Eigenfunction

Related

• Eigenvector

$\hat{A}f(x)=\lambda f(x),\lambda \in \mathbb{C}\leftrightarrow f(x)\text{ is an eigenfunction}$

• $\hat{A}$ is an operator

Discrete Spectra

The eigenfunctions also only have real eigenvalues. Eigenfunctions of discrete spectra form an orthogonal basis of all possible wave functions. In equation form for $\Psi$ in bound state:

$\hat{Q}\Psi =q\Psi \to q\in \mathbb{R}$

$⟨{\Psi }_n|{\Psi }_m⟩={\delta }_{nm}$

• ${\Psi }_n$ and ${\Psi }_m$ are eigenfunctions

$\text{span}({\Psi }_n)=\left\{\text{all possible }\Psi \right\}$

Continuous Spectra

Eigenfunctions of discrete spectra follow Dirac orthogonality and something similar to forming a basis of all possible wave functions. In equation form for $\Psi$ in scattering state:

$⟨{\Psi }_n|{\Psi }_m⟩=\delta (p-p^{\prime })$

$\Psi (x)={\int }_{-\infty }^{\infty }c(p){\Psi }_p(x)\mathrm{d}p$

$c(p^{\prime })=⟨{\Psi }_{p^{\prime }}|\Psi ⟩$