Quantum Harmonic Oscillator Calculator
Overview
The 1D quantum harmonic oscillator is a core model in quantum mechanics for systems bound by a restoring force proportional to displacement (the quantum analogue of an ideal spring). Unlike the classical oscillator, which can have any energy, the quantum oscillator has discrete (quantized) energy levels. This calculator uses the standard energy-level formula to compute the energy of level n from the oscillator frequency f.
Key formulas
The energy of the nth level is:
Text form: En = (n + 1/2) ħ ω
where:
- n is the quantum number (0, 1, 2, …)
- ħ is the reduced Planck constant (ħ ≈ 1.054571817 × 10−34 J·s)
- ω is the angular frequency (rad/s)
- f is the ordinary frequency (Hz = s−1)
Frequency conversion:
ω = 2πf
Energy spacing between adjacent levels is constant:
ΔE = En+1 − En = ħω
MathML (for clean rendering)
What this calculator returns
The calculator computes En in:
- Joules (J), the SI unit of energy
- Electronvolts (eV), using 1 eV = 1.602176634 × 10−19 J
Electronvolts are often more convenient for atomic and molecular energy scales (e.g., vibrational energies in spectroscopy).
Interpreting the results
- Ground state (n = 0): E0 = (1/2)ħω, called the zero-point energy. Even at absolute zero, the oscillator cannot have zero energy.
- Equal spacing: Each increase of n by 1 adds exactly ħω. This is a distinctive signature of the ideal harmonic oscillator model.
- Spectroscopy connection: In the simplest selection-rule picture (e.g., electric dipole transitions in an ideal harmonic oscillator), transitions are strongest for Δn = ±1, corresponding to photon energies near ħω.
Worked example
Given: f = 5 × 1013 Hz, n = 1.
- Compute angular frequency: ω = 2πf = 2π(5 × 1013) ≈ 3.141592654 × 1014 rad/s.
- Compute level factor: (n + 1/2) = (1 + 1/2) = 1.5.
- Compute energy in joules:
E1 = 1.5 × ħ × ω
≈ 1.5 × (1.054571817 × 10−34 J·s) × (3.141592654 × 1014 s−1)
≈ 4.97 × 10−20 J. - Convert to eV:
E1(eV) = E(J) / (1.602176634 × 10−19)
≈ 0.310 eV.
So for this molecular-scale frequency, the first excited level is on the order of a few tenths of an eV—typical of vibrational energies.
Comparison table (common related quantities)
| Quantity | Expression | Units | What it tells you |
|---|---|---|---|
| Angular frequency | ω = 2πf | rad/s | How fast the oscillator phase advances |
| Energy level | En = (n + 1/2)ħω | J (or eV) | Allowed quantized energies |
| Level spacing | ΔE = ħω | J (or eV) | Energy gap between adjacent levels |
| Zero-point energy | E0 = (1/2)ħω | J (or eV) | Minimum energy (cannot be zero) |
Assumptions and limitations
- Ideal harmonic potential: Real systems (especially molecular vibrations) can be anharmonic, so higher levels may deviate from equal spacing.
- 1D model: This formula is for the standard 1D quantum harmonic oscillator. Multi-dimensional oscillators can introduce degeneracies and multiple mode frequencies.
- Quantum number constraint: The physically valid values are integers n ≥ 0. Non-integer or negative n is not meaningful for energy eigenstates.
- Frequency definition: Input frequency f must be in Hz. The calculator uses ω = 2πf. If you already have ω, convert it to f first (f = ω/2π).
- No temperature/occupation model: The calculator outputs energy levels only; it does not compute thermal populations (Boltzmann factors) or partition functions.
- Constants: Conversion to eV uses the exact SI definition of the elementary charge (1 eV = 1.602176634 × 10−19 J).
References (optional reading)
- Standard quantum mechanics textbooks (e.g., Griffiths, Shankar) for the harmonic oscillator derivation
- NIST/CODATA values for physical constants (ħ, e)
Mini-Game: Zero-Point Groove
Guide a shimmering oscillator through quantized energy targets and feel how ℏω spacing shapes its motion. This playable vignette sits right below your calculation so that the numbers above come alive through rhythm and timing.
Design Deliverables
Chosen calculator & why it fits: The quantum harmonic oscillator calculator already speaks in energy quanta, and the spring-like motion begs for a tactile experience where you can nudge amplitudes and feel discrete levels. Its simple inputs map cleanly onto a resonant mini-game loop with gorgeous oscillatory motion.
Game concept pitch: “Zero-Point Groove” casts you as a caretaker of a glowing nano-spring. Tap to inject quanta, steer the bead into highlighted energy bands before measurement pulses arrive, and ride a swelling soundtrack of particles and easing motion. As the session flows from calm to frenetic, you internalize how level spacing grows with frequency.
Mechanic Breakdown
- Primary control: click/tap/press space to pump quanta, expanding amplitude; hold for a stronger burst, release to coast.
- Measurement waves sweep in procedurally with shifting target quantum numbers; align the bead’s radius with the highlighted band to bank score and streak multipliers.
- Feedback includes elastic screen shake, chromatic trails, sparkling quanta pickups, and easing flashes tied to your precision.
- Every 20 seconds, bonus “coherence orbs” drop — hit them to momentarily slow time or gain extra score, injecting delightful twists.
Technical Approach
- Responsive high-DPI canvas renderer with delta-timed physics, pause-on-blur behavior, and reduced-motion fallbacks.
- Shared energy spacing derived from the calculator input (
ℏω) feeds difficulty scaling, streak logic, and educational insight. - Procedural spawner manages measurement waves, coherence orbs, and adaptive difficulty based on precision streaks.
- LocalStorage persists best score; modular functions handle rendering, input smoothing, and end-screen UX.
Each success reveals how adjacent energy levels sit ℏω apart.