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.

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

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...

particle of mass m is moving in a one-dimensional potential V
(x) such that ⎧
⎨ mω2 x2 ifx>0 V (x) = 2
⎩ +∞ if x ≤ 0
(a) Consider the motion classically. What is the period of
motion in such potential and the corresponding cyclic
frequency?
(b) Consider the motion in quantum mechanics and show that the
wave functions of the levels in this potential should coincide with
some of the levels of a simple oscillator with the...

Exercise
3. Consider a particle with mass m in a
two-dimensional infinite well of length L, x, y
∈ [0, L]. There is a weak potential in the well
given by
V (x,
y) = V0L2δ(x −
x0)δ(y − y0)
.
Evaluate the first order correction to the energy of the ground
state.
Evaluate the first order corrections to the energy of the first
excited states for x0 =y0 = L/4.
For the first excited states, find the points...

1 - Write the one dimensional, time-independent Schrödinger
Wave Equation (SWE). Using the appropriate potential energy
functions for the following systems, write the complete time
independent SWE for:
(a) a particle confined to a one-dimensional infinite square
well,
(b) a one-dimensional harmonic oscillator,
(c) a particle incident on a step potential, and
(d) a particle incident on a barrier potential of finite width.
2 - Find the normalized wavefunctions and energies for the
systems in 1(a). Use these wavefunctions to...

which system does not have a zero point energy? a. particle in
one dimensional box (b). one dimensional harmonic oscillator. (c)
two particle rigid rotor d) hydrogen atom

As you know, a common example of a harmonic oscillator is a mass
attached to a spring. In this problem, we will consider a
horizontally moving block attached to a spring. Note that,
since the gravitational potential energy is not changing in this
case, it can be excluded from the calculations.
For such a system, the potential energy is stored in the spring
and is given by
U=12kx2,
where k is the force constant of the spring and
x is...

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