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 harmonic oscillator that might be a better model for real diatomic than the standard harmonic oscillator. On the right side of x = 0, the Hamiltonian is exactly the same as a harmonic oscillator Hamiltonian. The hard wall at x = 0, however, introduces a boundary condition. Use what you have learned about both the ordinary harmonic oscillator and particle in a box boundary conditions to answer the following questions.
a) what does this boundary condition require the wave function to do at x = 0?
b)
c)
d)
e)

Following is the text part of the solution. Please see the attached file for complete solution. Equations, diagrams, graphs and special characters will not appear correctly here. Thank you for using Brainmass.
===============================================================================

a. Boundary condition is that at x = 0, Wave function ψ(x) = ψ(0) = ...

Solution Summary

I have provided solution to this problem without the use of tedious mathematical formula. Solution is very well presented with explanations.

A particle of mass m is subject to the one-dimensional harmonicoscillator potential. Write down the first three normalised eigenfunctions ?_n (x) andthe 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 10-g particle is undergoing simple harmonic motion with an amplitude of 2.0 x 10^-3 m and a maximum acceleration of magnitude 8.0 x 10^-3 m/s^2. The phase angle is -pi/3 rad.
a) Write and equation for the force on theparticle as a function of time.
b) What is the period of the motion?
c) What is the maximum speed of the p

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

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 andtheir degeneracies?
(b) if a uniform electic field is applied, what are the new energy levels andtheir degeneracies?
******(c)if a uniform magnetic field is applied, what are th

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; andthe 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

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

Write the Lagrangian for a one-dimensional particle moving along the x-axis and subject to a force F=-kx (with k possitive). Find the lagrange eqn of the motion and solve it.
lagrange eqn 0=mX+Kx where X is the second erv with respect to T

Consider theharmonicoscillator, 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