Quantum Harmonic Oscillator

• The quantum harmonic oscillator as analog of the classical one is often used as an approximate model for the behavior of some quantum systems.
• It is one of the few quantum-mechanical systems for which an exact, analytical solution is known.
• The Hamiltonian for a particle of mass $m$ moving in one dimension in a potential $V(x)=1/2kx^2$ is

  $$\hat{H}=\frac{\hat{p}^2}{2m}+\frac{1}{2}k\hat{x}^2=\frac{\hat{p}^2}{2m}+\frac{1}{2}m\omega^2\hat{x}^2$$

  where $\hat{x}$ is the position operator, and $\hat{p}$ is the momentum operator (given by $\hat{p}=-i\hbar\partial / \partial x$ in the coordinate basis).
• The first term in the Hamiltonian represents the kinetic energy of the particle, and the second term represents its potential energy, as in Hooke's law.

• Then, Schrödinger equation becomes

  $$-\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2}+\frac{1}{2}kx^2\psi=E\psi$$

• with the change of variable, $q=(mk/\hbar^2)^{1/4}x$, this equation becomes

  $$-\frac{1}{2}\frac{d^2\psi}{dq^2}+\frac{1}{2}q^2\psi=\frac{E}{\hbar \omega}\psi$$

  where $\omega=\sqrt{k/m}$ is the angular frequency of the oscillator.
• This differential equation has an exact solution in terms of a quantum number $\nu=0,1,2,\ldots$:

  $$\boxed{\psi(q)=N_\nu H_\nu(q)e^{-q^2/2}}$$

  where $N_\nu=(\sqrt{\pi}2^\nu \nu!)^{-1/2}$ is a normalization constant.
• The function $H_\nu(q)$ is the physicists' Hermite polynomials of order $\nu$, defined by:

  $$H_\nu(q)=(-1)^\nu e^{q^2}\frac{d^\nu}{dq^\nu}\left( e^{-q^2} \right)$$

• The corresponding energy levels are

  $$E_\nu=\hbar \omega \left( \nu+\frac{1}{2}\right) =(2\nu+1)\frac{\hbar}{2} \omega$$

• Recursion formula:

  $$\boxed{H_{\nu+1}(q)=2qH_\nu(q)-2\nu H_{\nu-1}(q)}$$

  with the first two: $H_0=1$ and $H_1=2q$.

Example py-file: The program to find the harmonic oscillator wavefunctions/probability densities for up to 4 vibrational energy levels with the harmonic potential, $V=q^2/2$. QHO.py

Figure 5.20: Wavefunction representations for the first 5 bound eigenstates, $\nu = 0 - 4$.

Figure 5.21: Corresponding probability densities.