Schrodinger Equation

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Non-relativistic and doesn’t account for electron spin. The Dirac Equation accounts for these issues

Topics

• TISE
• TDSE
• Wave Function

Solution

$\Psi (\vec{r},t)=\psi (\vec{r})\varphi (t),\varphi (t)=e^{-iEt/\hslash }$

• Solved by separation of variables
• Must be square integrable

$\Psi (\vec{r},t)=\sum _{n=1}^{\infty }{c}_n{\psi }_n(\vec{r})e^{-i{E}_{n}t/\hslash }$

• 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
• ${\psi }_n(\vec{r})$ is the time-independent solution for the $n$th wave function
• $c_n$ is the wave function weighting coefficient
• $E_n$ is the energy of the $n$th wave function
• Each ${\psi }_n$ is orthogonality to each other

How to Solve

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

1. Solve the TISE $\hat{H}\psi =E\psi$
   1. $\psi (x)=\sum _n{c}_n{\psi }_n(x)$
   2. $c_n=\int {\psi }_n^{\ast }(x){\Psi }_{(x,0)}\mathrm{d}x$
2. Use $\Psi (x,t)=\sum _n{c}_n{\psi }_n(x)e^{-i{E}_{n}t/\hslash }=\sum _{n=1}^{\infty }{c}_n{\Psi }_n(x,t)$