Prof. Axon's full lecture notes are available in PDF format.
The main ideas for today are
We'll also look at the homework assigned for next week.
Review of the particle in a 1-D box
Recall that a particle trapped in an 1-D infinite potential well has a wave function which is quantized
and energy levels which are likewise quantized
Moving to two dimensions
Solving the Schroedinger solution for this 2-D box is not very hard. We can GUESS that the solution will be separable in the variables x and y.
When we plug a trial function of this form into the Schroedinger equation, we find that -- thank goodness! -- it produces a valid solution; and one that looks familiar, too.
The energy of the particle is again quantized
If the box is a square -- or SYMMETRIC -- then the expression for energy simplifies to
Let's look at the energy of the first few states, as both quantum numbers vary between 1 and 3. For convenience, we'll abbreviate this base energy as Q:
nx ny Energy
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1 1 (1 + 1) = 2 Q
2 1 (4 + 1) = 5 Q
1 2 (1 + 4) = 5 Q
2 2 (4 + 4) = 8 Q
3 1 (9 + 1) = 10 Q
1 3 (1 + 9) = 10 Q
3 2 (9 + 4) = 13 Q
2 3 (4 + 9) = 13 Q
3 3 (9 + 9) = 18 Q
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Note that some of these combinations yield the same energy. We call those levels degenerate in energy. If we measure the energy of the particle in this box and find that it has a value of 10 Q, we know that in one direction, the wavefunction has 3 wiggles, and in the other direction, it has 1 wiggle; but we can't say for sure whether the wave with 3 wiggles runs in the X-direction or the Y-direction.
Analogy: the cardboard square
What happens if the box is not a perfect square? If one side is even slightly longer than the other, then we say the symmetry is broken. Suppose, for example, that
In this case, we need to go back to the general expression to calculate the energy of any state.
Again, for convenience, abbreviate the base energy
nx ny Energy
-------------------------------------------------
1 1
2 1
1 2
2 2
3 1
1 3
3 2
2 3
3 3
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When we broke the symmetry of the box, we also lift the degeneracy of the energy levels.
Analogy: cardboard rectangle.
Moving to three dimensions
We can again break the energy degeneracy by making the box a different length in each direction.
So, we again see that degeneracy in energy is related to symmetry.
