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Returning to the Stark effect, let us examine the effect of an external electric
field on the energy levels of the states of a hydrogen atom.
There are four such states: an state, usually referred to as ,
and three states (with ), usually referred to as 2P. All
of these states possess the same unperturbed energy,
.
As before, the perturbing Hamiltonian is

(956) 
According to the previously determined selection rules (i.e., ,
and ), this Hamiltonian couples and .
Hence, nondegenerate perturbation theory breaks down when applied to
these two states. On the other hand, nondegenerate perturbation
theory works fine for the and states,
since these are not coupled to any other states by the perturbing
Hamiltonian.
In order to apply perturbation theory to the and states, we have to solve the
matrix eigenvalue equation

(957) 
where is the matrix of the matrix elements of between these states. Thus,

(958) 
where the rows and columns correspond to and , respectively. Here, we have again
made use of the selection rules, which tell us
that the matrix element of between two hydrogen atom
states is zero unless the states possess quantum numbers which differ by unity. It is easily demonstrated,
from the exact forms of the 2S and 2P wavefunctions, that

(959) 
It can be seen, by inspection, that the eigenvalues of
are
and
. The corresponding normalized eigenvectors are
It follows that the simultaneous eigenstates of and take the form
In the absence of an external electric field, both of these states possess the
same energy, . The firstorder energy shifts induced by
an external electric field are given by
Thus, in the presence of
an electric field, the energies of states 1 and 2 are shifted upwards and downwards, respectively, by an amount
. These states are orthogonal linear combinations of the
original and states. Note that the energy shifts
are linear in the electric fieldstrength, so this effectwhich is
known as the linear Stark effectis much larger than the quadratic effect described in Sect. 12.5.
Note, also, that the energies of the and states are not affected by the electric field to firstorder. Of course,
to secondorder the energies of these states are shifted by an amount
which depends on the square of the electric fieldstrength (see Sect. 12.5).
Next: Fine Structure of Hydrogen
Up: TimeIndependent Perturbation Theory
Previous: Degenerate Perturbation Theory
Richard Fitzpatrick
20100720