University Physics Notes: Quantum Mechanics – Table of Normalized Spherical Harmonics
The Schrodinger equation for the hydrogen atom takes the form
This equation is separable which means that while the solution is
a function of three variables, it is a product of three functions,
each one of which is a function of only one variable, The general
solution can be written
where
is
itself a product of a function of
and
with
a function of the form
where
and
are
integers.
|
|
|
|
|
0 |
0 |
|
|
1 |
0 |
|
|
1 |
|
|
|
2 |
0 |
|
|
2 |
|
|
|
2 |
|
|
Notice that in each of the
above
the degree of the polynomial in
is
equal toand
there is a complex exponential term
whereis
given in the table. There is one unique
for
each combination ofandand
obeys
the normalization condition
