# Orthogonality & Orthonormality of Wave function

Quantum Mechanics
Basic Quantum Mechanics

Orthogonality & Orthonormality | Quantum Mechanics

Orthogonality condition of wave functions:

Two wave functions $\psi_{m}\left(x\right)$ and $\psi_{n}\left(x\right)$ are said to be orthogonal to each other, if

$\intop_{-\infty}^{\infty}\psi_{m}^{*}(x)\psi_{n}(x)$dx=0 $(m\neq n)$

OR

$<\psi_{m}|\psi_{n}> = 0 (m \neq n)$

i.e. if a particle is in the state $\psi_{m}(x)$ , then the particle cannot be in the state $\psi_{n}(x)$ simultaneously together.

Orthonormality condition of wave functions:

Two wave functions $\psi_{m}\left(x\right)$ and $\psi_{n}\left(x\right)$ are said to be orthonormal to each other, if they are orthogonal and also normalized.

Orthonormal = Orthogonal + Normalized

$\intop_{-\infty}^{\infty}\psi_{m}^{*}(x)\psi_{n}(x)$dx=0 $(m\neq n)$

$\intop_{-\infty}^{\infty}\psi_{m}^{*}(x)\psi_{n}(x)$dx=1 $(m= n)$

OR

$<\psi_{m}|\psi_{n}> = 0 (m \neq n)$

$<\psi_{m}|\psi_{n}> = 1 (m = n)$