Question

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.

Answer #1

y

a)
For a 1D linear harmonic oscillator find the first order
corrections to the ground state due to the Gaussian perturbation.
b) Find the first order corrections to the first excited
state.
Please show all work.

Using perturbation theory, find the energy of two coupled harmonic
oscillators.

Consider a system of three non-interacting particles confined
by a one-dimensional harmonic oscillator potential and in thermal
equilibrium with a total energy of 7/2 ħw.
(a) what are the possible occupation numbers for this system
if the particles are distinquishable.
(b) what is the most probable energy for a distinquishable
picked at random from this system.

Consider a one-dimensional harmonic oscillator, in an energy
eigenstate initially (at t=t0), 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 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.
Find the average amount of work done by the force,...

Consider the three-dimensional harmonic
oscillator.
Indicate the energy of the base state if twelve
identical particles (spin 1) are placed in the system that do not
interact with each other.

A harmonic oscillator with the usual PE of V(x)= (.5)kx^2
perturbed by a small change to the spring constant k -->(1+E)k,
with E<<1.
1. Write the new energy eigenvalues, making sure any parameters
are clearly defined.
2. Expand the eigenvalue expression in a power series in E up to
the second order using a Taylor series expansion.
3. What is the perturbation Hamiltonian in the problem?

Which of the following systems is degenerate?
Question 4 options:
a)
Two-dimensional harmonic oscillator
b)
One-dimensional Infinite square well
c)
One-dimensional finite square well
d)
One-dimensional harmonic oscillator
e)
All time-dependent systems

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?

(a) Write down the energy eigenvalues for a 3-dimensional
oscillator with mass m and spring constant
kx= ky
=kz and quantum number nx,
ny and nz = 0, 1, 2, 3, 4
….
(b) Write down the degeneracy of the five lowest states of a
3-dimensional harmonic oscillator in terms of
nx, ny and
nz.
(c) Show that the number of degeneracy of a 3-dimensional
oscillator for the nth energy level is
1/2(n+1)(n+2).

Consider an electron bound in a three dimensional simple
harmonic oscillator potential in the n=1 state. Recall that the
e- has spin 1/2 and that the n=1 level of the oscillator
has l =1. Thus, there are six states {|n=1, l=1, ml,
ms} with ml= +1, 0, -1 and ms =
+/- 1/2.
- Using these states as a basis find the six states with
definite j and mj where J = L +s
- What are the energy levels...

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