Exactly-solvable quantum system with H = p²/2m + ½ mω²x². Spectrum E_n = ℏω(n + ½); ladder operators a, a† manipulate levels. Building block of QFT.
Quantum harmonic oscillator
Related concepts
- Canonical commutation relation
- Schrödinger equation
- Second quantisation
- Phonon
- Variational Monte Carlo (VMC)
- Water (H₂O)
- QHO partition function: Z = 1/(2 sinh(β ℏω/2)) per normal mode
- QHO Z = 1/(2 sinh(βℏω/2)) ≡ exp/geom form; βℏω=log 4 ⇒ Z = 2/3
- Quantum confinement (1D box): E_n = n²π²ℏ²/(2 m a²) ; E_n ∝ 1/a² (size scaling)
- Morse potential V(r) = D_e(1 - e^{-a(r-r_e)})² (canonical anharmonic diatomic)
- Harmonic PES V(x) = (1/2) k x² ; minimum at x=0 with curvature V''(0) = k
- Morse V(r_e) = 0, V(r_e + ln 2/a) = D_e/4 (quarter-well anchor exact)
- V(x) = k x²/2 has V'(0) = 0 and V''(x) = k (stationary-minimum signature)