Examples

This section describes complete, runnable examples demonstrating various capabilities of schr. All examples are available in the examples/ directory and include animated visualizations of quantum phenomena.

1D Examples

Free Particle Dispersion

File: examples/free_particle_1d.py

Physics:

A free particle is described by the time-dependent Schrödinger equation with zero potential:

\[i\hbar \frac{\partial \psi}{\partial t} = -\frac{\hbar^2}{2m} \frac{\partial^2 \psi}{\partial x^2}\]

For a Gaussian wavepacket with initial width \(\sigma_0\) and momentum \(p_0\):

\[\psi(x, 0) = \frac{1}{(\pi \sigma_0^2)^{1/4}} \exp\left(-\frac{x^2}{2\sigma_0^2} + \frac{ip_0 x}{\hbar}\right)\]

The wavepacket spreads in time due to quantum dispersion. The width evolves as:

\[\sigma(t) = \sigma_0 \sqrt{1 + \left(\frac{\hbar t}{2m\sigma_0^2}\right)^2}\]

The center of the wavepacket moves with group velocity \(v_g = p_0/m\), while the dispersion causes the uncertainty \(\Delta x\) to increase over time, satisfying the time-energy uncertainty relation.

Visualization:

The example generates an animated movie showing the probability density \(|\psi(x,t)|^2\) evolving in time, along with a plot comparing the numerical width evolution with the analytical prediction.

Quantum Tunneling

File: examples/tunneling_1d.py

Physics:

Quantum tunneling allows particles to penetrate classically forbidden regions. Consider a rectangular barrier:

\[\begin{split}V(x) = \begin{cases} V_0 & \text{if } |x - x_0| < a/2 \\ 0 & \text{otherwise} \end{cases}\end{split}\]

For a particle with energy \(E < V_0\), the wavefunction inside the barrier (\(E < V\)) decays exponentially:

\[\psi(x) \propto e^{-\kappa x}, \quad \kappa = \sqrt{2m(V_0 - E)}/\hbar\]

The transmission coefficient through a rectangular barrier of width \(a\) is:

\[T = \frac{1}{1 + \dfrac{V_0^2 \sinh^2(\kappa a)}{4E(V_0 - E)}}\]

For \(\kappa a \gg 1\), this simplifies to \(T \approx 16\dfrac{E}{V_0}\left(1 - \dfrac{E}{V_0}\right) e^{-2\kappa a}\).

The wavefunction exhibits interference between incident, reflected, and transmitted components, producing characteristic oscillations near the barrier edges.

Visualization:

The example creates an animated movie showing a Gaussian wavepacket encountering the potential barrier. The animation clearly shows the wavepacket splitting into reflected and transmitted parts, with the transmitted amplitude significantly smaller than classically expected.

Key Observable: Transmission and reflection coefficients compared with analytical predictions.

2D Examples

Double-Slit Experiment

File: examples/double_slit.py

Physics:

The double-slit experiment demonstrates wave-particle duality. A plane wave or wavepacket passes through two slits separated by distance \(d\), creating an interference pattern on a distant screen.

The interference pattern arises from the superposition:

\[\psi(x, y, t) = \psi_1(x, y, t) + \psi_2(x, y, t)\]

where \(\psi_1\) and \(\psi_2\) are contributions from each slit. The probability density \(|\psi|^2\) exhibits fringes with spacing:

\[\Delta y = \frac{\lambda L}{d}\]

where \(\lambda = 2\pi\hbar/p\) is the de Broglie wavelength, \(L\) is the distance to the screen, and \(p\) is the particle momentum.

Visualization:

An animated 2D visualization shows the wavefunction propagating through the double slit and forming interference fringes. Absorbing boundary conditions prevent reflections from domain edges.

Key Observable: Fringe spacing and visibility of interference pattern.

python examples/double_slit.py

Quantum Vortex

File: examples/quantum_vortex_2d.py

Physics:

A quantum vortex is a topological defect in a complex wavefunction where the phase winds by \(2\pi m\) around a singularity (\(m\) is the winding number). The wavefunction takes the form:

\[\psi(r, \theta) = f(r) e^{im\theta}\]

where \(f(r) \sim r^{|m|}\) near the core to ensure single-valuedness. In a harmonic trap \(V(r) = \frac{1}{2}m\omega^2 r^2\), the vortex state exhibits:

  1. Phase structure: The phase arg(\(\psi\)) winds \(m\) times around the vortex core

  2. Density depletion: \(|\psi|^2 \to 0\) at the core (\(r \to 0\))

  3. Angular momentum: The state carries angular momentum \(L_z = m\hbar\) per particle

The probability current circulates around the vortex:

\[\mathbf{j} = \frac{\hbar}{m} \operatorname{Im}(\psi^* \nabla \psi) = \frac{\hbar m}{mr^2} |\psi|^2 \hat{\boldsymbol{\theta}}\]

Visualization:

The example generates two animations:

  1. Probability density \(|\psi|^2\) showing the vortex core

  2. Phase structure revealing the \(2\pi m\) winding

Key Observable: Angular momentum expectation value \(\langle L_z \rangle\) and phase winding number.

QED Examples

Photon Field Basics

File: examples/photon_field.py

Physics:

A quantized photon field is described in Fock space using creation (\(a^\dagger\)) and annihilation (\(a\)) operators satisfying:

\[[a, a^\dagger] = 1\]

These operators act on number states \(|n\rangle\):

\[\begin{split}\begin{aligned} a^\dagger|n\rangle &= \sqrt{n+1}|n+1\rangle \\ a|n\rangle &= \sqrt{n}|n-1\rangle \end{aligned}\end{split}\]

The number operator \(\hat{n} = a^\dagger a\) counts photons:

\[\hat{n}|n\rangle = n|n\rangle\]

The Hamiltonian for a single mode is:

\[H = \hbar\omega\left(a^\dagger a + \frac{1}{2}\right)\]

Demonstration:

The example verifies:

  • Commutation relations

  • Action of operators on Fock states

  • Photon number expectation values

Key Observable: Verification of \([a, a^\dagger] = 1\) and proper normalization.

Jaynes-Cummings Dynamics

File: examples/jaynes_cummings.py

Physics:

The Jaynes-Cummings model describes a two-level atom coupled to a single photon mode. The Hamiltonian (in the rotating wave approximation) is:

\[\hat{H}_{\text{JC}} = \frac{\hbar\omega_a}{2}\hat{\sigma}_z + \hbar\omega_c \hat{a}^\dagger\hat{a} + \hbar g(\hat{a}\hat{\sigma}_+ + \hat{a}^\dagger\hat{\sigma}_-)\]

where:

  • \(\omega_a\) is the atomic transition frequency

  • \(\omega_c\) is the cavity mode frequency

  • \(g\) is the atom-field coupling strength

  • \(\hat{\sigma}_z = |e\rangle\langle e| - |g\rangle\langle g|\) is the Pauli z operator

  • \(\hat{\sigma}_+ = |e\rangle\langle g|\) raises the atom to the excited state

  • \(\hat{\sigma}_- = |g\rangle\langle e|\) lowers the atom to the ground state

For the resonant case (\(\omega_a = \omega_c\)) starting from \(|e,0\rangle\) (excited atom, vacuum field), the system undergoes Rabi oscillations:

\[P_e(t) = \cos^2(\Omega_R t/2)\]

where \(\Omega_R = 2g\) is the Rabi frequency. Energy is exchanged periodically between atom and field:

  • At \(t = 0\): atom excited, zero photons

  • At \(t = \pi/\Omega_R\): atom in ground state, one photon

  • At \(t = 2\pi/\Omega_R\): back to initial state

Visualization:

An animated plot showing the time evolution of:

  1. Excited state probability \(P_e(t)\)

  2. Average photon number \(\langle n \rangle(t)\)

Both oscillate with the Rabi frequency, demonstrating coherent energy exchange.

Key Observable: Rabi frequency \(\Omega_R\) extracted from oscillation period.

Running Examples

All examples can be run directly with uv:

# Ensure dependencies are synced
uv sync

# Navigate to examples directory
cd examples

# Run 1D examples
uv run python free_particle_1d.py
uv run python tunneling_1d.py

# Run 2D examples
uv run python double_slit.py
uv run python quantum_vortex_2d.py

# Run QED examples
uv run python photon_field.py
uv run python jaynes_cummings.py

# Run benchmarks
uv run python benchmark_gpu.py

Modifying Examples

All examples are designed to be easily modified. Key parameters to adjust:

Grid Parameters:

  • nx, ny: Number of grid points (powers of 2 recommended)

  • x_min, x_max: Spatial extent

Physical Parameters:

  • mass: Particle mass (default: 1.0 = electron mass)

  • hbar: Reduced Planck constant (default: 1.0)

  • Potential parameters: V0, omega, etc.

Numerical Parameters:

  • dt: Time step (reduce for better accuracy)

  • num_steps: Number of evolution steps

Visualization:

  • save_every: Frame saving frequency

  • dpi: Image resolution

  • cmap: Colormap (try ‘viridis’, ‘plasma’, ‘hot’, etc.)

Next Steps

  • Adapt examples for your research problem

  • Combine multiple techniques (e.g., tunneling + time-dependent potential)

  • Explore parameter space systematically

  • Validate against analytical solutions when available

For more details, see: