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The half harmonic oscillator

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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.

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Solution Preview

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

Therefore:

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 ...

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