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I am given the 3D wave packet:

$$\psi(x,y,z)=N\,\exp\left(\frac{-(x^2+y^2+2z^2)}{2a^2}\right).$$ I was asked to find N (easy enough). Then I was asked the probability that we measure $z$ greater than 0, which I solved to be $1/2$. Then I needed to find the probability we had $x$ and $y$ negative and $z$ positive; I solved this to be $1/8$.

The last question asks to calculate the the probability that a measurement of the momentum will yield a result included in the differential volume $dp_x dp_y dp_z$ centered around the point $p_x=p_y=0,p_z=\hbar/a$.

So I am wondering if (a), my solutions to the first two questions were correct (if necessary I can edit in how I did it) and (b) how do I want to approach the momentum problem. I am unsure if I need to transform the wave into momentum space and then just take $|\psi|^2$.

EDIT:::

So I find my wave packet in momentum space, $\psi(p_x,p_y,p_z)$ to find the probability mentioned around the points above, is it just

$$\int^a_{-a}\int^b_{-b}\int^{\hbar/a + c}_{\hbar/a -c}|\psi|^2dp_zdp_ydp_x$$

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(a) Your answers are correct.

(b) Yes,it is simpler to write out the wave packet in the momentum basis. (This is effectively equivalent to working out the three-dimensional Fourier transform of the given Gaussian wave packet in position basis.)

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