We now complete the series solution for the eigenvalue
equation by solving (15) for the 
, reconstructing the
and, finally, producing the wavefunctions
.
Equation (15) is solved by rewriting it as a linear recursion relation,
Applying (16) recursively for 
in
sequence, we find
Note that in this case, the even and odd terms form independent
sequences so that we may write 
as the sum of an even part 
and an odd part 
,
where 
and 
are even and odd functions of X
respectively. Note that the functions 
and 
may be
generated according to the above formulas for any value of 
, the
parameter describing the energy.
At last our solution is 
. At
this point is seems we
have identified two linearly independent solutions, 
(even) and 
(odd) for every (continuous) value of the
energy parameter 
. We know that all of these
solutions which we have found cannot represent pure energy states of
the SHO. First, all SHO
states are bound (
) and bound states
exhibit a discrete, not continuous, spectrum. Second, we expect (in
one dimension) only a single physical state for each allowed
energy, but here we appear to have found two. There must therefore be
a physical reason for rejecting many of the solutions we have found.
The physical reason for rejecting some of these solutions is
clearly illustrated when we look at the two solutions corresponding to
the ground state of the harmonic oscillator,
, or 
. From the
series solution we have
A plot of these two functions is provided in Figure 3.3.
While the even solution, 
, is well behaved and decays into
the forbidden regions at |X|>1 while curving away from the x-axis,
the odd solution enters the forbidden region with too great of a
slope and thus begins to grow exponentially as
. As in the case of the square well,
because this function is not be normalizable, we reject it as
unphysical.
We could have anticipated this behavior analytically with out
resorting to a plot. The key observation is that only when the
series for 
``terminates'' and the coefficients became zero
after some point leaving only a polynomial, then the wave function
drops sufficiently rapidly for large X
that it is normalizable. If the series does not terminate, the
function 
grows so rapidly with X that even after
multiplication by 
the wavefunction cannot be normalized.
In the next subsection we will proof that this is so. For now, we
will take this intuitively reasonable result as given.
The form of the recursion (16) is such that the series
terminates only for 
where n is a nonnegative integer. If
is not an integer, then neither 
nor 
will
terminate, and neither the even or odd solution is normalizable or
physical. Thus we see that 
must be an integer. Moreover, if
is an even (odd) integer, the series for 
(
)
will terminate at the point where 
but the series for 
(
) will not terminate because 
(
) contains
terms only for odd (even) n. If 
is even (odd), the only
allowable solution will be the finite polynomial 
(
).
Up to an overall normalization constant, the Hermite polynomials
are defined as these terminated 
and 
. Because
(8) is a linear equation, it does not determine the
overall normalization of the Hermite polynomials. By convention, this
normalization is set so that the highest order term in 
is
. Using this definition and the recursion
(16), we find the following for the first few Hermite
polynomials,
To sum up, we thus have,
where, the 
are normalization constants defined so that 
, which
can be show to
leave them with the values,
Note that these solutions also obey the rule that the 
excited
state is even or odd according to the value of n and has n nodes.
