A Probability flux integrated over all space is mean momentum?

euphoricrhino
Messages
22
Reaction score
7
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
 
Physics news on Phys.org
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
Great, thanks a lot.
I missed the integration by part trick. This is awesome!
 
Not an expert in QM. AFAIK, Schrödinger's equation is quite different from the classical wave equation. The former is an equation for the dynamics of the state of a (quantum?) system, the latter is an equation for the dynamics of a (classical) degree of freedom. As a matter of fact, Schrödinger's equation is first order in time derivatives, while the classical wave equation is second order. But, AFAIK, Schrödinger's equation is a wave equation; only its interpretation makes it non-classical...
I am not sure if this falls under classical physics or quantum physics or somewhere else (so feel free to put it in the right section), but is there any micro state of the universe one can think of which if evolved under the current laws of nature, inevitably results in outcomes such as a table levitating? That example is just a random one I decided to choose but I'm really asking about any event that would seem like a "miracle" to the ordinary person (i.e. any event that doesn't seem to...

Similar threads

Back
Top