Consider a hydrogen atom, for which the potential takes the specific form
The radial eigenfunction satisfies Equation (395), which can be written
Here, is the reduced mass, which takes into account the fact that the electron (of mass ) and the proton (of mass ) both rotate about a common centre, which is equivalent to a particle of mass rotating about a fixed point. Let us write the product as the function . The above equation transforms to
which is the one-dimensional Schrödinger equation for a particle of mass moving in the effective potential
The effective potential has a simple physical interpretation. The first part is the attractive Coulomb potential, and the second part corresponds to the repulsive centrifugal force.
and , with
Here, it is assumed that the energy eigenvalue is negative. Equation (398) transforms to
Let us look for a power-law solution of the form
Substituting this solution into Equation (402), we obtain
Equating the coefficients of gives
Now, the power law series (403) must terminate at small , at some positive value of , otherwise would behave unphysically as . This is only possible if , where the first term in the series is . There are two possibilities: or . The former predicts unphysical behavior of the wavefunction at . Thus, we conclude that . Note that for an state there is a finite probability of finding the electron at the nucleus, whereas for an state there is zero probability of finding the electron at the nucleus (i.e., at , except when ). Note, also, that it is only possible to obtain sensible behavior of the wavefunction as if is an integer.
For large values of , the ratio of successive terms in the series (403) is
according to Equation (405). This is the same as the ratio of successive terms in the series
which converges to . We conclude that as . It follows from Equation (401) that as . This does not correspond to physically acceptable behavior of the wavefunction, since must be finite. The only way in which we can avoid this unphysical behavior is if the series (403) terminates at some maximum value of . According to the recursion relation (405), this is only possible if
where the last term in the series is . It follows from Equation (400) that the energy eigenvalues are quantized, and can only take the values
is the ground state energy. Here, is a positive integer which must exceed the quantum number , otherwise there would be no terms in the series (403).
The properly normalized wavefunction of a hydrogen atom is written
is the Bohr radius, and is a well-behaved solution of the differential equation
that is consistent with the normalization constraint
Finally, the are spherical harmonics. The restrictions on the quantum numbers are , where is a positive integer, a non-negative integer, and an integer.
The ground state of hydrogen corresponds to . The only permissible values of the other quantum numbers are and . Thus, the ground state is a spherically symmetric, zero angular momentum state. The next energy level corresponds to . The other quantum numbers are allowed to take the values , or , . Thus, there are states with non-zero angular momentum. Note that the energy levels given in Equation (409) are independent of the quantum number , despite the fact that appears in the radial eigenfunction equation (415). This is a special property of a Coulomb potential.
In addition to the quantized negative energy states of the hydrogen atom, which we have just found, there is also a continuum of unbound positive energy states.
- Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)