Expectation Values

We can use operators to help us compute the expectation value of a physical variable. If a particle is in the state $\psi(x)$, the normal way to compute the expectation value of $f(x)$ is

$$\bgroup\color{black}\langle f(x)\rangle = \int\limits_{-\inft... ...= \int\limits_{-\infty}^\infty \psi^*(x)\psi(x) f(x) dx.\egroup$$

If the variable we wish to compute the expectation value of (like $p$) is not a simple function of $x$, let its operator act on $\psi(x)$

$$\bgroup\color{black}\langle p\rangle = \int\limits_{-\infty}^\infty\psi^*(x)p^{(op)}\psi(x) dx.\egroup$$

We have a shorthand notation for the expectation value of a variable $v$ in the state $\psi$ which is quite useful.

$$\bgroup\color{black}\langle\psi\vert v\vert\psi\rangle \equiv \int\limits_{-\infty}^\infty\psi^*(x)v^{(op)}\psi(x) dx.\egroup$$

We extend the notation from just expectation values to

$$\bgroup\color{black}\langle\psi\vert v\vert\phi\rangle \equiv \int\limits_{-\infty}^\infty\psi^*(x)v^{(op)}\phi(x) dx\egroup$$

and

$$\bgroup\color{black}\langle\psi\vert\phi\rangle \equiv \int\limits_{-\infty}^\infty\psi^*(x)\phi(x) dx\egroup$$

We use this shorthand Dirac Bra-Ket notation a great deal.