The relation between the Hamiltonian and the Casimir operatorsM^2 and
N^2 is
2 H(K^2 +L^2 +~^21 ) = 2H(2M^2 + 2N^2 +~^21 ) = 2H(4M^2 +~^21 ) =−me^41On irreducible representations ofso(3) of spinμ, we will have
M^2 =μ(μ+ 1)~^21for some half-integralμ, so we get the following equation for the energy eigen-
values
E=−
−me^4
2 ~^2 (4μ(μ+ 1) + 1)=−
−me^4
2 ~^2 (2μ+ 1)^2Lettingn= 2μ+ 1, forμ= 0,^12 , 1 ,...we getn= 1, 2 , 3 ,...and precisely the
same equation for the eigenvalues described earlier
En=−me^4
2 ~^2 n^2
One can show that the irreducible representations of the product Lie algebra
so(3)×so(3) are tensor products of irreducible representations of the factors, and
in this case the two factors in the tensor product are identical due to the equality
of the CasimirsM^2 =N^2. The dimension of theso(3)×so(3) irreducibles is
thus (2μ+ 1)^2 =n^2 , explaining the multiplicity of states one finds at energy
eigenvalueEn.
The states withE <0 are called “bound states” and correspond physically
to quantized particles that remain localized near the origin. If we had chosen
E >0, our operators would have satisfied the relations for a different real
Lie algebra, calledso(3,1), with quite different properties. Such states are
called “scattering states”, corresponding to quantized particles that behave as
free particles far from the origin in the distant past, but have their momentum
direction changed by the Coulomb potential (the Hamiltonian is non-translation
invariant, so momentum is not conserved) as they propagate in time.
21.3 The hydrogen atom
The Coulomb potential problem provides a good description of the quantum
physics of the hydrogen atom, but it is missing an important feature of that
system, the fact that electrons are spin^12 systems. To describe this, one really
needs to take as space of states two-component wavefunctions
|ψ〉=(
ψ 1 (q)
ψ 2 (q))
(or, equivalently, replace our state spaceHof wavefunctions by the tensor prod-
uctH⊗C^2 ) in a way that we will examine in detail in chapter 34.