Class: Quantum Mechanics
Quantum mechanics describes the behavior of nature at the scale of atoms and subatomic particles. The Schrödinger equation is a fundamental partial differential equation that governs how the quantum state of a physical system evolves over time.
This class simulates quantum mechanics by evolving the Schrödinger equation.
See the ComFiT Library Reference below for a complete list of class methods and their usage.
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 for plotting limits
# 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
The time-dependent Schrödinger equation for a particle of mass \(m_e\) in a potential \(V\) is given by:
where \(\psi\) is the wave function, \(\hbar\) is the reduced Planck constant.
To simplify the equation for numerical simulation, we can introduce dimensionless units. Dividing by the Hartree energy \(E_h = \frac{\hbar^2}{m_e a_0^2}\) (where \(a_0\) is the Bohr radius) yields:
Expressing time in units of \(\tau = \frac{\hbar}{E_h}\), potential energy \(V\) in units of \(E_h\), and length squared in units of \(a_0^2\), we obtain the dimensionless Schrödinger equation:
This is the form implemented in the QuantumMechanics class. The linear operator \(\omega\) and its Fourier space representation \(\omega_{\mathfrak f}\) are:
The non-linear part (in this context, the potential term which depends on \(\psi\) only through multiplication) is \(N = -\mathfrak i V \psi\).
| Atomic unit of | Value |
|---|---|
| Length (\(a_0\)) | 0.529 Å (Angstrom) |
| Energy (\(E_h\)) | 27.2 eV (electron volts) |
| Time (\(\tau\)) | 24.2 as (attoseconds) |
The Born rule
The Born rule states that the probability density of finding the particle at position \(\mathbf{r}\) at time \(t\) is given by \(|\psi(\mathbf{r}, t)|^2\). Consequently, the probability \(P\) of finding the particle within a volume \(\mathcal{V}\) is:
The total probability of finding the particle anywhere in space must be 1, leading to the normalization condition:
The Momentum representation
In quantum mechanics, the Fourier transform provides the connection between the position representation (\(\psi(\mathbf{r})\)) and the momentum representation (\(\phi(\mathbf{k})\)) of a quantum state. This relationship highlights the wave-particle duality inherent in quantum theory.
The momentum-space wave function \(\phi(\mathbf{k})\), where \(|\phi(\mathbf{k})|^2\) represents the probability density of finding the particle with wave vector \(\mathbf{k}\) (momentum \(\mathbf{p} = \hbar \mathbf{k}\)), is related to the position-space wave function \(\psi(\mathbf{r})\) via the Fourier transform:
where \(\psi_{\mathfrak f} (\mathbf{k})\) is the Fourier transform of \(\psi(\mathbf{r})\) as defined in the Base System documentation, and \(d\) is the spatial dimension. The factor \(\sqrt{(2\pi)^d}\) ensures that if \(\psi(\mathbf{r})\) is normalized in position space, then \(\phi(\mathbf{k})\) is normalized in momentum space:
Physical Significance of the Fourier Transform in QM
- Complementarity: The Fourier transform relationship mathematically embodies Heisenberg's uncertainty principle. A wave function highly localized in position space (narrow \(\psi(\mathbf{r})\)) corresponds to a widely spread momentum distribution (broad \(\phi(\mathbf{k})\)), and vice versa.
- Operator Correspondence: In the position representation, the momentum operator is \(\hat{\mathbf{p}} = -\mathfrak i\hbar\nabla\). The Fourier transform maps this differential operator in position space to a simple multiplicative operator (\(\hbar \mathbf{k}\)) in momentum space.
- Energy Eigenstates: For a free particle (\(V=0\)), the energy eigenstates are plane waves, \(\psi(\mathbf r) \propto e^{\mathfrak i \mathbf k \cdot \mathbf r}\), which are also momentum eigenstates.
A wave packet
A Gaussian wave function, often referred to as a wave packet, provides a localized representation of a particle in quantum mechanics, balancing position and momentum uncertainty. A normalized Gaussian wave packet centered at \(\mathbf{r}_0\) with width \(\sigma\) is given by:
such that the probability density \(|\psi(\mathbf{r})|^2 = (2\pi \sigma^2)^{-d/2} \exp\left ({-\frac{(\mathbf r - \mathbf r_0)^2} {2\sigma^2}}\right )\) is a Gaussian distribution with standard deviation \(\sigma\).
An initial average velocity \(\mathbf v_0\) (corresponding to an average momentum \(\hbar \mathbf{k}_0 = m_e \mathbf{v}_0\)) can be imparted to the wave packet by multiplying it with a complex phase factor \(e^{\mathfrak i \mathbf{k}_0 \cdot \mathbf r} = e^{\mathfrak i (m_e/\hbar)\mathbf v_0 \cdot \mathbf r}\). In dimensionless units where \(m_e=\hbar=1\), this simplifies to \(e^{\mathfrak i \mathbf v_0 \cdot \mathbf r}\).
Such an initial condition can be configured using the function qm.conf_initial_condition_Gaussian.