D.15 The hydrogen radial wave functions

This will be child’s play for harmonic oscillator,
{D.12}, and spherical harmonics,
{D.14}, veterans. If you replace the angular terms in
(4.33) by , and then divide the entire
equation by , you get

Since is nondimensional, all terms in this equation must
be. In particular, the ratio in the third term must be the reciprocal
of a constant with the dimensions of length; so, define the
constant to be the Bohr radius . It is convenient to also
define a correspondingly nondimensionalized radial coordinate as
. The final term in the equation must
be nondimensional too, and that means that the energy must take
the form , where is a
nondimensional energy. In terms of these scaled coordinates you get

or written out

where the primes denote derivatives with respect to .

Similar to the case of the harmonic oscillator, you must have solutions that become zero at large distances from the nucleus: gives the probability of finding the particle integrated over all possible positions, and if does not become zero sufficiently rapidly at large , this integral would become infinite, rather than one (certainty) as it should. Now the ODE above becomes for large approximately 0, which has solutions of the rough form for positive that do not have the required decay to zero. Zero scaled energy is still too much, as can be checked by solving in terms of Bessel functions, so you must have that is negative. In classical terms, the earth can only hold onto the moon since the moon’s total energy is less than the potential energy far from the earth; if it was not, the moon would escape.

Anyway, for bound states, you must have the scaled energy
negative. In that case, the solution at large takes the
approximate form . Only
the negative sign is acceptable. You can make things a lot easier for
yourself if you peek at the final solution and rewrite as
being 1 (that is not really cheating, since you are not at
this time claiming that is an integer, just a positive number.)
In that case, the acceptable exponential behavior at large distance
takes the form where .
Split off this exponential part by writing
where must remain
bounded at large . Substituting these new variables, the
ODE becomes

where the primes indicate derivatives with respect to .

If you do a power series solution of this ODE, you see that it must
start with either power or with power . The
latter is not acceptable, since it would correspond to an infinite
expectation value of energy. You could now expand the solution
further in powers of , but the problem is that tabulated
polynomials usually do not start with a power but with power zero
or one. So you would not easily recognize the polynomial you get.
Therefore it is best to split off the leading power by defining
, which turns
the ODE into

Substituting in a power series
, you get

The acceptable lowest power of is now zero. Again the series must terminate, otherwise the solution would behave as at large distance, which is unacceptable. Termination at a highest power requires that equals . Since and are integers, so must be , and since the final power is at least zero, is at least . The correct scaled energy 1 with has been obtained.

With identified, you can identify the ODE as Laguerre's associated differential equation, e.g. [40, 30.26], the -th derivative of Laguerre's differential equation, e.g. [40, 30.1], and the polynomial solutions as the associated Laguerre polynomials , e.g. [40, 30.27], the -th derivatives of the Laguerre's polynomials , e.g. [40, 30.2]. To normalize the wave function use an integral from a table book, e.g. [40, 30.46].

Putting it all together, the generic expression for hydrogen eigenfunctions
are, drums please:

Do keep in mind that different references have contradictory
definitions of the associated Laguerre polynomials. This book follows
the notations of [40, pp. 169-172], who define

In other words, is simply the -th derivative of , which certainly tends to simplify things. According to [25, p. 152], the

most nearly standardnotation defines

Combine the messy definition of the spherical harmonics (D.5) with the uncertain definition of the Laguerre polynomials in the formulae (D.8) for the hydrogen energy eigenfunctions above, and there is of course always a possibility of getting an eigenfunction wrong if you are not careful.

Sometimes the value of the wave functions at the origin is needed.
Now from the above solution (D.8), it is seen that