functions can allow for SIMD (Single Instruction, Multiple Data) design. This type of arithmetic essentially packs several numbers into a single register, so that operations take place on multiple pieces of data in a single instruction, which increases parallelism by making better use of the pipeline.

Figure 5. A table outlining the DSP intrinsics explored in development.

Branch Elimination via Predicated Operations

Embedded systems heavily rely on efficient pipelines to fetch, decode, and execute instructions at the same time. Conditional statements in C (IF/ELSE) are translated into assembly, and will either form an IT block or a conditional branch; both using a comparison operation to decide control flow. Unpredictable branch outcomes have the possibility to stall the pipelining as pipeline flushing or NOP operations may occur, which results in cycles being wasted.

Replacing conditional statements with predicate based branchless operations allow for the system to run the same operations without making a comparison, and the need for checking flags. Bit masking will allow these conditions to be converted into a mask, which can select, modify or negate values and run these operations every time, resulting in a more stable control flow.

When evaluating the functions using condition and predicated operations in this project, it was found that the conditional statements resulted in less cycles being used. The version using predicate based operations required a mask to be created and applied to several values, and having to manage intermediate results. These tasks were necessary to be applied during each loop iteration. This extra load on the ALU resulted in a larger cost than the small branch penalty in the unoptimized CORDIC algorithm.

Replacing Z-Table Calculation with Pre-Initialized Values

The initial code for the project used an implementation of the Z-table calculation algorithm to run before the CORDIC algorithm was used to populate the Z-values. Saving these values and simply assigning them to an array used 0.0683% of the cycles required to calculate them manually each time, from 105301 cycles required to 72.