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On 09/15/2012 02:55 PM, Kaitlyn.M.Logan wrote:
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cite="mid:CA765D0D95A04D449667AFA14377899C5560D746D6@MERCERMAIL.MercerU.local"
type="cite">
<pre wrap="">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
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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.<br>
<br>
<ol>
<li>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)</li>
<li>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.</li>
<li>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?</li>
</ol>
<p>Hope that helps...<br>
</p>
<br>
<pre class="moz-signature" cols="72">--
Andrew J. Pounds, Ph.D. (<a class="moz-txt-link-abbreviated" href="mailto:pounds@theochem.mercer.edu">pounds@theochem.mercer.edu</a>)
Associate Professor of Chemistry and Computer Science
Mercer University, Macon, GA 31207 (478) 301-5627
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