Identical particles 3879.4 Identical particles in quantum mechanics: Spin statistics
In 1922, an experiment carried out by Otto Stern and Walter Gerlach showed that
quantum particles possess an intrinsic property that, unlike charge and mass, has no
classical analog. When a beam of silver atoms was sent through an inhomogeneous
magnetic field with a field increasing from the south to north poles of the magnet, the
beam split into two distinct beams. The experiment was repeated in 1927 by T. E.
Phipps and J. B. Taylor with hydrogen atoms in their ground state in order to ensure
that the effect truly revealed an electronic property. The result of the experiment
suggests that the particles comprising the beam possess an intrinsic property that
couples to the magnetic field and takes on discrete values. This property is known as the
magnetic momentμˆMof the particle, which is defined in terms of a more fundamental
property calledspindenoted by the vector operatorˆS. These two quantities are related
byμˆM=γSˆ, where the constant of proportionalityγis thespin gyromagnetic ratio,
γ=−e/mec. The energy of a particle of spinˆSfixed in space but interacting with
a magnetic fieldBisE=−μˆM·B=−γSˆ·B. Unlike charge and mass, which are
simple scalar quantities, spin is expressed as a vector operator andcan take on multiple
values for a given particle. When the beam in a Stern–Gerlach experiment splits in the
magnetic field, for example, this indicates that there are two possible spin states. Since
a particle with a magnetic moment resembles a tiny bar magnet, the spin state that
has the south pole of the bar magnet pointing toward the north poleof the external
magnetic field will be attracted to the stronger field region, and theopposite spin state
will be attracted toward the weaker field region.
The three components of the spin operator vectorˆS= (Sˆx,Sˆy,Sˆz) satisfy the
commutation relations
[Sˆx,Sˆy] =i ̄hSˆz, [Sˆy,Sˆz] =ih ̄Sˆx, [Sˆz,Sˆx] =i ̄hSˆy. (9.4.1)These commutation relations are similar to those satisfied by the three components
of the angular momentum operatorLˆ=ˆr×pˆ. A convenient way to remember the
commutation relations is to note that they can be expressed compactly asˆS×Sˆ=i ̄hSˆ.
Since spin is an intrinsic property, a particle is said to be a spin-sparticle, wheres
can be either an integer or a half-integer. A spin-sparticle can exist in 2s+ 1 possible
spin states, which, by convention, are taken to be the eigenvectors of the operatorSˆz.
The eigenvalues ofSˆzthen range from−s ̄h,(−s+ 1) ̄h,...,(s−1) ̄h,s ̄h. For example,
the spin operators for a spin-1/2 particle can be represented by 2×2 matrices of the
form
Sˆx= ̄h
2(
0 1
1 0
)
, Sˆy=̄h
2(
0 −i
i 0)
, Sˆz=̄h
2(
1 0
0 − 1
)
, (9.4.2)
and the spin-1/2 Hilbert space is a two-dimensional space. The two spin states have
associated spin eigenvaluessz=m ̄h/2, wherem=− 1 /2 andm= 1/2, and the
corresponding eigenvectors are given by
|m= 1/ 2 〉≡|χ 1 / 2 〉=(
1
0
)
, |m=− 1 / 2 〉≡|χ− 1 / 2 〉=