Why does a free quantum wave packet spread over time?

A free-particle Gaussian wave packet spreads because it is composed of many plane-wave components (different momenta k), each propagating at a different phase velocity v_phase = hk/2m. The packet envelope � determined by the group velocity v_group = d?/dk = hk0/m � maintains its initial momentum expectation value, but the width grows as s(t) = s0v(1+(ht/2ms0�)�). This dispersion is a direct consequence of the quadratic dispersion relation E = p�/2m (equivalently ? = hk�/2m). Narrower initial packets (smaller s0) spread faster because their momentum uncertainty ?p = h/2s0 is larger by the Heisenberg uncertainty principle s_x�s_p = h/2.

What is quantum tunneling and how does the WKB approximation predict transmission probability?

Quantum tunneling occurs when a particle with energy E < V0 penetrates a classically forbidden potential barrier of height V0 and width a. The wave function does not vanish inside the barrier but instead decays exponentially as ? ? exp(-?x), where ? = v(2m(V0-E))/h. The WKB (Wentzel-Kramers-Brillouin) approximation gives the transmission probability T � exp(-2∫κ(x)dx), evaluated across the barrier. For a rectangular barrier: T � exp(-2?a). Tunneling is crucial in a-decay (Gamow factor), tunnel diodes, scanning tunnelling microscopy (STM), enzyme catalysis, and quantum computing gate operations. The simulator shows how increasing barrier height V0 or width dramatically reduces the transmitted wave intensity.

How does the split-operator FFT method solve the time-dependent Schr�dinger equation?

The split-operator method exploits the Trotter-Suzuki decomposition: exp(-iHdt/h) � exp(-iV^dt/2h)�exp(-iT^dt/h)�exp(-iV^dt/2h) + O(dt�). The kinetic propagator exp(-iT^dt/h) = exp(-ihk�dt/2m) is diagonal in momentum space, so: (1) multiply ?(x) by exp(-iV(x)dt/2h) in position space; (2) FFT to k-space; (3) multiply by exp(-ihk�dt/2m); (4) inverse FFT back; (5) multiply by exp(-iV(x)dt/2h) again. This scheme is unitary (exactly norm-conserving), time-reversible, and second-order accurate in dt. With N=512 grid points it runs at real-time or faster speed in JavaScript using the Cooley-Tukey radix-2 FFT algorithm (O(N log N) complexity per step).

Home▸Quantum Physics▸Quantum Wave Packet Simulator – Tunneling & Harmonic Oscillator

🧪 Quantum Wave Packet Simulator – Tunneling & Harmonic Oscillator

Interactive quantum wave packet simulator using split-operator FFT. Six presets: Free Particle spreading, Harmonic Oscillator coherent states, Barrier Tunneling, Double Well, Step Scattering, and Wave Packet Collision. Watch probability density evolve in real time. Explore the uncertainty principle, group velocity, and quantum tunneling.

Quantum Physics2DModerate60 FPS

⚙ Under the hood

This simulation models the behavior of a quantum wave packet, demonstrating concepts like tunneling and the harmonic oscillator. It illustrates fundamental principles of quantum mechanics at a macroscopic level.

Quantum Wave PacketTunnelingHarmonic Oscillator

2D · HTML5 Canvas 2D · 60 FPS target · runs fully client-side, no install