Matrix elements of the time development operator (position representation)

[siritachi]

Homework Statement

There is a freely moving particle of mass m. Prove that the matrix elements of the time development operator, T(t,0), in the position representation is <q'|T|q''> = [(m/2i*pi hbar t)^1/2] Exp[-m(q'-q'')^2/2it hbar]

Homework Equations

H = (p^2)/2m

The Attempt at a Solution

I know that for the momentum representation:

<p'|T|p''> = delta(p'-p'')Exp[-i(p'^2)t/2m hbar]

And I tried starting with:

<q'|p><p|q''> Exp[-i(p'^2)t/2m hbar]

but I am very new to this notation, and I'm not sure where to go next.

 

[mighty2000]

Can anyone offer some guidance?
Thank you for your post. I am a scientist and I would be happy to assist you with your problem.

The first step in solving this problem is to rewrite the momentum representation in terms of the position representation, since the problem asks for the matrix elements in the position representation. This can be done by using the Fourier transform:

<p'|q> = (1/(2*pi*hbar))^(1/2) Exp[i(p'q)/hbar]

Using this, we can rewrite the momentum representation as:

<p'|T|p''> = (1/(2*pi*hbar))^2 [(m/2i*pi*hbar t)^1/2] Exp[-i(p'-p'')^2t/2m hbar]

Next, we need to rewrite the operators in terms of the position representation. The Hamiltonian operator in the position representation is given by:

H = (-hbar^2/2m) d^2/dq^2

Where d/dq is the derivative with respect to position. Using this, we can rewrite the time development operator as:

T(t,0) = Exp(-iHt/hbar)

Substituting this into our expression for the matrix elements in the momentum representation, we get:

<p'|T|p''> = (1/(2*pi*hbar))^2 [(m/2i*pi*hbar t)^1/2] Exp[i(p'-p'')^2t/2m hbar] Exp[-iHt/hbar]

Using the position representation of the momentum states, we can rewrite this as:

<p'|T|p''> = (1/(2*pi*hbar))^2 [(m/2i*pi*hbar t)^1/2] Exp[i(p'-p'')^2t/2m hbar] Exp[-i(-hbar^2/2m) d^2/dq^2 t/hbar]

Finally, we can use the identity operator in the position representation, <q'|q''> = delta(q'-q''), to get the matrix elements in the position representation:

<q'|T|q''> = (1/(2*pi*hbar))^2 [(m/2i*pi*hbar t)^1/2] Exp[i(p'-p'')^2t/2m hbar] Exp[-i(-hbar^2/2m)