Energy Eigenvalue Equation

Another name for the Time independent Schrodinger equation

$$\hat{H}\phi (x)=E\phi (x)$$

The energy operator is given by $\hat{H}$, our eigenfunctions are given by $\phi (x)$, and our eigenvalue is given by $E$.

This can be extended to a general form where

$$\hat{A}{\phi }_a=a{\phi }_a$$

Where $\hat{A}$ can be any operator.

Example:

$$\hat{p}\left(\sqrt{\frac{2}{L}}\left(\frac{n\hslash x}{L}\right)\right)=\frac{\hslash }{i}\frac{d}{dx}\left(\sqrt{\frac{2}{L}}\sin \left(\frac{n\hslash x}{L}\right)\right)$$ $$=\frac{\hslash }{i}\sqrt{\frac{2}{L}}\frac{n\hslash }{L}\cos \left(\frac{n\hslash x}{L}\right)$$ $$\ne constantx\sin \left(\frac{n\hslash x}{L}\right)$$

Therefore, $\sin \left(\frac{n\hslash x}{L}\right)$ is not an eigenfunction of $\hat{p}$

The operator $\hat{H}$ is interesting because it is a linear operator Consider:

$$\psi (x)=c_1\phi (x)+c_2{\phi }_2(x)$$

Now if $\hat{H}{\phi }_n(x)=E_n{\phi }_n(x)$, then

$$\hat{H}{\Psi }_n(x)=\hat{H}[c_1{\phi }_1(x)=c_2{\phi }_2(x)]$$ $$=c_1{E}_1{\phi }_1(x)+c_2{E}_2{\phi }_2(x)\ne E[c_1{\phi }_1(x)+c_2{\phi }_2(x)]$$

In general, unless $E_1=E_2$