- #1
user3
- 59
- 0
I am working on the Hydrogen atom and I was trying to calculate [tex]\frac{d<r>}{dt}[/tex] using [tex]\frac{d<r>}{dt} = \frac{i}{\hbar} <[\hat{H} , \hat{r}]>[/tex]. Here [tex]r = \sqrt(x^2 + y^2 + z^2)[/tex] and [tex]H = \frac{p^2}{2m} + V[/tex] where [tex]p^2 = -\hbar^2 \nabla^2 [/tex]. Now according to Ehrenfest's theorem <r> should behave classically and give me some equivalent of velocity, and indeed I do get something but it does't resemble velocity: [tex]\frac{-\hbar^2}{2m} (2\nabla r \nabla f + f \nabla^2 r)[/tex]
where f is a test function.
Steps:
[tex][H ,r]f = [\frac{p^2}{2m} + V , r]f = \frac{p^2(rf)}{2m} + Vrf - \frac{rp^2(f)}{2m} - rVf = \frac{1}{2m}[p^2 ,r]f = \frac{1}{2m}[-\hbar^2\nabla^2 , r]f = \frac{-\hbar^2}{2m}[\nabla^2 , r]f [/tex] [tex]= \frac{-\hbar^2}{2m} (\nabla^2(rf) - r\nabla^2(f)) = \frac{-\hbar^2}{2m} (\nabla r\nabla f + r\nabla^2f + \nabla f \nabla r + f\nabla^2 r - r\nabla^2f) = \frac{-\hbar^2}{2m} (2\nabla r \nabla f + f \nabla^2 r) [/tex]
Am I doing something wrong?
where f is a test function.
Steps:
[tex][H ,r]f = [\frac{p^2}{2m} + V , r]f = \frac{p^2(rf)}{2m} + Vrf - \frac{rp^2(f)}{2m} - rVf = \frac{1}{2m}[p^2 ,r]f = \frac{1}{2m}[-\hbar^2\nabla^2 , r]f = \frac{-\hbar^2}{2m}[\nabla^2 , r]f [/tex] [tex]= \frac{-\hbar^2}{2m} (\nabla^2(rf) - r\nabla^2(f)) = \frac{-\hbar^2}{2m} (\nabla r\nabla f + r\nabla^2f + \nabla f \nabla r + f\nabla^2 r - r\nabla^2f) = \frac{-\hbar^2}{2m} (2\nabla r \nabla f + f \nabla^2 r) [/tex]
Am I doing something wrong?
Last edited: