Ethan's Notes

    Schrodinger Equation

    #Physics #Chemistry
    Universal Danker Video
    Non-relativistic and doesn't account for electron spin. The Dirac Equation accounts for these issues

    Topics

    Solution

    $\displaystyle \Psi(\vec{r},t)=\psi(\vec{r})\phi(t),\phi (t)=e^{-iEt/\hbar}$

    $\displaystyle \Psi(\vec{r},t)=\sum_{n = 1}^{\infty}c_{n}\psi_{n}(\vec{r})e^{-iE_{n}t/\hbar}$

    • A wave function can be the linear combination of a bunch of wave functions, which by Fourier's Theorem, can represent any type of wave function that satisfies the Schrodinger equation
    • $\displaystyle \psi_{n}(\vec{r})$ is the time-independent solution for the $\displaystyle n$th wave function
    • $\displaystyle c_{n}$ is the wave function weighting coefficient
    • $\displaystyle E_{n}$ is the energy of the $\displaystyle n$th wave function
    • Each $\displaystyle \psi_{n}$ is orthogonality to each other

    How to Solve

    Given $\displaystyle V(x)$ and $\displaystyle \Psi(x,0)$, solve $\displaystyle \Psi(x,t)$

    1. Solve the TISE $\displaystyle \hat{H}\psi=E\psi$
      1. $\displaystyle \psi(x)=\sum_{n}c_{n}\psi_{n}(x)$
      2. $\displaystyle c_{n}=\int \psi_{n}^{*}(x)\Psi_{(x,0)} , \mathrm{d}x$
    2. Use $\displaystyle \Psi(x,t)=\sum_{n}c_{n}\psi_{n}(x)e^{-iE_{n}t/\hbar}=\sum_{n = 1}^{\infty}c_{n}\Psi_{n}(x,t)$