My Internet search just told me how to find the expectation value of $x$ and $p$ for when the particle is in a box is in a particular eigenstate. However, how do we do the find the expectation values when the particle is in the superimposed state of $\sum_{n}c_{n}\psi _{n}$?
Also, as the $\psi _{n}$ are not a eigenfunction of $x$ and $p$ operators, can we interpret the result as the weighted sum of individual expectation values for individual $\psi _{n}$?
Also can anyone tell what exactly is a stationary state?
Best Answer
I am assuming your $\psi_n$s are functions of $x$. The expectation value of any operator $A(x,p)$ is given by, $$\langle A\rangle=\int dx\,\Psi^*(x)\,A\left(x,i\hbar\frac{\partial}{\partial x}\right)\Psi(x)=\sum_m\sum_n (c^*_mc_{n})\int dx \,\psi^*_m\,A\left(x,i\hbar\frac{\partial}{\partial x}\right)\psi_n$$ where, $$\Psi(x)=\sum_n c_n\psi_n$$
If the $\psi_n$s are eigenfunctions of $A$, and form an orthonormal set, then the expression can be further simplified. Say, $$A\psi_n=a_n\psi_n$$ Then we can write, $$\langle A\rangle=\sum_m\sum_n(c^*_mc_{n})\,a_n\int dx\,\psi_m^*\psi_n=\sum_m\sum_n(c^*_mc_{n})\,a_n\,\delta_{mn}=\sum_n|c_n|^2a_n$$
For your second question, I am sure any standard textbook will have a good answer.