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Hydrogen Atom
A hydrogen atom consists of an electron, of charge and mass ,
and a proton, of charge and mass , moving in the Coulomb
potential

(661) 
where is the position vector of the electron with respect to the
proton. Now, according to the analysis in Sect. 6.4, this twobody
problem can be converted into an equivalent onebody problem. In the
latter problem, a particle of mass

(662) 
moves in the central potential

(663) 
Note, however, that since
the difference
between and is very small. Hence, in the following,
we shall write neglect this difference entirely.
Writing the wavefunction in the usual form,

(664) 
it follows from Sect. 9.2 that the radial function satisfies

(665) 
Let , with

(666) 
where and are defined in Eqs. (678) and (679),
respectively.
Here, it is assumed that , since we are only interested in boundstates of the hydrogen atom. The above differential equation transforms
to

(667) 
where

(668) 
Suppose that
. It follows that

(669) 
We now need to solve the above differential equation in the domain to , subject to the constraint that be squareintegrable.
Let us look for a powerlaw solution of the form

(670) 
Substituting this solution into Eq. (669), we obtain

(671) 
Equating the coefficients of gives the recursion relation

(672) 
Now, the power series (670) must terminate at small , at
some positive value of , otherwise
behaves unphysically as
[i.e., it yields an that is not squareintegrable
as
]. From the above recursion relation, this is only possible if
, where the first term in the series is
. There are two possibilities:
or . However, the former possibility predicts unphysical behaviour of
at . Thus, we conclude that .
Note that, since
at small , there is a finite
probability of finding the electron at the nucleus for an state, whereas
there is zero probability of finding the electron at the nucleus for
an state [i.e., at , except when
].
For large values of , the ratio of successive coefficients in the power series (670)
is

(673) 
according to Eq. (672). This is the same as the ratio
of successive coefficients in the power series

(674) 
which converges to . We conclude that
as
. It thus follows that
as
. This does not correspond to physically acceptable behaviour of the wavefunction, since
must be finite.
The only way in which we can avoid this unphysical behaviour is if the
power series (670) terminates at some maximum value of . According to
the recursion relation (672), this is only possible if

(675) 
where is an integer, and the last term in the series is . Since the
first term in the series is
, it follows that must
be greater than , otherwise there are no terms in the series at all.
Finally, it is clear from Eqs. (666), (668), and (675) that

(676) 
and

(677) 
where

(678) 
and

(679) 
Here, is the energy of socalled groundstate (or lowest energy state) of the
hydrogen atom, and the length is known as the Bohr radius.
Note that
, where
is the dimensionless finestructure constant. The fact that
is the ultimate justification for our nonrelativistic treatment of the hydrogen atom.
We conclude that the wavefunction of a hydrogen atom takes the form

(680) 
Here, the
are the spherical harmonics (see Sect 8.7), and
is the solution of

(681) 
which varies as at small .
Furthermore, the quantum numbers , , and can only take values
which satisfy the inequality

(682) 
where is a positive integer, a nonnegative integer, and an integer.
Now, we expect the stationary states of the hydrogen atom to be orthonormal: i.e.,

(683) 
where is a volume element, and the integral is over all space. Of
course,
, where is an element
of solid angle. Moreover, we already know that the spherical
harmonics are orthonormal [see Eq. (615)]: i.e.,

(684) 
It, thus, follows that the radial wavefunction satisfies the
orthonormality constraint

(685) 
The first few radial wavefunctions for the hydrogen atom are listed below:



(686) 



(687) 



(688) 



(689) 



(690) 



(691) 
These functions are illustrated in Figs. 21 and 22.
Figure 21:
The
plotted as a functions of . The solid, shortdashed, and longdashed curves correspond to
, and , and , respectively.

Figure 22:
The
plotted as a functions of . The solid, shortdashed, and longdashed curves correspond to
, and , and , respectively.

Given the (properly normalized) hydrogen wavefunction (680),
plus our interpretation of as a probability density, we can calculate

(692) 
where the anglebrackets denote an expectation value.
For instance, it can be demonstrated (after much tedious algebra) that



(693) 



(694) 



(695) 



(696) 



(697) 
According to Eq. (676), the energy levels of the boundstates of a hydrogen atom
only depend on the radial quantum number . It turns out that this is a special
property of a potential. For a general central potential, , the
quantized energy levels of a boundstate depend on both and (see Sect. 9.3).
The fact that the energy levels of a hydrogen atom only depend on ,
and not on and , implies that the energy spectrum of a hydrogen
atom is highly degenerate: i.e., there are many different
states which possess the same energy. According to the inequality
(682) (and the fact that , , and are integers), for
a given value of , there are different allowed values of
(i.e.,
). Likewise, for a given value of ,
there are different allowed values of (i.e.,
). Now,
all states possessing the same value of have the same energy (i.e., they are degenerate). Hence, the total number of
degenerate states corresponding to a given value of is

(698) 
Thus, the groundstate () is not degenerate, the first excited
state () is fourfold degenerate, the second excited state
() is ninefold degenerate, etc. [Actually, when we take
into account the two spin states of an electron (see Sect. 10),
the degeneracy of the th energy level becomes .]
Next: Rydberg Formula
Up: Central Potentials
Previous: Infinite Spherical Potential Well
Richard Fitzpatrick
20100720