Consider a 3-dimensional, spherically symmetric, isotropic harmonic oscillator with a potential energy of [see the attachment for full equation]. The Hamiltonian in this case is: [attached]

a. Use the trial function [attached] and ﬁnd the value of the parameter a that the energy, and ﬁnd that minimum energy.
b. Repeat the process for the trial function [attached] and ﬁnd the parameter b and the minimum energy.
c. The actual minimum energy turns out to be [attached] (i.e. each dimension contributes the usual [attached]). Compare the values you calculated in parts a and b to this value. Which is closer? Why do you think that is the case?

The Hamiltonian is:
(1.1)
Which can be written as:
(1.2)
The expectation value of the energy is given by:
(1.3)
Where the integration is over the entire space.
First case:
(1.4)
Applying the Hamiltonian:

(1.5)
Then:
(1.6)
(1.7)
In our case this gives:

...

Solution Summary

The expert uses the trial function and finds the value of the parameter that the energy and the minimum energy is found.

A particle of mass m is subject to the one-dimensionalharmonicoscillator potential. Write down the first three normalised eigenfunctions ?_n (x) and the corresponding eigenvalues.
Initially the wavefunction is in a mixed state of the form
?(x)=(1/(7???))^(1?2) e^(-x^2/(2?)^2 ) ((3x)^2/(?)^2 +(x/?)-(3/2)+?2)
where ?=?(??m?).

A particle with mass m is in a one dimensional simple harmonicoscillator potential. At time t=0 it is described by the state
Ï?=b|Ï?_0+c|Ï?_1
Where Ï?_0 and Ï?_1 are normalised energy eigenfunctions corresponding to energies E_0 and E_1 and b and c are real constants.
Find b and c that (x)-(expectation value) is as

A particle of mass m moves in two dimensions under the influence of the potential V(x,y)=1/2 m?^2 (((6x)^2)-2xy+(6y)^2 ). Using the rotated coordinates u=(x+y)/?2 and w=(x-y)/?2 show that the Schrödinger equation in the new coordinates (u,w) is
-(?^2)/2m ((d^2/du^2) +(d^2/dw^2))?(u,w)+V ?(u,w)?(u,w)=E?(u,w)
Where V ?(u,w) sho

2) A force Fext(t) = F0[ 1?exp(?((alpha)(t)) ] acts, for time t > 0, on an oscillator which is at rest at x=0 at time 0. The mass is m; the spring constant is k; and the damping force is ?b[(x)dot]. The parameters satisfy these relations:
b = m*q , k = 4*m*q^2 (where q is a constant with units of inverse time)

(A) A damped oscillator is described by the equation:
m [(x)ddot] = ?b [(x)dot] ? kx
What is the condition for critical damping? Assume this condition is satisfied.
(B) For t < 0 the mass is at rest at x = 0. The mass is set in motion by a sharp impulsive force at t = 0, so that the velocity is v0 at time t = 0

Please read the attached file for complete description of the problem.
Consider the particle of mass subject to a one-dimensional potential of the following form:
V(x) = 1/2 kx^2 for x>0
V(x) = + infinity for x < 0
This is a combination of the particle in a box and the harmonicoscillator that might be a better model f

Consider the harmonicoscillator, for which the general solution is
x(t) = A cos(wt) + B sin(wt)
1. Express the energy in terms of A and B and show it is time independent.
2. Choose A and B such that x(0)=x1 and x(T)=x2.
3. Write down the energy in terms of x1, x2 and T.
4. Calculate the action S for the trajectory conn

Please see the attached file for the full problems.
1. Show by integration that the first two energy eigenstates for the HarmonicOscillator are orthogonal functions.
2. For eignstate ψ_2 (x) of the harmonicoscillator, show that...

a particle of mass m and electric charge q moves in a 3D isotropic harmonicoscillator potential V=1/2kr^2
(a) what are the energy levels and their degeneracies?
(b) if a uniform electic field is applied, what are the new energy levels and their degeneracies?
******(c)if a uniform magnetic field is applied, what are th