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Physics 3220
Homework 4 Solutions
Physics 3220
HW4_solutions.1
Fall 2011
1. Visualizing a possible harmonic oscillator ground state.
(5 points)
In PHYS 2170 and occasionally in class, we write the Schrödinger equation to
emphasize the curvature of the wave functions.
You can use the curvature to argue
what the shape of the wave function might be like. For example, here’s the
Schrödinger Equation for the Simple Harmonic Oscillator (SHO), written to
emphasize the curvature of the wave function:
2
22
0
21
2
dm
E
m
x
dx
Let’s see if we can use the curvature idea to get some idea of what the ground state
wave function might look like. We emphasized three special regions:
Assume that the wave function is positive valued.
First consider the regions beyond the classical turning points i.e., places where the
energy is BELOW the classical potential. Use what the Schrödinger Equation says
about the curvature (what DOES it say?) AND the fact that we must be able to
normalize the wave function (in other words how must the wave function behave as
you go to infinite distance?) to explain roughly what the wave function might look like
in these regions.
You should draw something that is positive valued, and concave up, but that also
decays toward zero as you get farther from the well. These areas show the wave
function decaying to zero, so it can be successfully integrated to infinity to get a
normalized result.
Next, (still assume a positive wave function) consider the points right at the classical
turning points. What does the Schrödinger Equation say about the curvature there?
Therefore, what shape must the function have right at the turning points?
The curvature at these turning points is zero. Therefore, the wave function must
be locally linear, or as is the case here, have an inflection point.
Third, consider the position in the middle of the well i.e., at x=0.
Again, for a
positive wave funtion, what does the Schrödinger Equation say about the curvature?
You’d have a wave function that is positive, and with downward curvature.
Now sketch a smooth symmetrical function that successfully pieces together all these
regions. What’s it look like to you?
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View Full DocumentPhysics 3220
Homework 4 Solutions
Physics 3220
HW4_solutions.2
Fall 2011
You should have pieced these things together to produce a symmetrical function
that looks very much like a Gaussian. Ta da!
2. Guessing about a possible harmonic oscillator wave function.
(25 points)
In class, we used the Schrödinger equation for the SHO to argue what the shape of
the wave function might be like. Here’s the Schrödinger Equation, written to
emphasize the curvature of the wave function:
2
22
0
21
2
dm
E
m
x
dx
We emphasized three special regions.
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