Harmonic Oscillator

A physical system that experiences a restoring force proportional to its displacement from an equilibrium position, leading to periodic, sinusoidal motion.

Classical

Hooke’s law: The restoring force is $F=-kx$, where $k$ is the spring constant and $x$ is displacement.
The classical oscillation frequency is $\omega =\sqrt{\frac{k}{\mu }}$. Here $\mu =\frac{m_1{m}_2}{m_1+m_2}$ is the reduced mass.

Quantum

Used to model molecular vibrations in Chemistry.
$-\frac{{\hslash }^2}{2\mu }\frac{d^2\psi }{d{x}^2}+\frac{1}{2}k{x}^2\psi =E\psi$
$E_v=\left(v+\frac{1}{2}\right)\hslash \omega \text{for }v=0,1,2,\dots$ where $v$ is the vibrational quantum number.

The first few energies are:

Quantum number	Energy
$v=0$	$\frac{1}{2}\hslash \omega$
$v=1$	$\frac{3}{2}\hslash \omega$
$v=2$	$\frac{5}{2}\hslash \omega$
$v=3$	$\frac{7}{2}\hslash \omega$
$\Delta E=\hslash \omega =\hslash \sqrt{k/\mu }$

The ground-state wavefunction as a Gaussian:
${\psi }_0(x)=N_0{e}^{-x^2/(2{\alpha }^2)}$
where $\alpha =\sqrt{\frac{\hslash }{\mu \omega }}$ controls the width.

To generalize the Gaussian ground state to all vibrational states, we introduce a new variable:
$y=\frac{x}{\alpha }$
Since $x$ and $\alpha$ both have units of length, $y$ is dimensionless.

The full wavefunction becomes:
${\psi }_v(x)=N_v{H}_v(y)e^{-y^2/2}$

• $N_v$ is a normalization constant to ensure total probability is one.
• $H_v(y)$ is a Hermite polynomial, which determines the nodes and shape.
• $e^{-y^2/2}$ is the Gaussian envelope, which makes the wavefunction decay far from equilibrium.

The normalization constant is:
$N_v=\frac{1}{\sqrt{2^{v}v!\sqrt{\pi }\alpha }}$