Multi-Particle QM Homework: Equations & Attempt at Solution

[binbagsss]

Homework Statement

Question attached:

Homework Equations

below

The Attempt at a Solution

I have completed this question and the needed commutator relation is that $[a(x),a^+(x')]=\delta(x-x')$

However I have it all with the integral i.e.

$\int dx_1 dx_2 a^+(x_1)a^+(x_2) \Psi (x_1,x_2) | 0>$

and

$\int dx_1 dx_2 a^+(x_1)a^+(x_2) \frac{-h^2}{2m}(\frac{\partial^2}{\partial x_1^2}+\frac{\partial^2}{\partial x_2^2} ) \Psi (x_1,x_2) |0>$ such terms etc

and am unsure of the final argument needed to explain why it holds without the integral.

Many thanks for your help.

 

[nrqed]

Homework Statement

Question attached:

View attachment 224316

Homework Equations

below

The Attempt at a Solution

I have completed this question and the needed commutator relation is that $[a(x),a^+(x')]=\delta(x-x')$

However I have it all with the integral i.e.

$\int dx_1 dx_2 a^+(x_1)a^+(x_2) \Psi (x_1,x_2) | 0>$

and

$\int dx_1 dx_2 a^+(x_1)a^+(x_2) \frac{-h^2}{2m}(\frac{\partial^2}{\partial x_1^2}+\frac{\partial^2}{\partial x_2^2} ) \Psi (x_1,x_2) |0>$ such terms etc

and am unsure of the final argument needed to explain why it holds without the integral.

Many thanks for your help.

It is hard to help without any details about your steps or the precise result of your final expression. I have the gut feeling that you applied H to Psi but that maybe you used the same coordinates in both H and Psi, which can lead to a problem. But again, more details would be helpful.