Assume a Stern-Gerlach type experiment is performed with an apparatus as shown in the figure.
A beam of silver atoms emerges from a furnace held at 1000K. It is collimated using
0.1mm slits, passed between the poles of a magnet, and directed onto a screen, where the
atoms are detected. The velocity of the silver atoms is approximately 500m/s (from
(3/2)kT=(1/2)mv2), k=1.38´10-23J/K).
Assume the apertures collimate the beam to approximately 0.1o angular spread.
The spread in the component of velocity perpendicular to the beam direction is at least
1m/s since 
.
We have 
.
The uncertainty principle requires that 
.
(Msilver=1.6´10-27kg´108.)
The position
and momentum of a particle must be treated quantum-mechanically when 
approaches the minimum value required by
the uncertainty principle. In a typical Stern-Gerlach type experiment this is not the
case. The external degrees of freedom of the silver atoms can be treated classically.
The
dimensions of the wave packet are much smaller that the characteristic dimensions of the
problem.
Assume each silver atom has a permanent magnetic moment m. We assume that m=gS, where S denotes the intrinsic angular momentum of the silver atom. g is called the gyromagnetic ratio. If m is a fundamental property of the atom then its total energy in an external magnetic field is
.
The torque on the atom is
and the force is
.
In our experiment B=(Bx, 0, Bz) is an inhomogeneous
field. In the region of interest, where the beam passes through the field, we have 
.
Therefore
, 
.
.
The component Sz is constant, while the component perpendicular to the z-axis rotates about the z-axis.
Consider an arbitrary vector A, rotating about the origin in the x-y plane with angular velocity w=dq/dt .
.
.
S^ rotates about the z-axis with angular velocity w=gBz.
The z-component of F acting on the silver atoms is
.
Fz produces a deflection of the atoms in the z-direction which
is proportional to mz. If 
is known, then measuring this deflection is equivalent to measuring mz.
Classically, we expect to find all values of mz between mz = +|m| and mz= -|m|.
In the experiment the silver atoms hit the screen in two distinct spots.
The
measurement yields only two possible values for mz. Sz is
therefore a quantized observable, whose discrete spectrum has only two eigenvalues, which
we will find to be 
.
In quantum mechanics, every measurable physical quantity is associated with a Hermitian operator whose eigenvectors form a basis for the state space. Assume that Sz is the observable associated with the z-component of the spin. The state space corresponding to the observable Sz of a Silver atom (a spin ½ particle) is therefore two-dimensional. We denote the eigenvectors of Sz by |+> and |->.
.
We assume that the eigenvectors are normalized.
.
Sz alone is a C.S.C.O. in the state space Es spanned by the
vectors |+> and |->. The closure relation is written as
|+><+| + |-><-| = I. The most general element of Es is 
.
If 
.
The matrix of Sz in the {|+>, |->} basis is
.
The complete state space for a particle is E=Espace
Espin.
However, if in a given problem the
spatial degrees of freedom can be treated classically, then we can predict the outcome of
measurements which depend on the internal degrees of freedom by considering only the state
vector in Espin.
Let Sx and Sy denote the observables associated
with the x- and y-components of the spin of a spin ½ particle, and 
the observable associated with the
component along a direction defined by the unit vector 
.
Es can also be spanned by the
eigenvectors 
, 
, and 
.
However, Sx, Sy, and Sz do
not commute. The eigenvectors of Si are not eigenvectors of Sj
and Sk. In matrix notation we find that the matrices of Sx
and Sy in the eigenbasis of Sz , {|+>, |->},
are not diagonal.
, 
.
We write 
. The matrices of the
three components of s in the {|+>, |->}
basis are called the Pauli matrices.
, 
, 
.
The eigenvalues of sx and sy are ± 1.
The eigenvectors of 
and the
eigenvectors of 
. These are also the
eigenvectors of Sx and Sy .
 |  .In general, Therefore
since The Pauli matrices anticommute. |
| Any arbitrary 2´2 matrix can be written as a linear
combination of the four matrices sx, sy, sz
and I. Let
(M) is a Hermitian matrix if a0, ax, ay, and az are real. |
.
The operator Su is defined through
.
The matrix of Su is
.
The eigenvectors of Su are
.
The most general normalized ket in the spin state space Es is 
with 
.
|y> is the eigenvector of some
operator Su with eigenvalue 
.
There exists a unit vector 
and an operator Su such that 
.
An arbitrary state vector can be written
as 
.
A system can be prepared in any state |y> by
placing a Stern-Gerlach apparatus such that its axis is directed along the unit vector 
and letting only the upward deflected
particles pass.
Link:
| Spin 1/2 Stern-Gerlach experiment |
Assume 
.
The potential energy of a
particle with intrinsic magnetic moment 
in this field is 
.
If
we are quantizing only the internal degrees of freedom, then the operator H, (the
Hamiltonian), which is the operator whose possible eigenvalues are the total energy of the
particle associated with those internal degrees of freedom, is found by letting Sz
become an operator.
.
H and Sz have the same eigenfunctions.
.
There are two energy levels, 
and 
, separated by 
.
They define the
Bohr frequency 
.
In general, the Bohr frequencies of a system whose energy eigenvalues are {En} are defined as
.
Note: If 
then 
. For silver atoms g
is negative, w0 is positive , E+>E-
.
Assume a system in an arbitrary state 
with 
and 
is placed into a uniform magnetic field 
.
The system evolves and we have 
,
where 
.
|y(t)> is an eigenvector of Su’
, (
is characterized by 
), while |y(0)>
is an eigenvector of Su , (
is characterized by 
).
The vector 
defining 
revolves around the z-axis with
angular frequency 
.
Its component
along the z-axis remains constant, while its component perpendicular to the z-axis
rotates with angular velocity w0. This is the
quantum mechanical analog of the rotation of the classical magnetic moment m
and is called Lamor precession.
.
Sz is a constant of motion.
.
.
.
Similarly,
.
Sx and Sy are not constants of motion.
Their
expectation values oscillate with the single Bohr frequency 
.
The mean values of Sx, Sy,
and Sz behave like the components of the classical angular momentum
undergoing Lamor Precession.
Link:
| Lamor Precession |
. At t=0 the particle is therefore
in the |+>x eigenstate of the Sx operator.
The
particle is confined to a region with a uniform magnetic field 
, its Hamiltonian is 
.
The eigenstates of H are |+> and |->, 
.
|+>x can be written
as a linear combination of eigenstates of H.(a) Find the probability of
measuring 
at t=T.
(b) What is the mean value of Sx, <Sx>, at t=T?
(c) Find the probability of measuring 
at t=T.
| Solution:
(a)
 .
 ,independent of t. |
.
, 
. These commutation relations
are the commutation relations for angular momentum.
.
We also find 
.
be an arbitrary 2´2 matrix.
.
.
.
.