Measuring momentum using position wavefunction

[Joao Victor]

I was solving an exercise from Cohen's textbook, but then I got stuck in this question.

"Let ψ(x,y,z) = ψ(r) the normalized wave function of a particle. Express in terms of ψ(r) the probability for:

b) a measurement of the component P_x of the momentum, to yield a result included between p₁ and p₂.
c) simultaneous measurements of X and P_z to yield
x₁ ≤ x ≤ x₂
p_z ≥ 0"

I've tried to work it out, but using the wave function in position space instead of using in momentum space really got me in trouble. I do now that they are related to each other by a Fourier Transform, but the expressions in terms of ψ(x,y,z) are a mess! I hope you guys can help me soon, so I can proceed on my study.

 

[George Jones]

I was solving an exercise from Cohen's textbook, but then I got stuck in this question.

Hmm ... I can't find this exercise in the book (on my shelf) "An Introduction to Hilbert Space and Quantum Logic" by Cohen. Or maybe Canadian singer/songwriter/poet Leonard Cohen wrote a a quantum book of which I am unaware. I can find this exercise in one of my books by an author who has a completely different name. Sorry for being harsh, but it is disrespectful to get an author's name so wrong.

Can you show your attempt for b?

 

[BvU]

@George Jones : in defence of the poster: in his first post he used the full regalia

Yesterday, I was solving an exercise from Cohen-Tannoudji's book - Quantum Mechanics

I fully subscribe your request to let the OP present his attempt at solution. In the thread above he did that correctly and already had good pointers from @nrqed

It is also long overdue to say: Hello Joao, $\quad$

and: use of the template is mandatory -- see guidelines

 

[Joao Victor]

Hello!

As BvU already pointed out, the complete name of the author is Claude Cohen-Tannoudji, one of the nobel prize winners on 1997. I also appreciate you guys reception, for it is only the second thread I create here on Physics Forums.
When I tried to solve b, I wrote and expression for the probability in terms of the wave function in momentum space Φ(p), and then i wrote Φ(p) in terms of ψ(r) (since they are related by a Fourier transform). However, the computation of the probability uses the modulus squared of Φ(p), which is Φ(p)Φ^*(p). We get, then, a product of integrals, which is a real mess. Is there a way to clean it up a bit?

 

[vela]

It would really help if you showed us your actual work rather than simply describing what you did because often the problem arises in the details of your work.

 

[Joao Victor]

Here is what I did for b.

We can express this probability using the wave function Φ(p) in momentum space. It is straightfoward - Px has to be between p₁ and p₂, while Py and Pz can be anything. So:

Prob = ∫∫∫ Φ(p) Φ^*(p) dp_x dp_y dp_z, where the last integral (in p_x) is from p₁ to p₂ and the other two integrals are from -∞ to ∞.

Now, we have to express that in terms of the wave function in position space: Φ(p) is related to ψ(r) by a Fourier Transform:

Φ(p) = (2πħ)^-3/2 ∫∫∫ e^-ip.r/ħ ψ(r) dxdydz, with the limits of integration of the three integrals being from -∞ to ∞.

Similarly, we can write an expression for Φ^*(p):

Φ^*(p) = (2πħ)^-3/2 ∫∫∫ e^ip.r/ħ ψ^*(r) dxdydz.

If we substitute this expressions on the equation for Prob, we get:

Prob = (2πħ)^-3 ∫∫∫ [(∫∫∫ e^-ip.r/ħ ψ(r) dxdydz) * (∫∫∫ e^ip.r/ħ ψ^*(r) dxdydz)] dp_x dp_y dp_z.

I'm sorry for the terrible format above. The equation for Prob is really a mess. Is there a way to "clean it up"?

 

[vela]

It's worth spending a little bit of time to learn LaTeX. PF has a primer here: https://www.physicsforums.com/help/latexhelp/.

Your work looks good so far, except you shouldn't be using $\vec{r}$ in both Fourier transform integrals in your expression for the probability. One should be $\vec{r}'$.