wyssard / Particle_in_a_Box

A small Python library to compute and visualize the time evolution of the quantum mechanical state of a particle in a one-dimensional box.

Pib Lib

A small Python library to compute and visualize the time evolution of the quantum mechanical state of a particle in a one-dimensional box.

Due to an issue concerning the integration of LaTeX in GitHub Markdown, some expressions are not rendered properly. For the time being, it is therefore suggested to also consider the raw data of any Markdown file in this repository.

Theoretical Background

Note: to understand this section, one should be familiar with basic quantum mechanics and the concept of self-adjointness (not to be confused with Hermiticity). To get a complete understanding of the library's capabilities, a perusal of the literature referred to below is likely to be necessary.

When discussing the behavior of a quantum mechanical particle in a one-dimensional box, i.e., one a finite interval $[-L/2, L/2]$, one usually imposes Dirichlet boundary conditions in order to quantize the energy eigenstates. That is, for the eigenfunctions $\psi_{l}$ of the Hamiltonian $H=-\frac{1}{2m}\partial_x^2$, we require $\psi_l(\pm\frac{L}{2})=0$, having in mind the conservation of the probability current. Besides that, these boundary conditions also provide the self-adjointness of the Hamiltonian which is necessary for the energy eigenstates to be orthogonal and the eigenvalues to be real (and thus for the Hamiltonian to describe a physical concept). It turns out, however, that much more general boundary conditions can be allowed that still provide the conservation of the probability current and the self-adjointness of the resulting Hamiltonian. These boundary conditions, parametrized by two real parameters $\gamma_+$ and $\gamma_-$, are so-called Robin Boundary Conditions which read: $$ \gamma_{\pm}\psi\left(\pm\tfrac{L}{2}\right) \pm\partial_x\psi\left(\pm\tfrac{L}{2}\right) = 0. $$ The aim of this library precisely consists in simulating the time evolution of states within the domain of a self-adjoint Hamiltonian described by the above boundary conditions. Notice that, e.g., Dirichlet boundary conditions correspond to the case where $\gamma_{\pm} \to \infty$. The boundary conditions supported by the library are given by:

• $\gamma_+ = \gamma_-$, referred to as symmetric boundary conditions (mostly based on analytic results)
• $\gamma_+ = -\gamma_-$, referred to as anti-symmetric boundary conditions (based on numeric results)
• Analytic implementations of the special cases:
  • $\gamma_+ = \gamma_- \to \infty$ (Dirichlet boundary conditions)
  • $\gamma_+ = \gamma_- = 0$ (Neumann boundary conditions)
  • $\gamma_+ = 0$ and $\gamma_- \to \infty$ (Mixed Neumann-Dirichlet boundaries)

The construction of such states with this library is illustrated in the examples section.

The time evolution of these states can be visualized in position and (new) momentum space. "New" thereby refers to a newly constructed self-adjoint operator $\hat{p}_R$, that describes the momentum of a particle on an interval. The necessity for such an operator is extensively discussed in the latter two recommended readings.

Further Reading

To get a more in-depth understanding of the concept of self-adjointness and the necessity of quantum mechanical operators to be self-adjoint to describe physical observables, consider the following two papers:

• Self-adjoint extensions of operators and the teaching of quantum mechanics
• From a Particle in a Box to the Uncertainty Relation in a Quantum Dot and to Reflecting Walls for Relativistic Fermions

This library was not primarily written to analyze the self-adjointness of the Hamiltonian but of the momentum operator as part of my bachelor thesis. The issues that arise in this context are discussed in:

• A New Concept for the Momentum of a Quantum Mechanical Particle in a Box
• [Bachelor Thesis] Self-Adjoint Energy and Momentum Operators with Local Boundary Conditions for a Particle in a Box

Installation

To install the library, download the repo as a zip file and run a pip install in your python environment (supported from version 0.2.0 onwards, earlier versions only persist for documentation purposes):

pip install <download-directory>/Particle_in_a_Box-<version>.zip

the library can then be accessed via:

>>> import pib_lib

Examples

To construct particle-in-a-box states, the main module needs to be imported via

>>> from pib_lib import particle_in_a_boy as pib

and states are then created by:

>>> state = pib.Particle_in_Box_State(boundary_condition, L, m, energy_states, amplitudes, gamma)

thereby, boundary_condition is a string specifying the boundary conditions according to one of the supported cases specified above. L corresponds to the length of the interval and m to the mass of the particle (both in natural units). energy_states is a list of integers that correspond to the energy quantum numbers of the states we want to superimpose to get the desired state with the respective amplitudes given by the list entries of amplitudes. The parameter gamma is only required when boundary_conditions is set to symmetric or anti-symmetric. Thereby gamma refers to $\gamma_+ = \gamma_-$ in the symmetric and to $\gamma_+ = -\gamma_-$ in the anti-symmetric case.

A full example, also including a visualization of the state via the update_plot.py module using matplotlib could look like this:

from pib_lib import particle_in_a_box as pib
from pib_lib import update_plot as up
import numpy as np
from matplotlib import pyplot as plt

plt.style.use('Solarize_Light2')

L = np.pi
m = 1
states = [1,2,3]
amps = [1,1,1]
case = "dirichlet_neumann"

state = pib.Particle_in_Box_State(case, L, m, states, amps)

fig = plt.figure(figsize=(6,5), tight_layout=True)
gs = fig.add_gridspec(nrows=2, ncols=1)
pos_ax = fig.add_subplot(gs[0,0])
momentum_ax = fig.add_subplot(gs[1,0])

position_plot = up.position_space_plot(state, fig, pos_ax)
momentum_plot = up.momentum_space_plot(state, fig, momentum_ax)
new_momentum_plot = up.new_momentum_space_plot(state, fig, momentum_ax)
new_momentum_plot["abs_square"].plot_config(color=up.light_blue)
plots = up.Update_Plot_Collection(fig, position_plot, new_momentum_plot, momentum_plot)

plots.set_n_bound(10)
plots.set_t(1)

plots.plot()

The top panel shows the absolute square of the position space wavefunction with the dotted line indicating the position expectation value. The bottom panel illustrates the projection onto momentum space (continuous line), i.e. the continuous momentum distribution, whereas the bar plot shows the discrete momentum distribution according to the new momentum concept.

A more thorough explanation of the library can soon be found in the demo notebook.