2023-10-10 Quantum Physics Lecture 12

Last class- -Functional Vector Space -Generalization of nth dimensional wave-vector spaces.

when a vector space is square-integrable it is known as a Hilbert Space

-Very important properties of wave-vector spaces -Normalize-able -Orthogonal -Can project $\Psi (x)$ onto ${\phi }_m(x)$ basis

The Energy Operator: Eigenvalue and Eigenfunction

We can try and understand the Energy Operator through the example of the Infinite square well. The basis vector, which is also the solution to the Schrodinger equation for this is given by:

$${\phi }_n(x)=\sqrt{\frac{2}{L}}sin\left(\frac{n\pi x}{L}\right)$$

Where the Schrodinger equation is then

$$\left[-\frac{{\hslash }^2}{2m}\frac{d^2}{d{x}^2}+V(x)\right]\phi (x)=E\phi (x)$$

The content within brackets, $\left[-\frac{{\hslash }^2}{2m}\frac{d^2}{d{x}^2}+V(x)\right]$, constitutes the Energy Operator, also denoted by $\hat{H}$.

Writing this in the standard Energy Eigenvalue 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 $E_1=E_2$