The Schrödinger theory of quantum mechanics extends the de Broglie
concept of matter waves by providing a formal method of treating the dynamics
of physical particles in terms of associated waves. One
expects the behavior of this wavefunction, generally called 
,
to be governed by a wave equation, which can be written
where the first term of the left represents the particle's kinetic energy, the second the particle's potential energy, and H is called the Hamiltonian of the system. Making the assertion that p and H are associated with differential operators,
this becomes
which is known as the time-dependent Schrödinger equation. For the specific case of a hydrogenic atom, the electron moves in a simple Coulomb potential, and hence the Schrödinger equation is
The solution proceeds by the method of separation of variables. First one writes the wavefunction as a product of a space component and a time component, for which the solution for the time part is easy and yields
Here E is the constant of the separation and is equal to the energy of the electron. The remaining equation for the spatial component is
and is called the time-independent Schrödinger equation. Due to the spherical symmetry of the potential, this equation is best solved in spherical polar coordinates, and hence one separates the spatial wavefunction as
The equations are more difficult but possible to solve and yield
where L is an associated Laguerre polynomial, and for convenience the
product of the angular solutions are written together in terms of a single
function, the spherical harmonic Y. With foresight the separation constants
and and l(l+1) were used. The meaning of the numbers n, l, and
will now be discussed.
The physics of the Schrödinger theory relies on the interpretation of the
wave function in terms of probabilities. Specifically, the absolute square
of the wavefunction, 
, is interpreted as the
probability density for finding the associated particle in the vicinity of
at time t. For this to make physical sense, the wavefunction
needs to be a well-behaved function of and t; that
is, should be a finite, single-valued, and continuous function.
In order to satisfy these conditions, the separation constants that appear
while solving the Schrödinger equation can only take on certain discrete
values. The upshot is, with the solution written as it is here, that the
numbers n, l, and , called quantum numbers of the electron,
can only take on particular integer values, and each of these corresponds to
the quantization of some physical quantity. The allowed values of the energy
turn out to be exactly as predicted by the Bohr theory,
The quantum number n is therefore called the principle quantum number.
To understand the significance of l and , one needs to consider the
orbital angular momentum of the electron. This is defined as
, or as an operator,
.
With proper coordinate transformations, one can write
the operators 
and the z-component of angular momentum 
in
spherical coordinates as
It can be shown that when these operators act on the solution , the
result is
It can also be shown that this means that an electron in a particular
state has orbital angular momentum of constant magnitude 
and constant projection onto the z-axis of 
. Since the electron
obeys the time-independent Schrödinger equation 
, and hence
has constant energy, one says that the wavefunction
is a simultaneous eigenstate of the operators H, , and
. Table 1 summarizes this information and gives the
allowed values for each quantum number. It is worth repeating that these
numbers can have only these specific values because of the demand that
be a well-behaved function.
Table 1: Some quantum numbers for the electron in the hydrogen atom.
It is common to identify a state by its principle quantum number n and a letter which corresponds to its orbital angular momentum quantum number l, as shown in table 2. This is called spectroscopic notation. The first four designated letters are of historical origin. They stand for sharp, primary, diffuse, and fundamental, and refer to the nature of the spectroscopic lines when these states were first studied.
Table 2: Spectroscopic notation.
Figure 1 shows radial probability distributions for some
different states, labelled by spectroscopic notation. The radial probability
density 
is defined such that
is the probability of finding the electron with radial coordinate between
r and r+dr. The functions are normalized so that the total probability of
finding the electron at some location is unity. It is interesting to note
that each state has n-l-1 nodes, or points where the probability goes to
zero. This is sometimes called the radial node quantum number and
appears in other aspects of quantum theory. It is also interesting that
for each n, the state with l=n-1 has maximum probability of being found
at 
, the radius of the orbit predicted by Bohr theory. This
indicates that the Bohr model, though known to be incorrect, is at least
similar to physical reality in some respects, and it is often helpful to
use the Bohr model when trying to
visualize certain effects, for example the spin-orbit effect, to be
discussed in the next section. The
angular probability distributions will not be explored here
, except to say
that they have the property that if the solutions with all possible values
of l and for a particular n are summed together, the result is
a distribution with spherical symmetry, a feature which helps to greatly
simplify applications to multi-electron atoms.
Figure 1: Radial probability distribution for an electron in some low-energy
levels of hydrogen. The abscissa is the radius in units of 
.