Why Wavefunctions for Different Energy Levels Are Orthogonal

1. They Come From a Hermitian Operator

The time-independent Schrödinger equation is:

Ĥ ψ = E ψ

Here, Ĥ (the Hamiltonian) is a Hermitian operator, which has two key properties:

• Its eigenvalues (energy levels) are real.
• Its eigenfunctions corresponding to different eigenvalues are orthogonal.

So if:

Ĥ ψ_n = E_n ψ_n
Ĥ ψ_m = E_m ψ_m

and E_n ≠ E_m, then:

<ψ_n | ψ_m> = 0

2. Orthogonality Prevents States From Overlapping

Different energy eigenstates are physically distinct. Orthogonality ensures:

• No energy state contains any component of another.
• Measurements of energy always yield one clear value.

3. It Comes From Conservation of Probability

Take two solutions of the Schrödinger equation, ψ_n and ψ_m. Multiply the equation for ψ_n by ψ_m* and the equation for ψ_m by ψ_n*, subtract, and integrate:

(E_n - E_m) ∫ ψ_m*(x) ψ_n(x) dx = 0

Since E_n ≠ E_m, the only solution is:

∫ ψ_m*(x) ψ_n(x) dx = 0

This is orthogonality.

4. Simple Example: Particle in a Box

Energy eigenfunctions are:

ψ_n(x) = √(2/L) sin(nπx / L)

Different sine modes are orthogonal:

∫_0^L sin(nπx / L) sin(mπx / L) dx = 0   (n ≠ m)

Like different notes on a guitar string—different vibrational modes don’t “mix”.

Summary

• The Hamiltonian is Hermitian → different eigenvalues → orthogonal eigenfunctions.
• Distinct energy states must not overlap physically.
• Orthogonality pops directly out of the integrated Schrödinger equation.
• In real systems (particle in a box, harmonic oscillator, hydrogen atom), this matches the behavior of different vibrational modes.