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Solution of the Schrödinger equation using Ritz method

In this post, we describe the solution of the Schrödinger equation using Ritz method and Hermite functions basis. This basis seems to be a good choice for the 1D Schrödinger equation since its an orthogonal basis over \((-\infty, \infty)\).

Transforming the equation to an algebraic one

We want so solve the time-independent Schrödinger equation

\begin{equation*} \left[-\frac{1}{2}\nabla^2 + V(x)\right] \psi = E\psi\, , \end{equation*}

where we are using natural units since they are a good choice for numeric calculations.

Solving equation is equivalent to solve the following variational equation

\begin{equation*} \delta \Pi[\psi] = 0\, , \end{equation*}

with

\begin{equation*} \Pi[\psi] = \frac{1}{2} \langle \nabla \psi, \nabla\psi\rangle + \langle \psi, V(x) \psi\rangle - E\langle \psi, \psi\rangle \, , \end{equation*}

being \(\psi\) the wave function, \(V(x)\) the potential and \(E\) the energy. This variational formulation is equivalent to the time-independent Schrödinger equation, and \(E\) works as a Lagrange multiplier to enforce that the probability over the whole domain is 1.

We can expand the wave function in an orthonormal basis, namely

\begin{equation*} \psi = \sum_{n=0}^N c_n u_n(x)\, , \end{equation*}

where \(u_n(x) \equiv \mu_n H_n(x) e^{-x^2/2}\) is a normalized Hermite function, \(\mu_n\) is the inverse of magnitude of the \(n\)th Hermite polynomial

\begin{equation*} \mu_n = \frac{1}{\sqrt{\pi^{1/2} n! 2^n}}\, , \end{equation*}

and \(c_n\) are the coefficients of the expansion. This representation is exact in the limit \(N \rightarrow \infty\).

If we replace the expansion in the functional, we obtain

\begin{equation*} \Pi_N = \sum_{m=0}^N\sum_{n=0}^N c_m c_n\left[ \frac{1}{2} \langle \nabla u_m, \nabla u_n\rangle + \langle u_m, V(x) u_n\rangle - E^N \delta_{mn}\right]\, . \end{equation*}

The integrand of the integral involving the two derivatives reads

\begin{align*} u_m' u_n' =& \left[2m \frac{\mu_{m-1}}{\mu_m}u_{m-1} - x u_m\right] \left[2n \frac{\mu_{n-1}}{\mu_n}u_{n-1} - x u_n\right]\\ =& 4mn\frac{\mu_{m-1} \mu_{n-1}}{\mu_m \mu_n} u_{n-1} u_{m-1} - 2m\frac{\mu_{m-1}}{\mu_{m}}x u_{m-1} u_n\\ &- 2n\frac{\mu_{n-1}}{\mu_{n}}x u_{n-1} u_m + x^2 u_m u_n \end{align*}

Thus, the kinetic energy term reads

\begin{align*} \langle \nabla u_m, \nabla u_n \rangle =& 4mn\frac{\mu_{m-1} \mu_{n-1}}{\mu_m \mu_n} \langle u_{n-1}, u_{m-1}\rangle - 2m\frac{\mu_{m-1}}{\mu_{m}} \langle u_{m-1}, x u_n\rangle\\ &- 2n\frac{\mu_{n-1}}{\mu_{n}} \langle u_{m}, x u_{n - 1}\rangle + \langle u_m, x^2 u_n\rangle\\ =& 4mn \frac{\mu_{m-1}^2}{\mu_m^2}\delta_{mn} - 2m \frac{\mu_{m-1}}{\mu_m} \alpha_{m-1, n} - 2n \frac{\mu_{n-1}}{\mu_n} \alpha_{m, n-1} + \beta_{mn} \, , \end{align*}

with

\begin{equation*} \alpha_{m,n} \equiv \langle u_{m}, x u_n\rangle = \begin{cases} \sqrt{\frac{n + 1}{2}} & m=n +1\\ \sqrt{\frac{n}{2}} & m=n - 1\\ 0 & \text{otherwise}\end{cases}\, , \end{equation*}

and

\begin{equation*} \beta_{m,n} \equiv \langle u_{m}, x^2 u_n\rangle = \begin{cases} \frac{\sqrt{n(n-1)}}{2} & m = n - 2\\ \frac{2n + 1}{2} & m = n \\ \frac{\sqrt{(n+1)(n+1)}}{2} & m = n + 2 \\ 0 & \text{otherwise}\end{cases}\, . \end{equation*}

The functional is rewritten as

\begin{align*} \Pi_N =& \sum_{m=0}^N \sum_{n=0}^N c_m c_n \left[2mn \frac{\mu^2_{m-1}}{\mu^2_m}\delta_{mn} - m\frac{\mu_{m-1}}{\mu_m}\alpha_{m - 1, n} - n\frac{\mu_{n-1}}{\mu_n}\delta_{m, n-1} \right.\nonumber \\ &\left. - \frac{1}{2}\beta_{mn} + \langle u_m, V(x) u_n\rangle - E^N \delta_{mn}\right] \, . \end{align*}

Taking the variation

\begin{equation*} \delta \Pi_N = 0\, , \end{equation*}

that in this case is equivalent to

\begin{equation*} \frac{\partial \Pi_N}{\partial c_i} = 0\quad \forall i=0, 1, \cdots N\, , \end{equation*}

yields to

\begin{equation*} [H]\lbrace c\rbrace = E^N\lbrace c\rbrace \, , \end{equation*}

where the components of the matrix \([H]\) are given by

\begin{equation*} H_{mn} = 2mn \frac{\mu^2_{m-1}}{\mu^2_m}\delta_{mn} - m\frac{\mu_{m-1}}{\mu_m}\alpha_{m - 1, n} - n\frac{\mu_{n-1}}{\mu_n}\delta_{m, n-1} - \frac{1}{2}\beta_{mn} + \langle u_m, V(x) u_n\rangle\, . \end{equation*}

The last integral can be computed using Gauss-Hermite quadrature. And we will need more Gauss points if we want to integrate higher-order polynomials. This method would work fine for functions that can be approximated by polynomials.

Examples

A Python implementation of this method is presented in this repo.

For all the examples we use the following imports

from __future__ import division, print_function
import numpy as np
from hermite_QM import *

Quantum harmonic oscilator

In this case the (normalized) potential is given by

\begin{equation*} V(x) = \frac{1}{2} x^2 \end{equation*}

and the exact normalized eigenvalues are given by

\begin{equation*} E_n = n + \frac{1}{2} \end{equation*}

The following snippet computes the first 10 eigenvalues and plot the corresponding eigenstates

potential = lambda x: 0.5*x**2
vals, vecs = eigenstates(potential, nterms=11, ngpts=12)
print(np.round(vals[:10], 6))
fig, ax = plt.subplots(1, 1)
plot_eigenstates(vals, vecs, potential);

with results

[ 0.5  1.5  2.5  3.5  4.5  5.5  6.5  7.5  8.5  9.5]
/images/hermite_ritz_harmonic.svg

Absolute value potential

potential = lambda x: np.abs(x)
vals, vecs = eigenstates(potential)
print(np.round(vals[:10], 6))
fig, ax = plt.subplots(1, 1)
plot_eigenstates(vals, vecs, potential, lims=(-8, 8));

with results

[ 0.811203  1.855725  2.57894   3.244576  3.826353  4.38189   4.895365
  5.396614  5.911591  6.421015]
/images/hermite_ritz_abs.svg

Cubic potential

potential = lambda x: np.abs(x/3)**3
vals, vecs = eigenstates(potential)
print(np.round(vals[:10], 6))
fig, ax = plt.subplots(1, 1)
plot_eigenstates(vals, vecs, potential, lims=(-6, 6));

with results

[ 0.180588  0.609153  1.124594  1.681002  2.272087  2.889805  3.530901
  4.191962  4.871133  5.566413]
/images/hermite_ritz_cubic.svg

Harmonic with quartic perturbation

potential = lambda x: 0.5*x**2 + 0.1*x**4
vals, vecs = eigenstates(potential, nterms=20, ngpts=22)
print(np.round(vals[:10], 6))
fig, ax = plt.subplots(1, 1)
plot_eigenstates(vals, vecs, potential, lims=(-5, 5))

with results

[  0.559146   1.769503   3.138624   4.628884   6.220303   7.899789
   9.658703  11.489094  13.38638   15.361055]
/images/hermite_ritz_pert_harm.svg

A Jupyter Notebook with the examples can be found here.

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