What does normalization mean in quantum mechanics?

A wavefunction ψ must satisfy ∫|ψ|² dV = 1 over all space. This ensures the total probability of finding the particle somewhere equals 100%. The normalization constant A scales ψ so this condition holds.

Why is the particle-in-a-box constant √(2/L)?

The unnormalized wavefunction is sin(nπx/L). Integrating sin²(nπx/L) from 0 to L gives L/2. Setting A²·(L/2) = 1 yields A = √(2/L), independent of quantum number n.

What is the harmonic oscillator length scale?

The oscillator length a₀ = √(ħ/mω) characterizes the spatial spread of the ground-state Gaussian. It is not the Bohr radius — you can enter any value appropriate to your system (e.g. for a diatomic molecule or a trapped ion).

How is the hydrogen 1s normalized?

The 1s wavefunction is ψ = A·exp(−r/a₀). Integrating 4π∫₀^∞ r²|ψ|² dr = 1 gives A = (πa₀³)^(−1/2) = 1/√(πa₀³). The calculator uses the physical Bohr radius a₀ = 5.29177×10⁻¹¹ m by default.

Does the normalization constant depend on the quantum number n?

For the particle in a box, no — the constant √(2/L) is the same for all n. For harmonic oscillator excited states the constant does change because higher-level Hermite polynomials have different norms. This tool covers the ground state (n=0) for the oscillator.

Wavefunction Normalization Calculator

Name: Wavefunction Normalization Calculator
Availability: InStock
Author: Nham Vu

Select a wavefunction type, enter the parameters, and get the exact normalization constant A so that ∫|ψ|²dV = 1.

Wavefunction Parameters

Model

Particle in a Box (1D)

ψ_n(x) = A sin(nπx/L), 0 ≤ x ≤ L

Harmonic Oscillator (ground state)

ψ₀(x) = A exp(−x²/2a₀²), −∞ < x < ∞

Hydrogen 1s

ψ(r) = A exp(−r/a₀), full 3D integral

Box Length L (m)

Typical atomic well: 1–5 Å = 1e-10 to 5e-10 m.

Quantum Number n

n ≥ 1; normalization constant is independent of n.

Select a model and enter its parameters, then click Calculate.

Summary

Select a wavefunction type, enter the parameters, and get the exact normalization constant A so that ∫|ψ|²dV = 1.

How it works

Choose a wavefunction model: particle in a box (1D), quantum harmonic oscillator ground state, or hydrogen 1s.
Enter the required parameter — box length L, oscillator length scale a₀, or Bohr radius a₀.
The calculator evaluates the normalization integral ∫|ψ(x)|² dx (or dV for 3D) analytically.
It returns the normalization constant A and the full normalized wavefunction ψ(x).
A verification step confirms A² × (analytic integral) = 1 to machine precision.

Use cases

Verify normalization constants for quantum mechanics homework.
Understand how box size affects probability density for particle in a box.
Compare ground-state spatial extents across different oscillator potentials.
Check that a trial wavefunction satisfies the normalization condition.
Quickly look up the hydrogen 1s radial norm factor for spectroscopy problems.
Teach or review the normalization procedure with a worked example.

Frequently Asked Questions

Last updated: 2026-07-01 · Reviewed by Nham Vu