Consider the corresponding problem for a particle confined to the right-hand half of a harmonic-oscillator potential:
V(x) = infinity, x< 0
V(x) = (1/2)Cx^2, x >= 0
a. Compute the allowed wave function for stationary states of this system with those for a normal harmonic oscillator having the same values of m and C.
b. What are the allowed quantized energies of the half-oscillator?
c. Devise a classical mechanical system that would be a macroscopic analog of this quantum-mechanical system.
The following formula for the potential energy of a harmonic oscillator is useful to remember:
V(x) = 1/2 m omega^2 x^2
where m is the mass , and omega is the angular frequency of the oscillator. You can see that the parameters are correct by writing down the classical equation of motion:
m d^2x/dt^2 = -dV/dx ----------->
d^2x/dt^2 = omega^2 x
which has the solution x(t) = A cos(omega t + phi), so we're indeed dealing with an oscillator with angular frequency omega and mass m.
This means that the constant C in V(x) = 1/2 C x^2 is given as:
C = m omega^2
Omega = sqrt(C/m)
The energy eigenstates of the harmonic oscillator are of the form:
psi_n(x) = A_n exp[-m omega^2/(2 hbar) x^2] H_n(m omega/hbar x)
where the A_n are normalization constants and the H_n are the Hermite polynomials. The energy eigenvalues are:
E_n = (n+1/2) hbar omega
In this problem we want to solve for the energy eigenstates and eigenvalues of the half harmonic oscillator. The potential energy is the same as that of ...
We give a detailed treatment of this problem.