## SHM p.3

We have found a length scale *l* and an energy scale
*e*, now we use these to transform Schrödinger's equation
into dimensionless form. We make a dimensionless position:
*x'*=*x/l* and a dimensionless energy:
*E'*=*E/e* and plug those into Schrödinger's equation.

The next step is to determine the large *x'* behavior
of , and then factor it out of .
The term *x'*^{2} obviously grows with *x'*;
one of the other terms must cancel it. Since *E'* is constant
it can't help, so there must be an approximate cancellation of the
first two terms for large *x'*. exp(-*x'*^{2}/2)
has the desired property: its second derivate (for large *x'*) is approximately
*x'*^{2}× itself. Thus we seek a solution
of the form: =*H(x')* exp(-*x'*^{2}/2).
(This is "without loss of generality" [wolog], i.e., for any
there is such an *H(x')*. We hope, however, that finding *H(x')*
will be easier than finding directly.)

If we now try to write *H* as a polynomial:

we can plug the polynomial form into the differential equation:

The result is a __two__ term recursion relation (i.e., the result
has just two *a*s so, for example, given *a*_{0} we can
calculate *a*_{2}, from which we can calculate *a*_{4},
etc. until we're done.) Note that the the recursion relation connects
even *k* to even *k* and odd *k* to odd *k*.
Thus the polynomials will have only even or odd terms.

It turns out that the polynomial must end if the
wavefunction is to be normalizable. One can show that the non-terminating
sum gets at least as big as exp(+*x'*^{2}), so
instead of going to zero for large *x'* gets
big like: exp(+*x'*^{2}/2). The only way the sum can end is if
the numerator of the recursion relation is zero, i.e., if *E'*=1+2*n*
then *a*_{n+2}=0 (and hence all further *a*s are zero).

Note that if we had tried for a polynomial solution for
itself (rather than factoring out the exp(-*x'*^{2}/2) first),
we would get an (unsolvable) __three__ term recursion relation.

The polynomials we have found are called Hermite polynomials
(see for example, Abramowitz & Stegun, §22 p. 771 or
Szegö Ch.V).
Below is a plot of a Hermite polynomial
(H_{10}(*x*)=-30240 + 302400 *x*^{2} -
403200 *x*^{4} + 161280 *x*^{6} -
23040 *x*^{8} + 1024 *x*^{10})
)
from *x*=-4 to 4 and *x*=-2 to 2

Note that *H*_{10}(*x*) is an even function with
ten real roots, and like any polynomial it becomes huge for
large |*x*|.

The corresponding position probability density and wavefunction
(i.e., with the exp(-*x'*^{2}/2) factor)
are plotted below.

Notice that we have a *discrete* set of energy levels, i.e.,
energy is quantized and the particle can have only the above energies.
Quantum mechanics says energies between these levels will not be seen
in nature. A discrete set of levels is expected if the
particle is confined to a region.

Notice that the energy levels are exactly evenly spaced.
(Energy *spacing* (i.e., differences)
are quite important as (1) that is what is observed in emitted photon
energies and (2) that is what is related to the classical period.)

Next