Quantum wavepackets: tunnelling, scattering & oscillators

A Gaussian wavepacket evolves under the time-dependent Schrödinger equation, solved live with the unitary split-step Fourier method. Fire it at a barrier and compare the measured transmission with the exact transfer-matrix prediction.

Wavefunction ψ(x, t)

Fill colour of |ψ|² = complex phase arg ψ · grey = potential V(x) (right axis) · dashed = mean energy ⟨E⟩

Momentum space |φ(k)|² and T(k)

Reflection & transmission vs time

Governing equation

Natural units $\hbar = m = 1$. The wavefunction obeys

$$ i\,\frac{\partial\psi}{\partial t} = -\frac{1}{2}\frac{\partial^2\psi}{\partial x^2} + V(x)\,\psi .$$

Numerical method: Strang split-step Fourier

$$ \psi(t+\Delta t) = e^{-iV\Delta t/2}\;\mathcal{F}^{-1}\!\left[e^{-ik^2\Delta t/2}\,\mathcal{F}\!\left[e^{-iV\Delta t/2}\,\psi(t)\right]\right] $$

Every factor is a pure phase, so the scheme is exactly unitary: $\int|\psi|^2dx$ stays 1 to machine precision (shown live). The kinetic step is applied in momentum space with a radix-2 FFT, giving spectral accuracy. Grid: 4096 points, Δx ≈ 0.098, Δt = 0.01, splitting error $O(\Delta t^3)$ per step.

Absorbing edges damp outgoing waves. The probability they remove is booked as reflected (left) or transmitted (right), so R + T stays meaningful.

Exact theory for comparison

For piecewise-constant potentials, write $\psi = A_j e^{iq_jx} + B_j e^{-iq_jx}$ with $q_j=\sqrt{2(E-V_j)}$ in each region and match ψ and ψ′ at every interface. The resulting transfer matrix gives the exact plane-wave $T(E)$. For a single barrier with $E<V_0$:

$$ T = \left[1 + \frac{V_0^2\sinh^2(\kappa a)}{4E(V_0-E)}\right]^{-1},\qquad \kappa=\sqrt{2(V_0-E)} .$$

A wavepacket contains a spread of momenta $|\phi(k)|^2 \propto e^{-(k-k_0)^2/2\sigma_k^2}$, with $\sigma_k = 1/2\sigma$. The prediction to compare with the simulation is therefore $\bar T = \int T(k)\,|\phi(k)|^2\,dk$. The test suite checks that the simulation matches this value to within 0.1%.

Things to try

Tunnelling: set V₀ above $k_0^2/2$. Part of the packet still gets through, and T falls off exponentially with width a.
Resonant tunnelling: in the double-barrier mode, look at the sharp T(k) peaks in the momentum plot and tune k₀ onto one. Use a wide packet (σ ≈ 20).
Above-barrier reflection: set E > V₀. A classical particle would always pass, but some of the wave reflects.
Harmonic oscillator: ⟨x⟩(t) follows the classical $x_0\cos\omega t$ (Ehrenfest's theorem). With σ = $1/\sqrt{2\omega}$ the packet is a coherent state and never spreads.
Measure x: collapses ψ to a narrow packet at a position drawn from |ψ|² (Born rule).