Classical limit of atomic motion

[mimi1234]

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!

 

[mark726]

Hello, thank you for reaching out for help with this exercise. I am happy to assist you. Let's break down each part of the exercise and address them one by one.

(1a) In this part, you are being asked to recall Ehrenfest's theorem and state the conditions for classicality of the trajectory of a quantum particle. Ehrenfest's theorem is a mathematical expression that relates the time derivative of the expectation value of an observable to the expectation value of the commutator of the Hamiltonian and the observable. The conditions for classicality of the trajectory of a quantum particle are that the particle's position and momentum must be well-defined and that the potential energy experienced by the particle can be approximated by a classical potential.

(1b) This part of the exercise asks you to consider a specific scenario involving an atom described by a wavepacket with a certain variance in position and momentum, and the scattering of light from an electromagnetic wave. The conditions for classicality in this situation are that the wavelength of the light must be much larger than the size of the atom, and the energy of the atom must be much smaller than the energy of the photon.

(1c) In this part, you are asked to consider the recoil velocity of an atom in a specific scenario and its relation to the thermal velocity from a 600K oven. In the classical limit, the recoil velocity of the atom is much smaller than the thermal velocity, meaning that the atom's motion can be considered classical and the effects of quantum mechanics can be neglected.

(1d) The result of this exercise is that in certain conditions, the behavior of a quantum particle can be approximated by classical mechanics, meaning that the particle's trajectory can be described by Newton's laws of motion. This is important because it allows us to better understand and predict the behavior of particles in different scenarios.

I hope this helps clarify the exercise for you. If you have any further questions, please don't hesitate to ask. Good luck!