Quantum Harmonic Oscillator

The quantum harmonic oscillator is one of the most important models in quantum physics because many systems behave “approximately harmonic” near a stable equilibrium position.

1. Model and energy spectrum

Near equilibrium, a restoring force of the form F = -kx corresponds to a quadratic potential:

V(x) = (1/2)kx² = (1/2)mω² x² where ω = square root of (k/m)

The allowed energies are:

Eₙ = (n + 1/2)ħω (n = 0,1,2,…)

Immediate consequences:

• Ground state is not zero: E₀ = (1/2)ħω (zero-point energy).
• Equal spacing: Eₙ₊₁-Eₙ = ħω (constant for all n).

2. Why the ground-state energy cannot be zero

The oscillator has both kinetic and potential energy. A standard H3 argument uses the uncertainty principle to estimate the minimum possible energy.

For a state centred on the equilibrium point with ⟨x⟩ = ⟨p⟩ = 0, write the energy in terms of its spreads:

E = ((Δ p)²)/2m + (1/2)mω²(Δ x)².

Using Δ x Δ p ≥ ħ/2 gives the lower bound

E(Δ x) ≥ ħ²/(8m(Δ x)²) + (1/2)mω²(Δ x)².

Minimise with respect to Δ x:

Substitute back to get the minimum energy scale:

Eₘᵢₙ ≥ (1/2)ħω.

The oscillator ground state is a minimum-uncertainty Gaussian and reaches this bound, so its exact energy is E₀ = ħω/2.

3. Common mistakes

1. Using the particle-in-a-box rule n = 1 as the ground state (for the harmonic oscillator, ground state is n = 0).
2. Mixing h and ħ ( ħ = h/2π).
3. Mixing frequency f and angular frequency ω ( ω = 2π f).

4. Useful checks

• If asked “why E₀ ≠ 0?”, use the uncertainty-principle wording: “a state cannot have both Δ x and Δ p arbitrarily small.”
• If asked for energy differences, go straight to Δ E = ħω.

5. Quick check

A. What is the spacing between adjacent energy levels?

B. If ω doubles, what happens to the ground state energy?