Quantum Harmonic Oscillator Energy Levels

The Quantum Harmonic Oscillator (QHO) The quantum harmonic oscillator is one of the most important exactly-solvable problems in quantum mechanics. It models vibrations of atoms in molecules, phonons in crystals, electromagnetic field modes, and many other physical systems. Unlike a classical oscillator, which can have any energy, the QHO has discrete (quantized) energy levels.

Energy Eigenvalues E_n = ℏω(n + ½) for n = 0, 1, 2, 3, … Where: ℏ = h/(2π) = 1.0546×10⁻³⁴ J·s (reduced Planck constant), ω = 2πf = angular frequency (rad/s), n = quantum number (non-negative integer). The energy levels are equally spaced by ℏω, unlike the hydrogen atom where the spacings shrink as n grows.

Zero-Point Energy Even at n = 0 (the ground state), the oscillator has energy E₀ = ½ℏω. This zero-point energy is a consequence of the Heisenberg uncertainty principle: a particle cannot simultaneously have a definite position and a definite momentum. Zero-point energy is real and measurable: it contributes to the Casimir effect, liquid helium’s resistance to solidifying at atmospheric pressure, and van der Waals forces.

Classical vs Quantum Oscillator Classical: E = ½kx² + ½mv², which can take any positive value. A pendulum at rest has E = 0. Quantum: E = ℏω(n + ½), so the minimum energy is ½ℏω, not zero. For large n (high energies), quantum predictions approach classical results (correspondence principle).

Ladder Operators The creation operator ↠raises n by 1: â†|n⟩ = √(n+1) |n+1⟩ The annihilation operator â lowers n by 1: â|n⟩ = √n |n−1⟩ These operators are used extensively in quantum field theory to create and destroy particles. The Hamiltonian is H = ℏω(â†â + ½) = ℏω(N̂ + ½) where N̂ = â†â is the number operator.

Applications of the QHO Molecular vibrations: diatomic molecules such as HCl vibrate at quantized frequencies, which is what infrared spectroscopy measures. Phonons: lattice vibrations in crystals are quantized harmonic oscillators, and they govern heat capacity. Photons: each mode of the electromagnetic field is a QHO. This leads directly to Planck’s blackbody radiation law. Quantum field theory: all quantum fields are composed of infinite collections of QHOs.

Selection Rules for Transitions For light absorption/emission: Δn = ±1 only (selection rule for harmonic oscillator). Transition frequency: f = ω/(2π) = the classical oscillation frequency. This is why infrared spectra of diatomic molecules show equally spaced peaks — each peak is one quantum jump.

Characteristic Length and Momentum The oscillator length x₀ = √(ℏ/mω) is the natural distance scale of the problem, and p₀ = √(ℏmω) is the matching momentum scale. Their product is x₀ × p₀ = ℏ, which follows straight from the definitions.

Those are not the ground-state uncertainties, and the difference matters. The actual standard deviations are smaller by a factor of √2: Δx = √(ℏ/2mω) and Δp = √(ℏmω/2). It is those two that multiply to Δx × Δp = ℏ/2, which is what exactly saturates the Heisenberg relation Δx·Δp ≥ ℏ/2. The oscillator ground state is a minimum-uncertainty state, one of the few places in quantum mechanics where the inequality becomes an equality.