Position and Momentum Operator Matrices

[kreil]

Homework Statement

Find the energy eigenvalues and eigenfunctions for the one-dimensional infinite square well. Calculate the matrices for the position and momentum operators, Q and P, using these eigenfunctions as a basis.

Homework Equations

The energy eigenvalues are

$$E_n = \frac{\pi^2 \hbar^2}{2Ma^2}n^2$$

and the eigenfunctions are,

$$\psi_n(x) = \sqrt{\frac{2}{a}} sin \left ( \frac{n \pi}{a} x \right )$$

The Attempt at a Solution

Finding the energy eigenvalues and eigenfunctions is straightforward. What is really confusing me is how to use the eigenfunctions as a basis to find the matrix representations of these operators. This is problem 4.2 in Ballentine's Quantum Mechanics, and I can't find any useful information in the chapter for doing this sort of problem. Here is the best attempt I have:

The position operator Q satisfies,

$$\hat Q \psi_n(x) = x_n \psi_n(x)$$

If we represent $\psi$ as a column matrix, we can write:

$$\psi_n(x) = \left[ \begin{array}{c} \psi_1(x) \\ \psi_2(x) \\ \vdots \\ \psi_n(x) \end{array} \right]$$

Using this representation,

$$\hat Q \left[ \begin{array}{c} \psi_1(x) \\ \psi_2(x) \\ \vdots \\ \psi_n(x) \end{array} \right] = \left[ \begin{array}{c} x_1 \psi_1(x) \\ x_2 \psi_2(x) \\ \vdots \\ x_n \psi_n(x) \end{array} \right]= \left[ \begin{array}{cccc} x_1 & 0 & \hdots & 0\\ 0 & x_2 & 0 \hdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & x_n \end{array} \right] \left[ \begin{array}{c} \psi_1(x) \\ \psi_2(x) \\ \vdots \\ \psi_n(x) \end{array} \right]$$

So Q would be this matrix with the x's along the diagonal:

$$\hat Q = \left[ \begin{array}{cccc} x_1 & 0 & \hdots & 0\\ 0 & x_2 & 0 \hdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & x_n \end{array} \right]$$

I feel like the answer is supposed to be more specific than this

 

[gabbagabbahey]

The position operator Q satisfies,

$$\hat Q \psi_n(x) = x_n \psi_n(x)$$

If you are talking about a one-dimensional harmonic oscillator, what the heck is $x_n$ supposed to mean? In one dimension, $\hat{Q}\psi_n(x)=x\psi_n(x)$.

The elements of a matrix $A$ in a basis $|v_i\rangle$ are given by

$$A_{ij}= \langle v_i|A|v_j\rangle$$

Apply that to find the matrix elements of the position operator in the $\psi_n\rangle$ eigenbasis (Hint: $\psi_n(x)=\langle x|\psi_n\rangle$ and $$\int |x\rangle\langle x|dx = \hat{1}$$ ).

 

[kreil]

got it, thanks!