AcademyQuantum Wave Functions
Academy
Harmonic Oscillator
Level 1 - Physics topic page in Quantum Wave Functions.
Principle
The quantum harmonic oscillator has equally spaced energy levels and a nonzero ground-state energy.
Notation
\(V(x)\)
oscillator potential energy
\(\mathrm{J}\)
\(m\)
particle mass
\(\mathrm{kg}\)
\(\omega\)
angular frequency
\(\mathrm{s^{-1}}\)
\(n\)
oscillator quantum number
1
\(E_n\)
nth oscillator energy
\(\mathrm{J}\)
\(x_0\)
ground-state length scale
\(\mathrm{m}\)
Method
Derivation 1: Oscillator potential
Near a stable equilibrium, many potentials are approximately quadratic.
Quadratic potential
\[V(x)=\frac12m\omega^2x^2\]
Time-independent equation
\[-\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2}+\frac12m\omega^2x^2\psi=E\psi\]
Derivation 2: Energy ladder
The oscillator spectrum rises in equal steps of \(\\hbar\\omega\).
Allowed energies
\[E_n=\left(n+\frac12\right)\hbar\omega\]
Adjacent spacing
\[E_{n+1}-E_n=\hbar\omega\]
Derivation 3: Ground-state width
The ground state balances kinetic energy from localization with potential energy from spreading.
Length scale
\[x_0=\sqrt{\frac{\hbar}{m\omega}}\]
Gaussian ground state
\[\psi_0(x)=\left(\frac{m\omega}{\pi\hbar}\right)^{1/4}e^{-m\omega x^2/(2\hbar)}\]
Rules
For the one-dimensional quantum harmonic oscillator:
Potential energy
\[V(x)=\frac12m\omega^2x^2\]
Energy spectrum
\[E_n=\left(n+\frac12\right)\hbar\omega\]
Ground-state length
\[x_0=\sqrt{\frac{\hbar}{m\omega}}\]
Examples
Question
What is the energy difference between
\[n=4\]
and \[n=1\]
?Answer
\[E_4-E_1=(4-1)\hbar\omega=3\hbar\omega\]
Checks
- The oscillator quantum number starts at \(n=0\).
- Energy spacing is constant even though the wave functions change shape.
- The ground-state probability density is centered at the equilibrium point.
- Large \(n\) states connect to classical turning-point behavior.