Expectation Values of $p$ and $x$

It is important to realize that we can just use the definition of $A$ to write $x$ and $p$ in terms of the raising and lowering operators.

$$\begin{eqnarray*} x&=&\sqrt{\hbar\over 2m\omega}(A+A^\dagger) \\ p&=&-i\sqrt{m\hbar\omega\over 2}(A-A^\dagger) \\ \end{eqnarray*}$$

This will allow for any computation.

* Example: The expectation value of $x$ for any energy eigenstate is zero.*
* Example: The expectation value of $p$ for any energy eigenstate is zero.*
* Example: The expectation value of $x$ in the state ${1\over \sqrt {2}}(u_0+u_1)$.*
* Example: The expectation value of ${1\over 2}m\omega ^2x^2$ for any energy eigenstate is ${1\over 2}\left(n+{1\over 2}\right)\hbar\omega$.*
* Example: The expectation value of ${p^2\over 2m}$ for any energy eigenstate is ${1\over 2}\left(n+{1\over 2}\right)\hbar\omega$.*
* Example: The expectation value of $p$ as a function of time for the state $\psi(t=0)={1\over\sqrt{2}}(u_1+u_2)$ is $-\sqrt{m\hbar\omega}\sin(\omega t)$.*