Question

- Consider a one-dimensional harmonic oscillator, in an energy
eigenstate initially (at t=t
_{0}), to which we apply a time dependent force F(t).- Write the Heisenberg equations of motion for x and for p.
- Now suppose F is a constant from time t
_{0}to time t_{0}+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. - 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?)
- Discuss the difference between the classical and quantum cases.

Answer #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**

,

Quantum mechanics:
Consider a particle initially in the ground state of the
one-dimensional simple harmonic oscillator. A uniform electric
field is abruptly turned on for a time t and then abruptly turned
off again. What is the probability of transition to the first
excited state?

Quantum mechanics problem: Consider a particle
initially in the ground state of the one-dimensional simple
harmonic oscillator. A uniform electric field is abruptly turned on
for a time t and then abruptly turned off again. What is the
probability of transition to the first excited state?

Consider a one dimensional Harmonic oscillator. Use perturbation
theory to find the energy corrections up to second order in the
perturbative parameter ? for a perturbative potential of the
kind:
a) V = ?x
b) V = ?x^3.

Consider the driven damped harmonic oscillator
m(d^2x/dt^2)+b(dx/dt)+kx = F(t)
with driving force F(t) = FoSin(wt).
Consider the overdamped case
(b/2m)^2 < k/m
a. Find the steady state solution.
b. Find the solution with initial conditions x(0)=0,
x'(0)=0.
c. Use a plotting program to plot your solution for
m=1, k=0.1, b=1, Fo=0.25, and w=0.5.

Classical Mechanics -
Let us consider the following kinetic (T) and potential (U)
energies of a two-dimensional oscillator :
?(?,̇ ?̇)= ?/2 (?̇²+ ?̇²)
?(?,?)= ?/2 (?²+?² )+???
where x and y denote, respectively, the cartesian displacements
of the oscillator; ?̇= ??/?? and ?̇= ??/?? the time derivatives of
the displacements; m the mass of the oscillator; K the stiffness
constant of the oscillator; A is the coupling constant.
1) Using the following coordinate transformations,
?= 1/√2 (?+?)
?= 1/√2...

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