The hint for #1 was to express in terms of and .

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- May 9th 2008, 10:49 AM #1

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## Qm

Hey topsquark,

Huge Quantum exam tomorrow! Few practice probs I didn't understand:

1.) Suppose that an atom in a solid is described by , where is the hamilotonian, and . Also is needed for the vibrations of an atom (in an object). is a real constant. Determine the -th energy eigenvalue of , assuming 's -th eigenvalue is also the -th[/tex] energy eigenstate of .

2a.) Using quantum mechanics, describe how you'd solve the system of a hydrogen atom. What are the physical results?

b.) Use separation of variables in coordinates in order to solve the time-indep. schrodinger equation for a particle (free) that's restricted to a 2-D square box. The PE (potential) of this system is:

.

c.) Determine all energy eigenfuntions and their energy eigenvalues. Then, determine what the degeneracy of the energy state . (Note, is the lowest level).

- May 9th 2008, 12:32 PM #2

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- May 9th 2008, 12:50 PM #3

- May 9th 2008, 04:39 PM #4

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- May 9th 2008, 05:32 PM #5
I don't want to rain on anyone's parade, but the solutions to all these questions can be found as worked examples in most quantum mechanics textbooks.

For 1.), since , you could write and treat it as a one-dimensional harmonic oscillator (the detailed solution to which is given in all quantum mechanics textbooks - spanning several pages).

Read this: Quantum harmonic oscillator - Wikipedia, the free encyclopedia

Or this: Quantum mechanics/harmonic oscillator/operator method - Physics

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For 2b.), you have a particle confined to a two-dimensional infinite square well. Since for and , the Schroedinger equation is:

.

Read this: Square Wells p.7

Or this: Particle in a box - Wikipedia, the free encyclopedia

- May 9th 2008, 07:04 PM #6

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- May 9th 2008, 08:04 PM #7

- May 10th 2008, 03:45 AM #8

- May 10th 2008, 03:49 AM #9
This is far too complicated a task to write down in a (single) post. The plan of attack:

The wavefunction in spherical-polar coordinates can be separated into a radial (r) and angular ( ) parts.

The radial wavefunction solution involves LaGuerre polynomials and the angular part solution deals with spherical harmonics. (The angular part of the wavefunction is also separable: the solution is merely harmonic, and the solution deals with associated Legendre polynomials.)

-Dan

- May 10th 2008, 03:53 AM #10

- May 10th 2008, 04:02 AM #11
You will find that

(where n and m denote the constants for the x and y parts of the solution.)

The problem is denoting the energy levels as E1, E2, ... in order of increasing energy. So

etc.

Carefully note some "strange" features of the energy spectrum such as

so be careful of simply using the "obvious" pattern to decide which energy level is higher than another.

-Dan