Quantum Mechanics Lab — Interactive Physics Experiments

Quantum mechanics is not an FFT. A state can be written in different bases, and its position and momentum descriptions are related by a Fourier transform. The FFT is only a fast way to evaluate that transform on a grid. Here it drives the solver in experiment 1 and the spectra in experiment 2 — a numerical tool, not the physics.

The Schrödinger equation

A Gaussian wavepacket moving through a potential you choose. This is a genuine numerical solution — split-step Fourier on 1024 points — so barriers reflect and tunnel, and wells hold the packet in.

Momentum distribution |φ(p)|²

time t: 0
⟨x⟩: 0
⟨p⟩: 0
spread σ_x: 0
beyond x = 0: 0
still in view: 1

height V₀

barrier width

launch momentum p₀

packet width σ

mass m

Click or tap the plot to relaunch the packet from that point. The dashed line is the packet’s average energy.

iℏ ∂ψ/∂t = −(ℏ²/2m) ∂²ψ/∂x² + V(x)ψ

Time evolution. How fast the complex state changes. The factor i makes that change a rotation of phase — watch the colours roll along the packet.

Kinetic energy. Curvature costs energy. A narrow packet contains many momenta, so it spreads; heavier particles spread more slowly.

Potential energy. Position-dependent phase. Barriers reflect and let some amplitude leak through, a step changes the wavelength, and a harmonic well pulls the packet back.

Try raising V₀ above the dashed energy line: a classical particle would bounce every time, yet some probability still appears on the far side. Reflection also shows up in the momentum strip as a second peak at negative p.

Units with ℏ = 1. Each step applies half the potential phase, the exact kinetic phase in momentum space (via FFT), then the other half. The edges absorb outgoing waves, so “still in view” falls once the packet leaves.

Position and momentum

The same state, seen twice. The right plot is the numerical Fourier transform of the left one. Squeeze the packet in position and it widens in momentum.

Position ψ(x)

Momentum φ(p)

Uncertainty product σ_xσ_p against the bound ℏ/2

σ_x: —
σ_p: —
σ_xσ_p: —
⟨p⟩: —

position width σ

carrier momentum p₀

separation d

φ(p) = (2πℏ)^−½ ∫ ψ(x) e^−ipx/ℏ dx

A plane wave e^ikx has one definite momentum p = ℏk, just as a pure tone occupies a single frequency bin. Localising the packet takes a mixture of many such waves.

Only the Gaussian reaches the bound exactly. Two packets interfere in momentum space; the fringe spacing is 2π/d. The box has sharp edges, so its momentum spread has no finite limit — the value shown is capped by the grid.

The chirp is the subtle one. It adds a phase that varies as x², so |ψ(x)|² does not change at all, yet the momentum spread grows. Phase carries physical information.

Unlike a signal spectrum, |φ(p)|² is a probability density for measured momentum, and position and momentum are operators that don’t commute. That is where the analogy with signal processing stops.

Energy levels

Confinement allows only standing waves that fit. Pick one energy eigenstate and |ψ|² stays frozen while its phase turns; mix in a second and the probability starts to slosh.

first state E: —
second state E: —
slosh period: —
⟨x⟩ now: —

first state

second state

share of second state

well width L

mass m

Set the share to 0 or 1 for a pure eigenstate. Animation time is rescaled so one cycle always takes about three seconds.

Pauli exclusion

Two identical particles in a box. The map is their joint wavefunction Ψ(x₁, x₂): colour is its sign, strength is its size. Fermions must change sign when mirrored across the diagonal.

Joint wavefunction Ψ(x₁, x₂)

Filling the box with N particles

under exchange: —
both in same half: —
closer than L/10: —
fermion / boson energy: —

particle 1 in state

particle 2 in state

particles in the box N

Put both fermions in the same state and the map goes blank: the wavefunction is zero everywhere.

Ψ(x₁, x₂) = −Ψ(x₂, x₁)

Swapping identical fermions flips the sign of the whole state. If both occupied the same one-particle state, swapping would change nothing — so the state must equal its own negative, which only zero does.

The same rule leaves a dark line along x₁ = x₂: fermions are never found at the same point, and they sit apart more often than distinguishable particles would. Bosons do the opposite and bunch up.

That is why matter has structure. Fermions stack up the ladder two per level, one for each spin, so the energy grows quickly with N; bosons can all share the lowest level.

The map assumes both particles have the same spin, so the spatial part alone carries the antisymmetry. The ladder counts spin ↑ and ↓ as separate states.

Measurement

Two packets heading towards each other, in superposition. Each measurement returns one position, drawn from |ψ|², and resets the state to a narrow packet there — which then spreads again.

predicted P(x < 0): —
measured x < 0: —
measurements: 0
last outcome: —

weight of left packet |α|²

separation

relative phase φ

The double slit

Single photons, one at a time. Each lands as a single dot at a random place; the fringes only appear as the dots accumulate. Add which-path information and they wash out.

fringe spacing: —
central envelope: —
fringe visibility: —
photons detected: 0

wavelength λ

slit separation d

slit width a

which-path information

Entanglement

A spin singlet is split between Alice and Bob. Each sees a random ±1, whatever the settings. The structure is only in how their results agree — and it breaks a limit that any local model must obey.

CHSH test |S|

angle between analyzers: —
predicted E: —
measured E: —
++ / +− / −+ / −−: 0 / 0 / 0 / 0

Alice’s analyzer θ_A

Bob’s analyzer θ_B

entanglement visibility

Measure 1000 pairs, move Bob’s analyzer, measure again. The dots trace out the quantum curve, not the straight classical one.

Quantummechanics lab

The Schrödinger equation

Position and momentum

Energy levels

Pauli exclusion

Measurement

The double slit

Entanglement

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