Wave Function Normalization Calculator

Normalizing a wave function means finding a constant N such that the total probability of finding the particle anywhere in space equals 1. Mathematically, this requires ∫|Ψ(x)|²dx = 1 over all space. The normalization constant N scales the wave function to satisfy this physical requirement.

For Ψ(x) = N·exp(−λx²/2), you integrate |Ψ|² over all x: ∫N²·exp(−λx²)dx = N²·√(π/λ) = 1. Solving gives N = (λ/π)^(1/4). Both positive and negative values of N are valid since the physical quantity is |Ψ|².

The physically observable quantity is the probability density |Ψ(x)|², not the wave function itself. Multiplying Ψ by an overall phase factor (including −1) does not change any measurable outcome. This is known as the global phase freedom in quantum mechanics.

For a particle in a 1D infinite potential well of length L in state n, Ψ(x) = N·sin(nπx/L). Integrating |Ψ|² from 0 to L gives N²·L/2 = 1, so N = √(2/L). This result is independent of the quantum number n.

In ideal, closed quantum systems governed by the Schrödinger equation with a Hermitian Hamiltonian, normalization is preserved over time — this is called unitarity. However, in approximate or dissipative models, re-normalization may be needed numerically.

For the quantum harmonic oscillator ground state Ψ₀(x) = N·exp(−mωx²/2ℏ), the normalization gives N = (mω/πℏ)^(1/4). In this calculator, the parameter λ represents mω/ℏ, so N = (λ/π)^(1/4), identical in form to the Gaussian result.

The units of N depend on the dimensionality. For a 1D wave function, N has units of m^(−1/2) so that |Ψ(x)|² has units of m^(−1) (probability per unit length), and the integral over length is dimensionless (a pure probability).

Normalization ensures the probabilistic interpretation of quantum mechanics is self-consistent. Without it, calculated expectation values of observables, transition probabilities, and measurement outcomes would be meaningless. Every physically valid quantum state must be normalized.