Quantum

Quantum Foundations

Start

Use the photoelectric effect and single-particle interference to motivate quantum ideas.

Wavefunctions

Next

Interpret amplitudes, probabilities, and measurement in a one-dimensional quantum state.

Quantization

Next

Use bound-state potentials to build intuition for discrete energies in quantum systems.

Time Evolution

Here

Evolve wavefunctions through free spreading, finite barriers, and two-dimensional interference.

Time Evolution

We introduced ψ(x)\psi(x) as a probability amplitude, and the time-independent Schrodinger equation gave us specific allowed states. The next question is how a wavefunction ψ(x,t)\psi(x,t) evolves over time.

In nonrelativistic quantum mechanics, wavefunction dynamics are given by the time-dependent Schrodinger equation:

iℏ∂ψ∂t=(−ℏ22m∇2+V)ψ.i\hbar \frac{\partial \psi}{\partial t} = \left(-\frac{\hbar^2}{2m}\nabla^2 + V\right)\psi.

This particular combination of spatial and temporal derivatives can seem a bit strange, especially if you haven’t encountered classical Hamiltonian mechanics.

The expression in parentheses is the Hamiltonian operator. At a high level, there is a kinetic energy term determined by curvature in the wavefunction and a potential energy term VV. Once you know the initial wavefunction and the potential landscape, the equation predicts the full future wavefunction.

Free Packets Spread

A localized wave packet is built from many momentum components. Even when no force acts (V=0V=0), those component waves propagate at different rates. The net result is that the packet’s center moves with the group velocity, while the envelope spreads.

For a free Gaussian packet in the dimensionless units, the spatial width follows

σ(t)=σ01+(t2σ02)2.\sigma(t)=\sigma_0\sqrt{1+\left(\frac{t}{2\sigma_0^2}\right)^2}.

The narrower the packet starts (smaller uncertainty in position), the faster it spreads. This is Heisenberg’s uncertainty principle naturally arising from wave dynamics.

Free Gaussian Packet

Dispersion

A free packet translates at its group velocity while its width grows because different momentum components carry different phase speeds.

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Why Free Packets Spread

Show how the zero-potential Schrodinger equation makes each momentum component accumulate a different phase.

With V=0V=0 in one dimension, the time-dependent Schrodinger equation becomes

iℏ∂ψ∂t=−ℏ22m∂2ψ∂x2.i\hbar \frac{\partial \psi}{\partial t} = -\frac{\hbar^2}{2m}\frac{\partial^2 \psi}{\partial x^2}.

Try one momentum component,

ψk(x,t)=Aei(kx−ωt).\psi_k(x,t)=A e^{i(kx-\omega t)}.

Substitution gives

ℏω=ℏ2k22m,ω(k)=ℏk22m.\hbar \omega = \frac{\hbar^2 k^2}{2m}, \qquad \omega(k)=\frac{\hbar k^2}{2m}.

So the component evolves as

ψk(x,t)=Aei(kx−ℏk22mt).\psi_k(x,t)=A e^{i\left(kx-\frac{\hbar k^2}{2m}t\right)}.

A localized packet is a sum of many nearby kk values:

ψ(x,t)=∫a(k)ei(kx−ω(k)t) dk.\psi(x,t)=\int a(k)e^{i(kx-\omega(k)t)}\,dk.

Because ω(k)\omega(k) is quadratic, nearby momentum components do not keep the same relative phase. The center moves at the group velocity vg=dω/dk=ℏk/mv_g=d\omega/dk=\hbar k/m, while the changing relative phases make the envelope spread.

Scattering From A Barrier

When a packet reaches a finite potential barrier,

V(x)={V0a<x<b0otherwiseV(x) = \begin{cases} V_0 & a < x < b \\ 0 & \text{otherwise} \end{cases}

we would expect a classical particle to either be reflected or transmitted, dependent on the particle’s kinetic energy. In quantum mechanics, the wavefunction has both a reflected and transmitted component after interaction with the barrier. When the barrier is higher than the packet’s energy scale, the transmitted part is an example of tunneling.

Finite Barrier Scattering

V0

Scattering

The barrier is higher than the packet energy scale, but the tail can still leak through because the wavefunction penetrates the forbidden region.

Controls

Two-Dimensional Evolution

In two dimensions, the time dependent Schrodinger equation becomes

i∂ψ(x,y,t)∂t=[−12(∂2∂x2+∂2∂y2)+V(x,y)]ψ.i\frac{\partial \psi(x,y,t)}{\partial t} = \left[-\frac{1}{2}\left(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}\right)+V(x,y)\right]\psi.

The simulator below uses a split-step Fourier method in dimensionless units with ℏ=1\hbar=1 and m=1m=1. It alternates between the potential update in position space and the kinetic update in momentum space. Wavelike phenomena like diffraction, reflection, transmission, spreading, and interference emerge naturally from the dynamics described by this equation.

Note that it’s difficult to handle boundaries in a finite simulation, so there are some non-physical effects once the wavefunction begins “interacting” with the screen edges.

2D Time Evolution Simulator

Evolve a wavefunction through slits, free space, and finite barriers using the time-dependent Schrodinger equation.

Problem Solving

Packet Spreading Estimate

Problem

A free Gaussian packet begins with width σ0=0.50\sigma_0=0.50 in dimensionless units. Estimate the width at t=1.0t=1.0.

Use

σ(t)=σ01+(t2σ02)2.\sigma(t)=\sigma_0\sqrt{1+\left(\frac{t}{2\sigma_0^2}\right)^2}.

Solve

σ(1)=0.501+(12(0.50)2)2=0.501+22≈1.12.\sigma(1)=0.50\sqrt{1+\left(\frac{1}{2(0.50)^2}\right)^2} =0.50\sqrt{1+2^2} \approx 1.12.

Check

The packet more than doubles in width because it began tightly localized.

Time Evolution Checkpoint

Question 1 of 2

Why does a free localized wave packet spread over time?

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