The total Hamiltonian is,
Eq. (43) is the eigenvalue problem of a one particle central-potential. We consider the factorizable solution,
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
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,
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
but that such state is not an eigenstate of ,