Quantum Harmonic Oscillator Wavefunction Calculator

Compute the harmonic oscillator wavefunction ψ_n(ξ), probability density, and energy level from quantum number n and dimensionless position ξ, with a plotted waveform.

Input

Enter the quantum number n and dimensionless position ξ to compute the harmonic oscillator wavefunction ψ_n(ξ), its probability density, and the energy level.

Quantum number n

Enter an integer from 0 to 20.

Dimensionless position ξ

ξ = x √(mω/ħ). Negative values are allowed.

Result

Wavefunction ψ_2(ξ = 0.5)

-0.234359

Probability density |ψ|² = 0.054924

Energy level E_n

2.5 ħω

Number of nodes

Normalization N_n

0.265563

Classical turning point ξ_t

± 2.236068

ψ_n(ξ)=N_n H_n(ξ)e^(−ξ²/2), N_n=1/√(2ⁿ n! √π), E_n=(n+1/2)ħω. H_n is the Hermite polynomial and the number of nodes equals n.

How it works

The wavefunction is computed as ψ_n(ξ)=N_n H_n(ξ)e^(−ξ²/2), where ξ is the dimensionless position variable (ξ = x √(mω/ħ)).
H_n is the Hermite polynomial (physicists definition), obtained from the recurrence H_(k+1)=2ξH_k−2kH_(k−1).
The normalization constant is N_n = 1/√(2ⁿ n! √π).
The energy level is E_n=(n+1/2)ħω, shown here in units of ħω. Even the ground state has a zero point energy of 1/2 ħω.
The number of nodes (zero crossings) of the wavefunction equals the quantum number n.
The classical turning points are ξ_t=±√(2n+1); beyond them lies the classically forbidden region.
Enter an integer quantum number n between 0 and 20.

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