(We can't have any a_1 y term, because then it wouldn't be purely even. I put a prime on a_0 because it's a different a_0 from the one that appeared in the ground state wavefunction)
Now this can work, i.e. it can solve the TISE, but only if that last
term vanishes, AND
.
If the last term vanishes, then
(Again, there's one overall normalization constant still to be determined) And so on! The math gets more and more painful, but each time, we discover a new possible eigenfunction, and corresponding eigenvalue.
Note that
is marching steadily upwards (1, 3, 5, and this pattern does indeed continue).
Alpha is proportional to energy, so we are finding progressively higher and
higher energy solutions.
Let's put this all together, and start studying our solutions:
The Hermite polynomials are just what we've seen (derived!) above.
H_n(y) is a polynomial of order n. It is even for even n, and odd for odd n. They are not especially intuitive, beyond that. You really have to work them out, as we have. Here are the first few, with purely conventional normalizations:
The constant in the front of any of the wave functions can be found by
normalization (it's a bit of a pain, and NOT obvious!):
(Ack!)
Note that I've changed notation a bit - just to make the power series "n" match up with the "n" of the wave function, I'm now starting at n=0. (Before, my ground state was E_1, now it's E_0. This is just a label, it doesn't matter)
So our qualitative picture has now come out mathematically!
A Gaussian wave function (with non zero "zero point energy") It does indeed fall off "sort of exponentially", only faster! (Not Exp[-x/a], but Exp[-x^2/2a^2]) Near the origin, however, it's quite flat!
Here, odd parity, slightly wider than the first (the "x" out front is what does both of these things)
You can calculate where these functions have a sign flip in the second derivative (take 2 derivs, set it to zero, solve for x), and you will discover it's precisely where it should be -- the classical turning point! You don't have to make any special separation of the wave function at any point, everything is totally smooth and continuous.
Once again, we have found discrete allowed energies, and a zero point energy. (All of which is decidedly non-classical!)
Notice that the energy difference between the E_n's is just
,
a constant. This is quite different from the hydrogen atom (where E_n ~ 1/n^2,
i.e. the levels get closer and closer), and also from the rigid box (where E_n
~ n^2, i.e. the levels get farther and farther apart). So the harmonic
oscillator is "just so", not as rigid as a box, nor as soft as an atom. This
has interesting consequences in experiments, as we shall see.
Finally, we should ask about the correspondence principle. What happens as n -> large? That math is a bit hairy, but here's the result:
A is the classical turning point (E_n = (1/2) C A^2), and the solid line is the classical probability for a pendulum (or spring) with amplitude A, which you derived in a homework earlier. It goes slow at x=+/- A, so spends lots of time there, that's why the probability gets larger out there.
If you average the wildly wiggling quantum probability over any kind of small sized region of x, the average will agree well with the classical probability. So indeed, correspondence works fine (and gets better for large n, 10 is not really enough to work well right at the very edges...)
Next: Application of Harmonic Oscillators:
Vibrations of diatomic molecules.
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