Quantum Lab 03 · path amplitudes

Single-Particle Interference

How do unpredictable hits accumulate into stable fringes? Each particle lands at one random spot, but the chance of each spot comes from adding the amplitudes for both paths and squaring the sum. Many independent hits sample that pattern and slowly build up steady fringes. Reduce the coherence and the fringes fade.

Each particle lands at a single spot, but where it is likely to land depends on the complex amplitudes from both paths together. Send many particles and the unpredictable hits build up a stable interference pattern. Lower the coherence and the fringes wash out.

Detected particles0

Fringe visibility100%

Fringe spacing0.100 L

RegimeCoherent

Start with the essentials

Focus question: If particles are emitted one at a time and never collide, where does the fringe pattern come from?
One-sentence intuition: Each hit lands at random according to the squared magnitude of the sum of two path amplitudes, and many independent hits trace out that probability distribution.

Core mathematical model

Path-amplitude superposition

P(x)=|\psi_1(x)+\psi_2(x)|^2

The cross term contains the relative phase of the two paths and produces the bright and dark fringes.

Fringe visibility

\mathcal V=\frac{I_{\max}-I_{\min}}{I_{\max}+I_{\min}}

As $\mathcal V$ falls from 1 to 0, coherent interference disappears.

Common difficulties

Particles do not need to collide

Typical misconceptionMany particles must pass simultaneously for interference to occur.

Better mental modelEven when hits arrive far apart in time, each one still shows interference between its own two path amplitudes.

Path detection is not merely a mechanical kick

Typical misconceptionFringes vanish only because the detector knocks particles aside.

Better mental modelIf the path could in principle be identified, the particle becomes entangled with its environment, and that entanglement suppresses the cross term.

Run the experiment

01
Build a distribution hit by hit
Emit fully coherent particles at a low rate.
What to observe: Each hit is unpredictable, but the accumulated pattern settles into stable fringes.
02
Reduce coherence
Move $\mathcal V$ slowly from 1 to 0.
What to observe: Hits keep arriving, but the contrast between bright and dark fades.
03
Change the geometry
Vary $\lambda$ and $d$ separately.
What to observe: Larger $\lambda$ spreads fringes; larger $d$ compresses them.