Expectation value of momentum in symmetric 2D H.O

[ma18]

Homework Statement

Consider the following inital states of the symmetric 2D harmonic oscillator

ket (phi 1) = 1/sqrt(2) (ket(0)_x ket(1)_y + ket (1)_x ket (0)_y)

ket (phi 2) = 1/sqrt(2) (ket(0)_x ket(0)_y + ket (1)_x ket (0)_y)

Calculate the <p_x (t)> for each state

Homework Equations

The Attempt at a Solution

I am not sure how to work with these kets, I know that for the expectation value using you would do
<phi_1 | p_x (t) | phi_2>

but I don't know how to represent p_x (t) in the notation used in the ketsAny help would be much appreciated.

 

[DEvens]

The expectation value for an operator on a state is <state 1| operator |state 1>. You wrote <state 1| operator |state 2>. Note the 2 in the last ket. That's a transition amplitude as induced by the operator.

Look in your text for a representation of the momentum operator. You should find it near he Heisenberg uncertainty formula, or the Schrödinger equation or some such. You probably want a derivative with some constants.

Notice that the problem says "initial states." So these are not the wave functions. These are the t=0 values. It does not give the x dependence, nor the t dependence. You will need to do some reading in your text to find the 2-D harmonic oscillator and the general solution for it.