Quantum Harmonic Oscillator

Quantum Harmonic Oscillator

The quantum harmonic oscillator models a particle in a quadratic potential well, representing small oscillations around a stable equilibrium. Unlike a classical oscillator confined between its turning points, a quantum stationary state exhibits nonzero probability density tails extending into the classically forbidden region.

In terms of the dimensionless coordinate q = x × √(mω / ℏ), the normalized eigenstates are ψ_n(q) = H_n(q) exp(−q²/2) / (π^1/4 √(2ⁿ n!)), where H_n(q) are physicists' Hermite polynomials (H₀ = 1, H₁ = 2q, H₂ = 4q² − 2, H₃ = 8q³ − 12q).

Worked Example

Consider the first excited state (n = 1) evaluated at dimensionless position q = 1:

1. Identify the Hermite polynomial: For n = 1, H₁(q) = 2q, so H₁(1) = 2.
2. Calculate the wavefunction: The normalized wavefunction is ψ₁(1) = 2 exp(−1/2) / (π^1/4 √2) ≈ 0.64428836511348.
3. Calculate probability density: Squaring the wavefunction gives |ψ₁(1)|² ≈ 0.41510749742059.
4. Calculate energy and variance: Both dimensionless energy E / (ℏω) and position variance equal n + 1/2 = 3/2 = 1.5.
5. Determine turning points: The classical turning points occur at q = ±√(2n + 1) = ±√3 ≈ ±1.7320508075689, placing q = 1 inside the classically allowed region.

Experiment Instructions

Interact with the Quantum Harmonic Oscillator simulation to inspect nodes, turning points, and classically forbidden tails:

1. Select State n = 0 and Position q = 0 to observe the ground state with zero-point energy 0.5 and peak density π^−1/2 ≈ 0.564189583548.
2. Keep State n = 0 and set Position q = 2. Because |q| = 2 exceeds the turning point magnitude 1, the probe region becomes "Classically forbidden" while the probability density remains positive (0.010333492677).
3. Change State n to 2 and Position q to 1/sqrt2. Both the wavefunction and probability density vanish at this exact root of H₂(q) = 4q² − 2.

Model Limitations

This module evaluates single stationary energy eigenstates (n = 0, 1, 2, 3) in dimensionless coordinates using full-line analytic normalization over (−∞, ∞). It does not model time-dependent coherent-state superpositions, anharmonic potential perturbations, or relativistic corrections.

Further References

For the derivation of the Schrödinger equation solutions, Hermite polynomials, and zero-point energy in a parabolic potential well, see:

• OpenStax University Physics Volume 3, Section 7.5: The Quantum Harmonic Oscillator