Quantum Harmonic Oscillator Explorer

Quantum Harmonic Oscillator

Energy Level (n) 0

Angular Frequency (ω) 1.0

Particle Mass (m) 1.0

Show Potential

Energy Levels

The allowed energy levels are quantized according to:

Eₙ = (n + ½)ħω

Physical Explanation

The quantum harmonic oscillator is one of the most important model systems in quantum mechanics. It describes systems with a quadratic potential, such as molecular vibrations or quantum fields.

The wavefunctions are given by Hermite polynomials multiplied by a Gaussian:

ψₙ(x) = (1/(2ⁿn!))^(1/2) (mω/πħ)^(1/4) Hₙ(√(mω/ħ)x) e^(-mωx²/2ħ)

Key features:

• Equally spaced energy levels (ħω apart)
• Non-zero ground state energy (zero-point energy)
• Wavefunctions have n nodes
• Tunneling into classically forbidden regions