Harmonic Oscillator Calculator

Visualize wavefunctions, energy levels, and probability distributions

Oscillator Parameters

Adjust parameters to visualize the quantum harmonic oscillator

N/m
kg
J·s
Energy Level (En)
0.5 ħω
Energy of quantum state
Angular Frequency (ω)
1.00 rad/s
√(k/m)
Zero-point Energy
0.50 ħω
Ground state energy
Characteristic Length
1.00
√(ħ/mω)

Understanding Quantum Harmonic Oscillators

The quantum harmonic oscillator is a quantum version of the classical harmonic oscillator. It describes systems bound to an equilibrium position by a force proportional to displacement.

1

Energy Quantization: En = ħω(n + ½)

Discrete energy levels with equal spacing ħω

2

Zero-Point Energy: E0 = ½ħω > 0

Minimum energy even at absolute zero temperature

3

Wavefunctions: Gaussian multiplied by Hermite polynomials

Higher states have more nodes and spread out further

4

Applications: Molecular vibrations, quantum optics, solid-state physics

Key Concepts

Zero-Point Energy
E0
Minimum energy at n=0
Energy Spacing
ΔE
Constant ħω between levels
Hermite Polynomials
Hn(ξ)
Determine wavefunction shape
Characteristic Length
√(ħ/mω)
Natural length scale

Note: The quantum harmonic oscillator is a fundamental model in quantum mechanics. Unlike classical oscillators, quantum oscillators have discrete energy levels and a minimum "zero-point" energy.

Quantum vs Classical

Quantum Oscillator
  • Discrete energy levels
  • Zero-point energy
  • Probability distribution depends on state
  • Wavefunction nodes
Classical Oscillator
  • Continuous energy
  • Minimum energy = 0
  • Probability highest at turning points
  • Definite position and momentum
Harmonic Oscillator Formulas
En = ħω(n + 1/2)
ψn(x) = (1/√(2nn!)) (mω/πħ)1/4 Hn(ξ) e-ξ²/2
ξ = x √(mω/ħ)
Where:
n = Quantum number (0,1,2,...)
ħ = Reduced Planck's constant
ω = Angular frequency = √(k/m)
Hn = Hermite polynomial
m = Particle mass
k = Spring constant
Harmonic Oscillator Applications
System Quantum Number Energy Level Description
Molecular Vibration 0-5 0.001-0.1 eV Vibrational states in diatomic molecules
Quantum Optics 0-100 0.1-10 eV Photon states in optical cavities
Solid State Physics 0-10 0.01-0.1 eV Phonons in crystal lattices
Quantum Computing 0,1 10⁻⁵-10⁻³ eV Qubit states in superconducting circuits
Atomic Traps 0-20 10⁻¹⁰-10⁻⁸ eV Atoms in optical lattices
Quantum Oscillator Facts
  • The harmonic oscillator has equally spaced energy levels
  • Ground state (n=0) has Gaussian wavefunction
  • Classically forbidden regions show quantum tunneling
  • Used to model molecular vibrations in IR spectroscopy
  • Basis for quantum field theory and second quantization

Zero-point energy is the minimum energy a quantum system can have:

  • E0 = ½ħω for harmonic oscillator
  • Consequence of Heisenberg uncertainty principle
  • System cannot be completely at rest
  • Observed in molecular vibrations and quantum fields

Example: Liquid helium remains liquid at absolute zero due to zero-point energy preventing freezing.

Harmonic oscillators have equally spaced energy levels due to:

  • Quadratic potential V(x) = ½kx²
  • Schrödinger equation solutions yield En = ħω(n + ½)
  • Energy difference between levels: ΔE = ħω
  • All transitions have same energy difference

This property makes harmonic oscillators useful for:

  • Molecular spectroscopy
  • Quantum optics
  • Quantum computing
  • Phonons in solids

Wavefunctions evolve with quantum number n:

  • n=0: Gaussian function (no nodes)
  • n=1: Odd function (one node at x=0)
  • n=2: Even function (two nodes)
  • Higher n: More nodes, spread further from center

General properties:

  • Number of nodes = n
  • Parity alternates (even for even n, odd for odd n)
  • Amplitude decreases with |x|
  • For large n, probability density approaches classical distribution

Quantum harmonic oscillators model many physical systems:

  • Molecular Vibrations: Diatomic molecules vibrate as harmonic oscillators
  • Quantum Optics: Electromagnetic fields modeled as harmonic oscillators
  • Solid-State Physics: Phonons (quantized lattice vibrations)
  • Quantum Computing: Superconducting qubits behave as harmonic oscillators
  • Atomic Physics: Trapped ions in harmonic potentials
  • Quantum Field Theory: Fields quantized as infinite harmonic oscillators