Appendix A — The quantum harmonic oscillator

As an example, here we consider the quantum harmonic oscillator. Classically, the harmonic oscillator is defined as a system subject to the force \(\mathbf{F} = - k \mathbf{r}\), where \(k\) is the elastic constant. In other words, the force is proportional to the displacement from a stable point (in this case the origin).

Following the relation \(\mathbf{F} = - \nabla V(\mathbf{r})\), we can say that the corresponding potential is \(V(\mathbf{r}) = k/2 \ \mathbf{r}^2\). The solution of the Schrodinger equation

\[ i \hbar \frac{\partial}{\partial t} \Psi(\mathbf{r}, t) = -\frac{\hbar^2}{2 m} \nabla^2 \Psi(\mathbf{r}, t) + V(\mathbf{r}) \Psi(\mathbf{r}, t) \, , \]

where \(\hbar\) is the reduced Planck constant, \(m\) is the mass of the particle, and \(\nabla^2\) is the Laplacian operator, gives us the eigenstates of the system. Considering only the one-dimensional case, we obtain the following eigenstates for the quantum harmonic oscillator:

\[ \psi_n(x) = \frac{1}{\sqrt{2^n n!}} \left(\frac{m\omega}{\pi \hbar}\right)^{1/4} e^{-\frac{m\omega x^2}{2\hbar}} H_n\left(\sqrt{\frac{m\omega}{\hbar}} x\right) \, , \tag{A.1}\]

where \(\omega = \sqrt{k / m}\) is the resonance frequency of the oscillator and \(H_n\) is the \(n\)-th Hermite polynomial.

A useful way to describe the quantum harmonic oscillator is by using the ladder operators

\[\begin{eqnarray} \hat{a} &=& \sqrt{\frac{m \omega}{2 \hbar}} \left( \hat{x} + i \frac{1}{m \omega} \hat{p} \right) \\ \hat{a}^\dagger &=& \sqrt{\frac{m \omega}{2 \hbar}} \left( \hat{x} - i \frac{1}{m \omega} \hat{p} \right) \, , \end{eqnarray}\]

note that the position \(\hat{x}\) and conjugate momentum \(\hat{p}\) are operators too. If we now write the eigenstates in Equation 6.1 in the bra-ket notation (\(\psi_n \to \ket{n}\)), the ladder operators allow us to move from one eigenstate to the next or previous one:

\[\begin{eqnarray} \label{eq1: destroy operator action} \hat{a} \ket{n} &=& \sqrt{n} \ket{n-1} \\ \label{eq1: creation operator action} \hat{a}^\dagger \ket{n} &=& \sqrt{n+1} \ket{n+1} \, , \end{eqnarray}\]

and it is straightforward to recognize the creation (\(\hat{a}^\dagger\)) and annihilation (\(\hat{a}\)) operators. In this framework the system Hamiltonian of the quantum harmonic oscillator becomes \[ \hat{H} = \hbar \omega \left( \hat{a}^\dagger \hat{a} + \frac{1}{2} \right) \, . \]

import numpy as np
import matplotlib.pyplot as plt
from scipy.special import eval_hermite, factorial

# Physical parameters
m = 1.0
k = 1.0
w = np.sqrt(k/m)
alpha = -np.sqrt(2)   # coherent‐state parameter

# Grid
bounds = 6.0
x = np.linspace(-bounds, bounds, 1000)

# n-th eigenfunction of the HO (hbar=1)
def psi(n, x):
    Hn = eval_hermite(n, np.sqrt(m*w) * x)
    norm = (m*w/np.pi)**0.25 / np.sqrt(2**n * factorial(n))
    return norm * Hn * np.exp(-m*w*x**2/2)

# Build the first six eigenstates and energies
psi_n = [psi(n, x) for n in range(6)]
E_n   = [(n + 0.5) * w for n in range(6)]

# Coherent-state wavefunction (real alpha ⇒ no overall phase)
psi_coh = (m*w/np.pi)**0.25 * np.exp(- (x - np.sqrt(2)*alpha)**2 / 2)

# Plotting
fig, ax = plt.subplots()

# 1) potential
ax.plot(x, 0.5*k*x**2, 'k--', lw=2, label=r'$V(x)=\tfrac12 k x^2$')

# 2) coherent state
ax.fill_between(x, psi_coh, color='gray', alpha=0.5)
ax.plot(x, psi_coh, color='gray', lw=2, label='Coherent state')

# 3) eigenstates offset by E_n
lines = []
for n in range(6):
    y = psi_n[n] + E_n[n]
    line, = ax.plot(x, y, lw=2, label=fr'$|{n}\rangle$')
    lines.append(line)

# Cosmetics
ax.set_ylim(0, 7)
ax.set_xlabel(r'$x$')

# State labels on the right
for n, line in enumerate(lines):
    ax.text(4.5, E_n[n] + 0.2, rf'$|{n}\rangle$', color=line.get_color())

ax.text(-3.5, 6.5, r"$V(x)$")
ax.text(-3, 0.9, r"$\ket{\alpha}$", color="grey")
ax.annotate("", xy=(-5.5,3.5), xytext=(-5.5,2.5), arrowprops=dict(arrowstyle="<->"))
ax.text(-5.4, 3, r"$\hbar \omega$", ha="left", va="center")

plt.show()

Figure A.1: First eigenstates of one-dimensional the quantum harmonic oscillator, each of them vertically shifted by the corresponding eigenvalue. The grey-filled curve corresponds to a coherent state with \(\alpha=-\sqrt{2}\). The used parameter are \(m=1\), \(\omega=1\), and \(\hbar=1\).

It is worth introducing the coherent state \(\ket{\alpha}\) of the harmonic oscillator, defined as the eigenstate of the destroy operator, with eigenvalue \(\alpha\), in other words, \(\hat{a} \ket{\alpha} = \alpha \ket{\alpha}\). It can be expressed analytically in terms of the eigenstates of the quantum harmonic oscillator \[ \begin{aligned} \ket{\alpha} = e^{-\frac{1}{2} \abs{\alpha}^2} \sum_{n=0}^{\infty} \frac{\alpha^n}{\sqrt{n!}} \ket{n} \, , \end{aligned} \]

and it can be seen as the most classic-like state since it has the minimum uncertainty \(\Delta x \Delta p = \hbar / 2\).

Figure A.1 shows the first eigenstates of the quantum harmonic oscillator, each of them vertically shifted by the respective energy, while the grey-filled curve is a coherent state with \(\alpha = - \sqrt{2}\). The black dashed curve is the potential, choosing \(k = 1\), \(m = 1\), and \(\hbar = 1\). It is worth noting that also the groundstate \(\ket{0}\) has a nonzero energy (\(E_0 = \hbar \omega / 2\)).