Tensors for Physics

13.2 Magnetic Sub-states 241

13.2 Magnetic Sub-states

13.2.1 Magnetic Quantum Numbers and Hamilton Cayley

The spin operator possesses sub-states, which are eigenstates of one of its Cartesian
components. Let this preferential direction be parallel to the unit vectorh.The
symbolhalludes to the direction of a magnetic fieldH. Frequently, thez-direction
of a coordinate system is chosen parallel toh. The eigenvalues are referred to as
magnetic quantum numberssince they determine the strength of the interaction of
a magnetic moment, which is parallel or anti-parallel to the spin operators,inthe
presence of a magnetic field. More specifically, the energy of a magnetic momentm
in the presence of a magnetic fieldB=Bhis−m·B. With the magnetic moment
given bym=γs, whereγis thegyromagnetic ratio, the corresponding Hamilton
operator isHmag=−γBh·s.
The eigenvalues ofh·sare denoted bym. These magnetic quantum numbers
mare of relevance, even when no magnetic field is applied. Due to theRichtungs-
Quantelung, the allowed values formare

−s,−s+ 1 ,...,s− 1 ,s,

for a spins.Themare integer or halve-integer numbers, depending on whethersis
an integer or a halve-integer number. Altogether, there are 2s+1 magnetic quantum
numbers and magnetic sub-states. Clearly, the smallest non-zerosiss=^12.
TheHamilton-Cayleyrelation for the magnetic sub-states is

∏s

m=−s

(h·s−m)= 0. (13.7)

This is a polynomial of degree 2s+1, inh·s. Thus(h·s)(^2 s+^1 )is equal to a linear
combination of lower powers ofh·s. The same applies for(h·s)p, when the power
pis larger than 2s+1. The Hamilton-Cayley relation also implies, that symmetric
traceless tensors of rank, constructed from the Cartesian components of the spin
operator, are non-zero only up to rank= 2 s, see the following section.

13.2.2 Projection Operators into Magnetic Sub-states

The projection operatorP(m)into the sub-state with the eigenvaluemis defined via

P(m)h·s=mP(m). (13.8)