The total Hamiltonian is,

(43) |

Eq. (43) is the eigenvalue problem of a one particle central-potential. We consider the factorizable solution,

(44) |

This equation could be solved by first transforming it into the *associated
Laguerre equation*, for which solutions are well-known. Here, however,
we limit the presentation to note that Eq. (44) has solutions that are
finite, single valued and square integrable only when

or | (45) |

where ,
and
is the *Bohr radius*.

These are the bound-state energy levels of hydrogen-like atoms responsible
for the *discrete* nature of the absorption spectrum. In particular,
the wavenumbers of the spectral lines are

**Degeneracy: **Since the energy E depends only on the principal
quantum number ,
and the wave function
depends on ,
and ,
there are
possible states with the same energy. States with the same energy are called
*degenerate
states*. The number of states with the same energy is the *degeneracy*
of the energy level.

n=1, 2, 3, ...

l=0, 1, 2, ... n-1 } these are n states

m=-l, -l+1, ..., 0, 1, 2, ...l } these are 2 l + 1 states

**Exercise 38**: Prove that the degeneracy of the energy level
is .

The complete hydrogen-like bound-state wave functions with quantum numbers , and are,

Therefore,

**Example 2:** The possible wave functions with
are:

**Exercise 39**: Show that,

**Exercise 41**: Use perturbation theory to first order to compute
the energies of states , ,
and
when a hydrogen atom is perturbed by a magnetic field ,
according to ,
where .
(The splitting of spectroscopic lines, due to the perturbation of a magnetic
field, is called *Zeeman
effect*).

**Radial Distribution Functions**

The probability of finding the electron in the region of space where is between to , between to and between and is,

In order to visualize other radial distribution functions of an electron
in a hydrogen atom click here.

Pictures of atomic orbitals with n<=10 are available here.

**Real Hydrogen-like Functions**

Any linear combination of degenerate eigenfunctions of energy is also an eigenfunction of the Hamiltonian with the same eigenvalue . Certain linear combinations of hydrogen-like wavefunctions generate real eigenfunctions. For example, when ,

Function is zero in the plane, positive above such plane, and negative below it. Functions and are zero at the and planes, respectively. and are eigenfunctions of with eigenvalue . However, since and are eigenfunctions of with different eigenvalues (e.g., with eigenvalues and , respectively), linear combinations , and , are eigenfunctions of but not eigenfunctions of .

**Exercise 42**: (A) What is the most probable value of r, for the
ground state of a hydrogen atom? Such value is represented by .

(B) What is the total probability of finding the electron at a distance ?

(C) Verify the orthogonality of functions and.

(D) Verify that the ground state of the hydrogen atom is an eigenstate
of ,
but that such state is not an eigenstate of ,
or .