AP Physics 2 · Unit 7

Wave–Particle Duality & Quantum Mechanics

Standards: 15.1.1 · 15.1.2 · 15.1.3
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Double-Slit Simulation

λ Wavelength400 nm
d Slit Sep.1.50 μm
Δy Fringe Spacing—
Hits0
Mode
Wavelength λ
400 nm
Slit Sep. d
1.50 μm
How to Use

The Double-Slit Experiment

Fire light or electrons through two narrow slits at a detector. Classical physics predicts two bright bands. What you actually get shatters that prediction.

  1. Wave mode: animated interference — bright and dark bands from constructive/destructive overlap.
  2. Particle mode: each dot is one quantum arriving at a random spot. Unpredictable individually.
  3. Let 1,000+ particles accumulate — the quantum interference pattern emerges from randomness.

Try it: Drag λ slider to see fringe spacing change live. Drag d — fringes compress. This is exactly d sinθ = mλ.

The Big Picture

Why This Is Radical

Each particle passes through both slits simultaneously — then collapses to a point when detected. Before measurement: wave. After measurement: particle. This is not a limitation of instruments. It is quantum reality.

Thomas Young (1801)

The Original Experiment

Young used sunlight through a pinhole to create a coherent source, then directed it at two closely spaced slits carved into a card. Alternating bright and dark bands appeared on a screen — explainable only by wave interference.

Revolutionary: Newton believed light was particles. Young's fringes proved it was a wave. A century-long debate began.

Electron Double-Slit (1961)

Jönsson's Confirmation

Claus Jönsson fired individual electrons through a double slit. Same interference pattern as light — even one electron at a time. Each electron interferes with itself, passing through both slits simultaneously.

Voted "most beautiful experiment in physics" in a 2002 Physics World poll.

Timeline
1801
Young's double-slit
Sunlight + card slits → fringes. Proved light is a wave.
1905
Einstein — photons
Photoelectric effect proves light is also discrete particles.
1924
de Broglie — matter waves
All matter has wavelength λ = h/p.
1961
Jönsson — electrons
Single electrons make interference pattern. Wave-particle duality confirmed for matter.
Standard 15.1.1

Light as a Wave

Each slit becomes a new point source (Huygens' principle). Circular wavefronts spread out and overlap.

Constructive: path difference = mλ → bright fringe

Destructive: path difference = (m+½)λ → dark fringe

Key Equations

Fringe Conditions

Bright fringe
\[ d \sin\theta = m\lambda \]
Fringe spacing (small angle)
\[ \Delta y = \frac{\lambda L}{d} \]
dSlit separation (m)
θAngle to m-th fringe
mOrder: 0, ±1, ±2…
λWavelength (m)
LSlit-to-screen distance (m)
ΔyFringe spacing (m)

AP Tip: sin θ ≈ y/L for small angles. Keep everything in meters.

Standard 15.1.2

Photon Energy

Planck-Einstein
\[ E = hf = \frac{hc}{\lambda} \]
EPhoton energy (J or eV)
h6.626 × 10⁻³⁴ J·s
fFrequency (Hz)
c3.00 × 10⁸ m/s
λWavelength (m)
Born Rule

Probability

Probability density
\[ P(y) \propto |\psi(y)|^2 \]

Individual hits: random. Many hits: the quantum interference pattern emerges statistically.

Standard 15.1.3

Wave–Particle Duality

🌊 WAVE

Interference, diffraction. Spread across space. No definite position before measurement.

⚛ PARTICLE

Localized. Discrete energy E = hf. Definite position when measured.

de Broglie

Matter Wavelength

de Broglie
\[ \lambda = \frac{h}{p} = \frac{h}{mv} \]
λWavelength (m)
h6.626×10⁻³⁴ J·s
pMomentum (kg·m/s)
mMass (kg)
vSpeed (m/s)

Measuring which slit collapses the wave function — interference vanishes. Fundamental, not instrumental.

Must Know

Key Equations

  • Bright fringes: d sin θ = mλ
  • Fringe spacing: Δy = λL/d
  • Photon energy: E = hf = hc/λ
  • de Broglie: λ = h/p = h/mv
Sample FRQ

AP-Style

Electrons: d = 2.0×10⁻⁶ m, L = 1.0 m, first fringe y = 0.15 m. Find (a) λ, (b) p, (c) effect of higher speed.

  1. sin θ ≈ y/L = 0.15
  2. λ = d·sinθ = (2×10⁻⁶)(0.15) = 3.0×10⁻⁷ m = 300 nm
  3. p = h/λ = 6.626×10⁻³⁴/3×10⁻⁷ ≈ 2.2×10⁻²⁷ kg·m/s
  4. Higher v → larger p → smaller λ → fringes compress
Common Mistakes
  • Larger d compresses fringes (Δy = λL/d)
  • Higher f → shorter λ (c = fλ)
  • Heavier particle → larger p → shorter λ

Quick Reference — Double-Slit

Bright Fringe

d sin θ = mλ

d and λ in meters. m = 0 is center. sin θ ≈ y/L for small angles.

Fringe Spacing

Δy = λL / d

Larger λ or smaller d → wider fringes.

Photon Energy

E = hf = hc/λ

h = 6.626×10⁻³⁴ J·s. Divide by 1.6×10⁻¹⁹ for eV.

de Broglie

λ = h / mv

Fast/heavy particles → tiny λ → quantum effects undetectable.

Born Rule

P(y) ∝ |ψ(y)|²

Random individually. Many particles → interference pattern.

History

Young (1801)

Sunlight + card slits → fringes visible to the naked eye. Proved light is a wave.

Photoelectric Effect — Vacuum Tube Circuit

hf Photon E—
φ Work Function—
Kmax=hf−φ—
Vstop—
Photocurrent I0.00 μA
e⁻ Ejected0
Raise frequency above threshold to eject electrons
Frequency f
5.0×10¹⁴
Angle θ
40°
Intensity
5
Stopping V
0.00 V
Metal (φ)
Hertz & Hallwachs (1887–88)

Accidental Discovery

Hertz noticed that UV light striking metal electrodes made sparks jump more easily. Hallwachs confirmed: UV causes negative charge to leave a metal surface. No classical explanation existed.

Lenard's Vacuum Tube (1902)

How Electrons Are Actually Measured

Lenard built exactly what this simulation shows — a sealed vacuum tube with a photocathode and collector, connected to a battery and galvanometer. He discovered:

  • Light produces a measurable current — electrons crossing the vacuum gap
  • Applying reverse voltage slows electrons down
  • At the stopping voltage Vs, current drops to exactly zero
  • Brighter light = more current, but NOT faster electrons

Lenard won the Nobel Prize in 1905 — the same year Einstein explained his own results.

Einstein (1905) & Millikan (1916)

Explanation and Proof

Einstein's photon theory: E = hf per photon. One photon → one electron. Kmax = hf − φ. Nobel Prize 1921.

Millikan spent a decade trying to disprove Einstein. His Vs vs f plot gave a straight line with slope h/e — the most accurate measurement of h at the time. He confirmed it instead. Nobel Prize 1923.

1887
Hertz discovers the effect
UV light causes sparks — no classical explanation.
1902
Lenard — vacuum tube
V_stop, photocurrent, frequency threshold measured precisely.
1905
Einstein — photon theory
K_max = hf − φ. Nobel 1921.
1916
Millikan — verification
V_s vs f straight line, slope = h/e. Nobel 1923.
The Circuit

How Electrons Are Measured

The experiment takes place inside a sealed vacuum tube — no air to stop electrons. Four components:

  • Photocathode: metal plate illuminated by light. Ejected electrons leave its surface.
  • Anode/collector: electrode that collects arriving electrons, completing the circuit.
  • Battery (Vstop): applies electric field. Forward accelerates electrons; reverse opposes them.
  • Galvanometer: sensitive current meter. Needle deflects proportional to electron flow.

The electric field between the plates (visible in the sim when Vstop > 0) steers electrons and allows Kmax to be measured.

Stopping Voltage Method

Measuring Kmax

  1. Apply a reverse (retarding) voltage — field opposes electron motion toward collector.
  2. Gradually increase reverse voltage until galvanometer reads zero.
  3. That voltage is Vs. It stopped even the fastest electrons.
  4. Therefore Kmax = eVs — work done against field = kinetic energy lost.

Try it: Raise the Stopping V slider — watch field lines appear and electrons curve back. At Vs ≈ Kmax, galvanometer reads zero.

Core AP Equation

Einstein's Photoelectric Equation

AP Physics 2 — must memorize
\[ K_{\max} = hf - \phi \]
KmaxMax KE of ejected electron (J or eV)
h4.136×10⁻¹⁵ eV·s
fFrequency of incident photon (Hz)
φWork function of metal (eV)

Ejection condition: hf ≥ φ. Below this: zero electrons, period.

Derived Equations
Threshold frequency
\[ f_0 = \frac{\phi}{h} \]
Stopping potential
\[ K_{\max} = eV_s \]

Shortcut: Kmax in eV = Vs in volts numerically. e.g. Kmax = 1.5 eV → Vs = 1.5 V.

Why Classical Physics Failed

Prediction vs. Reality

🌊 CLASSICAL

Any frequency works with enough intensity. Dim light builds up energy over time. Intensity → electron speed.

⚛ ACTUAL

Frequency threshold is absolute. Ejection is instant. Intensity only changes electron count, never speed.

Every classical prediction was wrong. This single experiment proved light must be quantized.

The Key Distinction

Intensity vs. Frequency

  • Frequency: determines if ejection occurs and Kmax of each electron
  • Intensity: determines photons/sec → electrons/sec → photocurrent magnitude
AP Exam Checklist
  • Kmax = hf − φ
  • Threshold: f₀ = φ/h
  • Stopping voltage: Kmax = eVs
  • Intensity → electron count (not energy)
  • Below f₀ → zero electrons always
  • Vs (volts) = Kmax (eV) numerically
Sample FRQ

f = 8.0×10¹⁴ Hz, φ = 2.0 eV. (a) Kmax? (b) Vs? (c) Halve frequency — ejection?

  1. E = hf = (4.136×10⁻¹⁵)(8.0×10¹⁴) = 3.31 eV
  2. Kmax = 3.31 − 2.0 = 1.31 eV
  3. Vs = 1.31 V
  4. f/2 → E = 1.65 eV < φ → No ejection
Common Mistakes
  • Brighter ≠ faster electrons — only more of them
  • At f₀: Kmax = 0, Vs = 0
  • Angle does NOT change Kmax
  • Ejection is always instantaneous above threshold

Quick Reference — Photoelectric Effect

Core Equation

Kmax = hf − φ

h = 4.136×10⁻¹⁵ eV·s. Ejection only when hf ≥ φ. Nobel Prize 1921.

Threshold Freq.

f₀ = φ / h

Below f₀: no electrons ever. At f₀: Kmax = 0 eV exactly.

Stopping Voltage

Kmax = eVs

Increase reverse V until I = 0. That Vs in volts = Kmax in eV.

The Circuit

Vacuum Tube

Cathode + Anode + Battery + Galvanometer, all in series, inside vacuum glass tube.

Intensity → I

Brighter = more e⁻

More photons/sec → more electrons/sec → higher photocurrent. Kmax unchanged.

Millikan (1916)

Vs vs f is linear

Slope = h/e. Intercept = −φ/e. This graph proves Einstein's equation experimentally.