Explore BrainMass

Explore BrainMass

    The half harmonic oscillator

    Not what you're looking for? Search our solutions OR ask your own Custom question.

    This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

    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.

    © BrainMass Inc. brainmass.com December 24, 2021, 11:04 pm ad1c9bdddf

    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


    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 ...

    Solution Summary

    We give a detailed treatment of this problem.