wang
(Wang)
#1
Thus, we conclude thatb∈RangeAif and only if
(v,b) = 0 for all v∈KerA (10.30)
If this condition is satisfied, equation (10.25) admits at least one solution which we denote
byx 0. The general solution is then
x=x 0 +u for any u∈KerA (10.31)
The above discussion readily extends to infinite-dimensional Hilbert spaces, as long as the
operatorAis bounded. In general, ifAis unbounded, subtle new issues may arise.
10.4 The Stark effect for the ground state of the Hydrogen atom
Faraday investigated the effect of electric and magnetic fields on light, in particular on
its polarization and wavelength (color). He discovered that polarization is influenced by
magnetic fields, but he found no change in color. Actually, both electric and magnetic
fields change the energy levels of atoms and molecules, and will thus change the frequencies
of radiation emitted and absorbed. The effects are so small, however, that more refined
experimentation than Faraday disposed of in the mid 1800’s to observe the effects. The
effect of electric fields was established by Stark. We shall study it here as an example of how
to use perturbation theory, both non-degenerate and degenerate.
The correction to the Hamiltonian of an electron in the presence of an electric fieldE
along thezaxis is given by
H 1 (x) =eEz (10.32)
wherezis the coordinate along thez-axis, and−eis the charge of the electron. The electric
field also interacts with the electrically positive nucleus, but since themass of the nucleus is
much larger than that of the electron, this effect may be safely neglected. Spin will similarly
be neglected. We shall concentrate on atoms with only a single electron, whose states are
labeled by the quantum numbersn,ℓ,mof the Coulomb problem, so the states are denoted
by|n,ℓ,m〉withn≥ℓ+ 1, and|m|≤ℓ.
Forn ≥ 2, each energy level is degenerate, so we shall have to develop degenerate
perturbation theory to handle this reliably. For the ground state,first order perturbation
theory gives
E|^11 , 0 , 0 〉=〈 1 , 0 , 0 |H 1 | 1 , 0 , 0 〉=eE〈 1 , 0 , 0 |z| 1 , 0 , 0 〉= 0 (10.33)
The above matrix element vanishes by rotation invariance of the ground state. Second order
perturbation theory of the ground state always lowers the energy on general principles. We
cannot quite calculate this effect here, though, because the summation over all states, which
enters second order perturbation theory, will involve here the bound state part, but also the
continuous spectrum part.
