[CHM 331] question 9.54

[Andrew J. Pounds]

On 09/15/2012 02:55 PM, Kaitlyn.M.Logan wrote:
> HI Dr. Pounds
> Question 9.54 asks us to find the quantum energy levels for a three dimensional harmonic oscillator with a potential energy function. I have no idea where to even begin. We never went through the solution to the Scrodinger equation for a 1D harmonic oscillator, we were just given the solutions, so I am not sure how to find the solutions for 3D. Also, once we figure out the 3D energy functions, do we just add the potential energy to it?
> -Katie Logan
You are correct -- we never did a 3D harmonic oscillator in class -- but 
we did go through the 1D case for the harmonic oscillator and also 
discussed the 1D, 2D, and 3D particle in a box.   I want to address 
several things about your questions.

 1. When we set up and solved the 1D harmonic oscillator we included the
    potential function in the Schrodinger equation.  When we therefore
    got an expression for the 1D eigenvalues the result included the
    potential (remember the Hamiltonian gives us the total energy --
    potential + kinetic)
 2. When we proceeded to write down the 3D particle in a box eigenvalues
    we noted how they were just summing the energies from each of the
    different dimensions.  You should take a similar approach here.
 3. When we wrote down the energy levels for a 3D PIB we recognized that
    there was a quantum number in each dimension and something tied to
    the coordinate (the length of the box) in each dimension.  In your
    3D harmonic oscillator energy expression you should also have a
    quantum number in each dimension and something tied to the
    coordinate.   Look back at equation 9.116 in your text.  Can you
    tell for the 1D case which of the variables is tied to a coordinate
    and which it tied to a quantum level?

Hope that helps...


-- 
Andrew J. Pounds, Ph.D.  (pounds at theochem.mercer.edu)
Associate Professor of Chemistry and Computer Science
Mercer University,  Macon, GA 31207   (478) 301-5627

-------------- next part --------------
An HTML attachment was scrubbed...
URL: <http://theochem.mercer.edu/pipermail/chm331/attachments/20120916/2bffd54c/attachment.html>