# Classical limit of atomic motion

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1. Apr 28, 2017

### mimi1234

• Moved from a technical forum, so homework template missing
Hello, I need same help with the following exercise:
(1a)Recall Ehrenfest’s theorem and state the conditions for classicality of the trajectory of a quantum particle.
(1b) Consider an atom whose state is described by a wavepacket with variance ∆x^2 in position and ∆p^2 in momentum. The atom scatters light from an electromagnetic wave of wavelength λ. State the conditions under which, in this situation, the atom can be treated classically.
(1c) As we have seen in question (2a) of Sheet 4, the recoil velocity is much smaller than the thermal velocity from a 600K oven. In the classical limit considered before, what can we conclude about the relation between the ∆v of the atom and the recoil velocity due to the scattering process?
(1d) Interpret the result.
(1a):
The Ehrenfest's theorem is
$${\frac {d}{dt}}\langle O\rangle ={\frac {i}{\hbar }}\langle [H,O]\rangle +\left\langle {\frac {\partial O}{\partial t}}\right\rangle$$right?
For a particle with${\frac {\partial p}{\partial t}}=0$ and${\frac {\partial x}{\partial t}}=0$ we have:
$m{\frac {d}{dt}}\langle x\rangle =\langle p\rangle \ ,\quad {\frac {d}{dt}}\langle p\rangle =-\langle \nabla V(x)\rangle$ and hence:
$m{\frac {d^{2}}{dt^{2}}}\langle x\rangle =-\langle \nabla V(x)\rangle =\langle F(x)\rangle$
This equation is equivalent to the classical equation, if $\langle F(x)\rangle \approx F(\langle x\rangle )$

I don't know what I'm supposed to do in (1a) ...do I have to insert the radiationforce $F_{rad}= \frac{d N}{d t}\frac{2h}{\lambda}$?
So the velocity would be ${\frac {d}{dt}}\langle x\rangle=\frac{2Nh}{\lambda m}+p_0/m$?

Thanks!

2. May 3, 2017

### PF_Help_Bot

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