Quantum Harmonic Oscillator Calculator
Enter the angular frequency and quantum number to compute the energy levels, zero-point energy, and level spacing of a quantum harmonic oscillator. ω=1e14, n=0 → E=5.27e-21 J.
Input Data
Results
At a glance:Quantum harmonic oscillator (Heisenberg 1925, Schrödinger 1926): a particle of mass m in a quadratic potential V=½kx²=½mω²x² has quantized energy levels E_n=(n+½)ℏω, n=0,1,2,…. Derivation: the time-independent Schrödinger equation −ℏ²/(2m)·d²ψ/dx²+½mω²x²ψ=Eψ with boundary condition ψ(∞)=0 yields Hermite-polynomial solutions ψ_n=N_n·H_n(αx)·exp(−α²x²/2) and eigenenergies E_n=(n+½)ℏω. Key properties: (1) equally spaced levels ΔE=ℏω, differing from the classical prediction of continuously variable energy; (2) a non-zero zero-point energy E₀=½ℏω even at n=0, a direct consequence of the Heisenberg uncertainty principle; (3) at large n the probability distribution approaches the classical result (correspondence principle). History: Planck's 1900 blackbody model assumed equally spaced oscillator energies hν (equivalent to E_n=nℏω, missing the zero-point term); Heisenberg's matrix mechanics later gave the correct E_n=(n+½)ℏω. Applications: (1) diatomic molecular vibration spectra (infrared, Raman); (2) lattice phonons (Debye model); (3) quantum field theory, where each field mode is a harmonic oscillator; (4) lasers; (5) superconducting qubits.
Formula
Energy level: E_n = (n + ½)·ℏω
Zero-point energy: E₀ = ½·ℏω
Level spacing: ΔE = ℏω
Equally spaced ladder: E_{n+1} − E_n = ℏω
Correspondence principle: n ≫ 1 → probability distribution approaches the classical
$$E_n = \left(n + \frac{1}{2}\right)\hbar\omega, \quad E_0 = \frac{1}{2}\hbar\omega, \quad \Delta E = \hbar\omega$$How to Use
- Enter the angular frequency ω (rad/s; molecular vibration ≈1e14).
- Enter the quantum number n (≥0; ground state 0).
- The calculator returns the level energy E_n, the zero-point energy E₀, and the level spacing ℏω.
Case Studies
Diatomic molecular vibration
A diatomic molecule such as HCl approximates a harmonic oscillator; its vibration frequency shows up as an infrared absorption line. HCl vibrates at ω≈5.6e14 rad/s (2886 cm⁻¹).
Infrared spectrometers identify chemical bonds via vibrational absorption; this is routine in university chemistry departments in Hong Kong. Energy quantization makes the spectrum discrete rather than continuous.
Real molecular vibrations are not strictly harmonic; higher levels crowd closer (Morse potential), and HCl dissociates around n≈14.
Lattice phonons and specific heat
Each lattice vibration mode is a harmonic oscillator whose energy quantum is called a phonon, with energy ℏω.
The Debye model predicts the low-temperature T³ specific-heat law: only low-frequency phonons are excited at low T, while higher modes are frozen out. The Einstein model assumes a single frequency — correct at high T but failing at low T.
Phonons are central to semiconductor physics and BCS superconductivity; the superconducting critical temperature relates to the phonon spectrum (McMillan formula).
Content review: Calculatorism Science Team. Results are for reference only; please refer to the relevant authorities for the official figures.