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

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Homework Help Overview

The discussion revolves around a harmonic oscillator subjected to a time-dependent force represented by a perturbation. The original poster seeks to understand the classical interpretation of this force and its implications for the oscillator's state transitions.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • The original poster attempts to relate classical mechanics to the quantum scenario but expresses uncertainty about the explanation. Some participants suggest using Hamilton's equations to clarify the relationship between force and momentum. Others question the reasoning behind the first-order correction involving only the state n=1, with references to matrix elements and ladder operators.

Discussion Status

The discussion is ongoing, with participants exploring different interpretations of the perturbation's effects on the oscillator's states. Some guidance has been offered regarding the use of Hamilton's equations and the properties of matrix elements, but no consensus has been reached on the reasoning behind the specific state transitions.

Contextual Notes

Participants note the complexity of the problem and the challenges in conveying the concepts online. There is an indication of imposed homework constraints, as well as a focus on the mathematical formalism involved in perturbation theory.

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

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

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