Class: Quantum Mechanics

Quantum mechanics is one of the most classic examples of field theories in physics. The Schrödinger equation is a partial differential equation that describes how the quantum state of a physical system changes over time.

In this class, we simulate quantum mechanics through the evolution of the Schrödinger equation.

file: comfit/models/quantum_mechanics.py 
class: QuantumMechanics

See the ComFiT Library Reference below for a complete list of class methods and their usage.

ComFiT Library Reference

Example

The following example demonstrates how to set up a 1D quantum system with a Gaussian wave packet and a potential barrier. It runs smoothly with comfit 1.8.4.

import comfit as cf
import matplotlib.pyplot as plt
import numpy as np

# Set up a 1D quantum system
qm = cf.QuantumMechanics(1, xlim=[-50,50], xRes=1001, dt=0.1)

# Initialize a Gaussian wavepacket at x=5 with velocity=1
qm.conf_initial_condition_Gaussian(position=5, width=1, initial_velocity=1)

# Add a potential barrier (optional)
qm.V_ext = 0.5 * (qm.x > 10) * (qm.x < 12)  # Barrier from x=10 to 12

height = np.max(abs(qm.psi))  # Get the maximum value of the wavefunction

# Optional: Animate it
for n in range(61):
    qm.evolve_schrodinger(5)
    fig, ax = qm.plot_complex_field(qm.psi)
    qm.plot_field(qm.V_ext, fig=fig, ax=ax, ylim=[0,height], xlim=[0,20])
    qm.plot_save(fig, n)
cf.tool_make_animation_gif(n)  # Creates animation.gif

The Schrödinger equation

Evolving according to the Schrödinger equation with electron mass

\[ \mathfrak i \hbar \partial_t \psi = \left [-\frac{\hbar^2}{2m_e} \nabla^2 + V \right ] \psi. \]

Dividing by the Hartree energy \(E_h = \frac{\hbar^2}{m_e a_0^2}\)

\[ \mathfrak i \frac{\hbar}{E_h} \partial_t \psi = \frac{1}{E_h}\left [-\frac{\hbar^2}{2m_e} \nabla^2 + V \right ] \psi. \]

When expressing time in units of \(\frac{\hbar}{E_h}\), potential energy in units of \(E_h\) and length squared in units of

\[ \frac{\hbar^2}{E_h m_e} = \frac{\hbar^2}{\frac{\hbar^2}{m_e a_0^2} m_e} = a_0^2, \]

we get the Schrödinger equation in its dimensionless form

\[ \partial_t \psi = \mathfrak i\left [\frac{1}{2} \nabla^2 - V \right ] \psi. \]

So

\[ \omega = \mathfrak i\frac{1}{2} \nabla^2 \Rightarrow \omega_{\mathfrak f} = -\mathfrak i \frac{1}{2} \mathbf k^2 \]

Atomic unit of	Value
Length	0.529 Å (Angstrom)
Energy	27.2 eV (electron volts)
Time	24.2 aS (atto seconds)

The Born rule

The Born rule states that the probability \(p\) of measuring a particle in the interval \([a,b]\) is given by

\[ p = \int_a^b dx |\psi(x)|^2. \]

A wave packet

A Gaussian wave function is often called a wave packet and can visualize the position and motion of a particle in a quantum mechanical system. An initial wave function with a widt of \(\sigma\) is given by

\[ \psi(\mathbf r) = \sqrt{( 2\pi \sigma )^{-d/2} \exp\left ({-\frac{(\mathbf r - \mathbf r_0)^2} {(2\sigma^2)}}\right )} , \]

so that \(|\psi|^2\) is a Gaussian distribution. An initial velocity \(\mathbf v_0\) can be given to the wave packet by multiplying with a complex phase \(e^{\mathfrak i \mathbf v_0 \cdot \mathbf r}\). Such an initial condition can be configured by the function qm.conf_initial_condition_Gaussian.