Magnetic moment
Charged particles which have orbital angular momentum have a magnetic moment
μ
which is proportional to their angular momentum L.
Electrons, protons and neutrons are spin 1/2 particles and they have an
intrinsic magnetic moment proportional to their spin S.
Since Sz is quantized, the z-component of the magnetic moment is
quantized.
The square of the magnitude of an electrons spin is always (3/4)ħ2 and the z-component is equal to ±ħ/2. For an electron the direction of μ is opposite to that of S and μz = ±μB, where μB is the Bohr magneton, μB = 9.27*10-24 A m2.
Forces and torques
In an inhomogeneous magnetic field B a force acts on a magnetic dipole.
In a constant magnetic field a magnetic dipole experiences no net force.
In any magnetic field B a magnetic dipole μ
not aligned with the field experiences a torque, trying to align it with the
field. The magnitude of the torque is τ = μBsinθ,
where θ is the smallest angle between the directions
of the vectors μ and B.
Mathematical description
We denote the eigenstates of the Sz operator by |+> and |->. Sz|+> =
ħ/2|+>, Sz |-> =
-ħ/2|->. A spin 1/2 particle does not have
to be in an eigenstate of Sz, it can be in a state described by a linear
combination of those eigenstates, a|+> + b |->. If a spin 1/2 particle is
in the state a|+> + b |->, then the probability of measuring Sz =
ħ/2
is |a|2 and the probability of measuring Sz =
-ħ/2 is |b|2.