Probability flux integrated over all space is mean momentum?

In summary, the conversation discusses the relationship between probability flux and mean momentum in Sakurai Modern Quantum Mechanics. The expectation value of momentum is obtained through an integral involving the operator for momentum and a partial integration trick. This leads to the equation (2.192) in the book. The conversation concludes with the speaker expressing gratitude for the explanation and acknowledging that they had missed the integration by parts trick.
  • #1
euphoricrhino
22
6
In Sakurai Modern Quantum Mechanics, I came across a statement which says probabiliy flux integrated over all space is just the mean momentum (eq 2.192 below). I was wondering if anybody can help me explain how this is obtained.
I can see that ##i\hbar\nabla## is taken as the ##\mathbf{p}## operator, but I don't see how the integration gives the mean of ##\mathbf{p}##.
Thanks in advance!

Screen Shot 2022-08-11 at 15.28.15.png
 
  • Like
Likes gentzen
Physics news on Phys.org
  • #2
The expectation value of the momentum is
$$\langle \vec{p} \rangle = \langle \psi|\hat{\vec{p}}|\psi \rangle = \int_{\mathbb{R}^3} \mathrm{d}^3 x \psi^*(t,\vec{x}) (-\mathrm{i} \hbar \vec{\nabla}) \psi(t,\vec{x}).$$
Now you can add the same expression with the ##\nabla## put to ##\psi^*## by partial integration and divide by 2:
$$\langle \vec{p} \rangle = \frac{1}{2} \int_{\mathbb{R}^3} \mathrm{d}^3 x (-\mathrm{i} \hbar) [\psi^*(t,\vec{x}) \vec{\nabla} \psi(t,\vec{x}) - \psi(t,\vec{x}) \vec{\nabla} \psi^*(t,\vec{x})].$$
Comparing this with Eq. (2.191) of the book you get immediately Eq. (2.192).
 
  • Like
Likes Lord Jestocost, gentzen, PeroK and 1 other person
  • #3
Great, thanks a lot.
I missed the integration by part trick. This is awesome!
 

Similar threads

  • Quantum Physics
2
Replies
56
Views
3K
Replies
13
Views
2K
Replies
6
Views
635
Replies
153
Views
13K
Replies
11
Views
1K
  • Quantum Physics
Replies
3
Views
2K
  • Electromagnetism
Replies
5
Views
4K
  • Quantum Physics
Replies
3
Views
1K
  • Quantum Physics
Replies
5
Views
2K
  • Differential Geometry
Replies
10
Views
2K
Back
Top