1.25 Spin 1/2 and other 2 State Systems
The angular momentum algebra defined by the commutation relationsbetween the operators requires
that the total angular momentum quantum number must either be an integer or a half integer.
The half integer possibility was not useful for orbital angular momentum because there was no
corresponding (single valued) spherical harmonic function to represent the amplitude for a particle
to be at some position.
The half integer possibility is used to represent the internal angularmomentum of some particles.
The simplest and most important case is spin one-half (See section 18.8). There are just two
possible states with different z components of spin: spin up
(
1
0
)
, with z component of angularmomentum + ̄h 2 , and spin down
(
0
1
)
, with− ̄h 2. The corresponding spin operators areSx=̄h
2(
0 1
1 0
)
Sy=̄h
2(
0 −i
i 0)
Sz=̄h
2(
1 0
0 − 1
)
These satisfy the usual commutation relations from which we derived the properties of angular
momentum operators.
It is common to define the Pauli Matrices,σi, which have the following properties.
Si ≡̄h
2σi.S~ = ̄h
2~σσx=(
0 1
1 0
)
σy =(
0 −i
i 0)
σz=(
1 0
0 − 1
)
[σi,σj] = 2iǫijkσk
σ^2 i = 1
σxσy+σyσx=σxσz+σzσx = σzσy+σyσz= 0
{σi,σj} = 2δijThe last two lines state that the Pauli matrices anti-commute. Theσmatrices are theHermitian,
Traceless matricesof dimension 2. Any 2 by 2 matrix can be written as a linear combination of
theσmatrices and the identity.
1.26 Quantum Mechanics in an Electromagnetic Field
Theclassical Hamiltonianfor a particle in an Electromagnetic field is
H=
1
2 m(
~p+e
cA~
) 2
−eφwhereeis defined to be a positive number. This Hamiltonian gives the correct Lorentz force law.
Note that the momentum operator will now include momentum in the field, not just the particle’s
momentum. As this Hamiltonian is written,~pis the variable conjugate to~rand is related to the
velocity by~p=m~v−ecA~.