# Measuring momentum using position wavefunction

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 Px of the momentum, to yield a result included between p1 and p2.
c) simultaneous measurements of X and Pz to yield
x1 ≤ x ≤ x2
pz ≥ 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.

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George Jones
Staff Emeritus
Gold Member
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
Homework Helper
2019 Award
@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

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
Staff Emeritus
Homework Helper
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.

BvU
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) dpx dpy dpz, where the last integral (in px) 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 ∫∫∫ eip.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) * (∫∫∫ eip.r/ħ ψ*(r) dxdydz)] dpx dpy dpz.

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
Staff Emeritus
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 $\vec{r}'$.