The Hamiltonian operator for the hydrogen atom acts trivially on theC^2
factor, so the only effect of the additional wavefunction component is to double
the number of energy eigenstates at each energy. Electrons are fermions, so
antisymmetry of multi-particle wavefunctions implies the Pauli principle that
states can only be occupied by a single particle. As a result, one finds that when
adding electrons to an atom described by the Coulomb potential problem, the
first two fill up the lowest Coulomb energy eigenstate (theψ 100 or 1Sstate atn=
1), the next eight fill up then= 2 states (two each forψ 200 ,ψ 211 ,ψ 210 ,ψ 21 − 1 ),
etc. This goes a long ways towards explaining the structure of the periodic table
of elements.
When one puts a hydrogen atom in a constant magnetic fieldB, for reasons
that will be described in section 45.3, the Hamiltonian acquires a term that acts
only on theC^2 factor, of the form
2 e
mcB·σThis is exactly the sort of Hamiltonian we began our study of quantum mechan-
ics with for a simple two-state system. It causes a shift in energy eigenvalues
proportional to±|B|for the two different components of the wavefunction, and
the observation of this energy splitting makes clear the necessity of treating the
electron using the two-component formalism.
21.4 For further reading
This is a standard topic in all quantum mechanics books. For example, see
chapters 12 and 13 of [81]. Theso(4) calculation is not in [81], but is in some
of the other such textbooks, a good example is chapter 7 of [5]. For extensive
discussion of the symmetries of the^1 rpotential problem, see [38] or [39].