The harmonic oscillator wavefunctions


The harmonic oscillator is often used as an approximate model for the behaviour of some quantum systems, for example the vibrations of a diatomic molecule. The Schrödinger equation for a particle of mass $m$ moving in one dimension in a potential $V(x) = \frac{1}{2}kx^2$ is $$ -\frac{\hbar^2}{2m}\frac{\mathrm{d}^2\psi}{\mathrm{d}x^2} + \frac{1}{2}kx^2\psi = E\psi. $$ With the change of variable, $q = (mk/\hbar^2)^{1/4}x$, this equation becomes $$ -\frac{1}{2}\frac{\mathrm{d}^2\psi}{\mathrm{d}q^2} + \frac{1}{2}q^2\psi = \frac{E}{\hbar\omega}\psi, $$ where $\omega = \sqrt{k/m}$. This differential equation has an exact solution in terms of a quantum number $v=0,1,2,\cdots$: $$ \psi(q) = N_vH_v(q)\exp(-q^2/2), $$ where $N_v = (\sqrt{\pi}2^vv! )^{-1/2}$ is a normalization constant and $H_v(q)$ is the Hermite polynomial of order $v$, defined by: $$ H_v(q) = (-1)^ve^{q^2}\frac{\mathrm{d}^v}{\mathrm{d}q^v}\left(e^{-q^2}\right). $$ The Hermite polynomials obey a useful recursion formula: $$ H_{n+1}(q) = 2qH_n(q) - 2nH_{n-1}(q), $$ so given the first two: $H_0 = 1$ and $H_1 = 2q$, we can calculate all the others.

The following code plots the harmonic oscillator wavefunctions (PLOT_PROB = False) or probability densities (PLOT_PROB = True) for vibrational levels up to VMAX on the vibrational energy levels depicted within the potential, $V = q^2/2$.

Harmonic oscillator wavefunctions Harmonic oscillator probability distributions

import numpy as np
from matplotlib import rc
from scipy.misc import factorial
import matplotlib.pyplot as plt
rc('font', **{'family': 'serif', 'serif': ['Computer Modern'], 'size': 14})
rc('text', usetex=True)

# PLOT_PROB=False plots the wavefunction, psi; PLOT_PROB=True plots |psi|^2

# Maximum vibrational quantum number to calculate wavefunction for
VMAX = 6

# Some appearance settings
# Pad the q-axis on each side of the maximum turning points by this fraction
# Scale the wavefunctions by this much so they don't overlap
# Colours of the positive and negative parts of the wavefunction
COLOUR1 = (0.6196, 0.0039, 0.2588, 1.0)
COLOUR2 = (0.3686, 0.3098, 0.6353, 1.0)

# Normalization constant and energy for vibrational state v
N = lambda v: 1./np.sqrt(np.sqrt(np.pi)*2**v*factorial(v))
get_E = lambda v: v + 0.5

def make_Hr():
    """Return a list of np.poly1d objects representing Hermite polynomials."""

    # Define the Hermite polynomials up to order VMAX by recursion:
    # H_[v] = 2qH_[v-1] - 2(v-1)H_[v-2]
    Hr = [None] * (VMAX + 1)
    Hr[0] = np.poly1d([1.,])
    Hr[1] = np.poly1d([2., 0.])
    for v in range(2, VMAX+1):
        Hr[v] = Hr[1]*Hr[v-1] - 2*(v-1)*Hr[v-2]
    return Hr

Hr = make_Hr()

def get_psi(v, q):
    """Return the harmonic oscillator wavefunction for level v on grid q."""
    return N(v)*Hr[v](q)*np.exp(-q*q/2.)

def get_turning_points(v):
    """Return the classical turning points for state v."""
    qmax = np.sqrt(2. * get_E(v + 0.5))
    return -qmax, qmax

def get_potential(q):
    """Return potential energy on scaled oscillator displacement grid q."""
    return q**2 / 2

fig, ax = plt.subplots()
qmin, qmax = get_turning_points(VMAX)
xmin, xmax = QPAD_FRAC * qmin, QPAD_FRAC * qmax
q = np.linspace(qmin, qmax, 500)
V = get_potential(q)

def plot_func(ax, f, scaling=1, yoffset=0):
    """Plot f*scaling with offset yoffset.

    The curve above the offset is filled with COLOUR1; the curve below is
    filled with COLOUR2.

    ax.plot(q, f*scaling + yoffset, color=COLOUR1)
    ax.fill_between(q, f*scaling + yoffset, yoffset, f > 0.,
                    color=COLOUR1, alpha=0.5)
    ax.fill_between(q, f*scaling + yoffset, yoffset, f < 0.,
                    color=COLOUR2, alpha=0.5)

# Plot the potential, V(q).
ax.plot(q, V, color='k', linewidth=1.5)

# Plot each of the wavefunctions (or probability distributions) up to VMAX.
for v in range(VMAX+1):
    psi_v = get_psi(v, q)
    E_v = get_E(v)
    if PLOT_PROB:
        plot_func(ax, psi_v**2, scaling=SCALING*1.5, yoffset=E_v)
        plot_func(ax, psi_v, scaling=SCALING, yoffset=E_v)
    # The energy, E = (v+0.5)
    ax.text(s=r'$\frac{{{}}}{{2}}\hbar\omega$'.format(2*v+1), x=qmax+0.2,
            y=E_v, va='center')
    # Label the vibrational levels.
    ax.text(s=r'$v={}$'.format(v), x=qmin-0.2, y=E_v, va='center', ha='right')

# The top of the plot, plus a bit.
ymax = E_v+0.5

    ylabel = r'$|\psi(q)|^2$'
    ylabel = r'$\psi(q)$'
ax.text(s=ylabel, x=0, y=ymax, va='bottom', ha='center')

ax.set_xlim(xmin, xmax)
ax.set_ylim(0, ymax)

plt.savefig('sho-psi{}-{}.png'.format(PLOT_PROB+1, VMAX))
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