Question

1. Consider a one-dimensional harmonic oscillator, in an energy eigenstate initially (at t=t₀), to which we apply a time dependent force F(t).
   1. Write the Heisenberg equations of motion for x and for p.
   2. Now suppose F is a constant from time t₀ to time t₀+τ(tau), and zero the rest of the time. Find the average position of the oscillator <x(t)> as a function of time, after the force is switched off.
   3. Find the average amount of work done by the force, and discuss the limit ωτ <<1. (Hint: Why does the average work vanish if the τ is an integral number of periods?)
   4. Discuss the difference between the classical and quantum cases.

Answer 1

1. The harmonic oscillator is described by position x(t) and momentum p(t) .And energy E of a particle with position x and momentum p is

  

Here  is a constant with inverse of time,is related to the period of oscillation T by

  

The classical harmonic oscillator aries when a mass m is free to move along x axis is attached a spring with constant k.The restoring force F= -k x ,the angular frequency

And the system define in terms of position and momentum operators

=ih1

,