What is the zero-point energy?

Zero-point energy is the ground-state energy E0 = ½ħω. Even at absolute zero the oscillator retains this energy — it cannot be removed because the uncertainty principle forbids a perfectly still particle.

Why are the energy levels equally spaced?

The harmonic potential V = ½mω²x² produces a Schrödinger equation whose solutions are Hermite polynomials. The eigenvalue spectrum En = (n + ½)ħω yields a constant gap ħω between every adjacent pair of levels.

What units should ω be in?

Angular frequency ω must be in radians per second (rad/s). If you have ordinary frequency f in Hz, convert with ω = 2πf. Typical molecular vibrations are in the range 10¹² – 10¹⁴ rad/s.

ħ is the reduced Planck constant, equal to h / (2π) ≈ 1.054571817 × 10⁻³⁴ J·s. It sets the scale of quantum effects and appears throughout quantum mechanics.

Can n be any positive integer?

Yes. n = 0 is the ground state, n = 1 is the first excited state, and so on. There is no upper bound theoretically, though very high levels approach classical behavior.

Quantum Harmonic Oscillator Calculator

Enter angular frequency ω and quantum number n to compute energy level En = (n + ½)ħω, zero-point energy, and level spacing.

Parameters

Angular Frequency ω (rad/s)

Typical molecule: 10¹² – 10¹⁴ rad/s. Use scientific notation.

Quantum Number n

n = 0 is the ground state.

Quick presets

Energy E_n

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Zero-Point E₀

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Level Spacing ħω

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Formula

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Energy Ladder (first 8 levels)

n	Energy (J)	Energy (eV)	Coeff (n+½)

Enter ω and n, then click Calculate.

Summary

The quantum harmonic oscillator is a foundational model in quantum mechanics. Its energy levels are En = (n + ½)ħω, where n is a non-negative integer and ħω is the uniform spacing between levels. This calculator computes any level instantly and displays the full energy ladder so you can see how excited states build up from the ground state.

How it works

Enter the angular frequency ω in rad/s (or use the preset values for common systems).
Enter the quantum number n (0 = ground state, 1 = first excited state, etc.).
The calculator applies En = (n + ½)ħω using ħ = 1.054571817 × 10⁻³⁴ J·s.
Results display in joules and electron-volts simultaneously.
The energy ladder table shows the first several levels for comparison.

Use cases

Verify homework or exam answers for quantum mechanics courses.
Explore how energy levels scale with frequency for diatomic molecules.
Compare zero-point energy across different oscillator systems.
Understand the uniform spacing ħω between adjacent energy levels.
Convert between joule and eV representations for lab or lecture notes.
Quick reference for spectroscopy: infrared absorption energies.

Frequently Asked Questions

Last updated: 2026-06-13 · Reviewed by Nham Vu