Explaining the Interaction of a Time-Dependent Force and an Oscillator

[bon]

Homework Statement

A harmonic oscillator starts in its ground state (n=0) at t=-infinity. A perturbation H = -xF(t) is applied between t= -infinity and t = T.

(a) by considering the corresponding classical interaction explain why this represents the application of a time dependent force F(t) to the oscillator.

Homework Equations

The Attempt at a Solution

Not sure what to say other than that in classical mechanics wd = force x distance moved. How do i explain this. thanks.!

 

[ideasrule]

Do you know Hamilton's equations? Force is equal to the time derivative of momentum, so try to show that according to Hamilton's equations, the time derivative of momentum is equal to F(t).

 

[bon]

Sorry haven't done these equations and can't see how it is meant to work :(

 

[bon]

Ok well I've given up on that part, anyone else able to help me on it?

Stuck on the next part now:

It says: calculate the ket at time T correct to first order in the perturbation. Why does the first order correction involve only the state n=1?

So basically i can't see why it would include only n=1.
using the standard formula for time dependent perturbations, the amplitudes

am = 1/(i hbar) times the integral from -infinity to T of e^i(Em-E0)t/hbar times <Em|-x|E0> F(t) dt

But I can't see why that matrix element would equal 0 unless n=1? Any ideas?

Hard question to explain online but I hope you understand what I am asking..

Thanks

 

[Faulks]

BUMP

I need help on the exact same question. Did you figure it out in the end? Or does anyone have any new ideas?

 

[jsen]

I think it's because <Em|-x|E0> only non zero when m=1.

If you rewrite x in terms of the ladder operators of the harmonic oscillator the E0 ket goes to E1.

but then that's m, not n... change notation?

 

[grey_earl]

Yes, it doesn't matter how the states are named. m, n, superfragilistic, ... what's important is that it is the first state higher in energy than E_0.