Ethan's Notes

    Eigenfunction

    #Math

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

    • $\displaystyle \hat{A}$ 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 $\displaystyle \Psi$ in bound state:

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

    $\displaystyle \braket{ \Psi_{n} | \Psi_{m} }=\delta_{nm}$

    • $\displaystyle \Psi_{n}$ and $\displaystyle \Psi_{m}$ are eigenfunctions

    $\displaystyle \text{span}(\Psi_{n})=\left{ \text{all possible }\Psi \right}$

    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 $\displaystyle \Psi$ in scattering state:

    $\displaystyle \braket{ \Psi_{n} | \Psi_{m} }=\delta(p-p')$

    $\displaystyle \Psi(x)=\int_{-\infty}^{\infty} c(p)\Psi_{p}(x) , \mathrm{d}p$

    $\displaystyle c(p')=\braket{ \Psi_{p'} | \Psi }$