Quantum Harmonic Oscillator

Energy levels and Hermite wave functions

Parameters
Energy Diagram
Wave Function ψ(x)
Hamiltonian

The harmonic oscillator is one of the most important models in quantum mechanics. It describes any system performing small oscillations around equilibrium.

$$ \hat{H} = -\frac{\hbar^2}{2m}\frac{d^2}{dx^2} + \frac{1}{2}m\omega^2 x^2 $$
Energy Spectrum

Energy levels of the harmonic oscillator are equally spaced — a unique property that distinguishes it from the quantum well:

$$ E_n = \hbar\omega\left(n + \frac{1}{2}\right), \quad n = 0, 1, 2, \ldots $$

Important: the ground state (n=0) energy is not zero — E₀ = ℏω/2. This is the "zero-point energy" — a fundamental result of quantum mechanics.

Wave Functions

Wave functions are expressed via Hermite polynomials (Hₙ):

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

x₀ = √(ℏ/mω) — characteristic oscillator length. Hermite polynomials: H₀=1, H₁=2x, H₂=4x²-2, H₃=8x³-12x, ...

Real Systems
Systemω (rad/s)ℏω (meV)
GaAs კვანტური წერტილი~10¹³~6.6
C-H ქიმიური ბმა~9×10¹³~59
H₂ მოლეკულა~8×10¹⁴~530
InAs კვანტური წერტილი~5×10¹²~3.3
Step-by-Step Guide
1
Enter Frequency
Enter ω in units of ×10¹⁴ rad/s. E.g.: 1.0 = 10¹⁴ rad/s (typical electronic system).
2
Choose Particle
Electron — for nanophysics. Proton — for molecular vibrations. Custom mass — for any system.
3
Energy Diagram
The plot shows the parabolic potential V(x) = ½mω²x² with equally spaced energy levels inside.
4
Hermite Wave Functions
Use n= buttons (n=0,1,2...) to view ψₙ(x). n=0 — Gaussian, n=1 — one node, n=2 — two nodes, etc.
Interpreting Results

E₀ = ℏω/2 — zero-point energy, exists at any temperature. ΔE = ℏω — equal spacing between all levels (unlike the quantum well where ΔE increases). Doubling the frequency → energies double.

Self-Assessment

Q1. What is the ground state (n=0) energy of the harmonic oscillator?

A 0
B ℏω
C ℏω/2
D 2ℏω

Q2. What is the energy spacing between adjacent levels of the harmonic oscillator?

A ℏω (constant)
B n·ℏω (increasing)
C ℏω/n (decreasing)
D n²·ℏω

Q3. How many nodes (zeros) does the n=2 wave function have?

A 1
B 2
C 0
D 3