Actual and 'expected' classical results from the Stern-Gerlach experiment
Above: the apparatus used in the Stern-Gerlach experiment. In
reality, since the magnetic field weakens  on either side of the
beam, two arcs are formed, resulting in an approximate
ellipse-shape as shown on the left-hand side of the diagram on
the left. Classically we would expect any value within the range to
occur, if the angular momentum of atoms was not quantised, in
which case a solid ellipse would result as shown on the
right-hand side of the diagram on the left.
Angular momentum in Quantum Mechanics
The Stern-Gerlach Experiment

In the Stern-Gerlach experiment, a sample of caesium metal is vaporised in the oven and the emerging beam
collimated to form a narrow beam in which all the atoms are travelling more-or-less in one direction (otherwise they do
not pass through the holes in the collimators). This beam then passes through a pair of magnetic poles. These
magnets are shaped so that the magnetic form between them is not uniform and so accelerates the atoms by
deflecting them by an amount that is proportionate to the intrinsic angular momentum of the atoms (and to the strength
of the field). Since there are only two such values, the beam splits into two and then strikes a detector (shown here as
a screen). [Note in the diagram below we have chosen z as the vertical axis, from S to N, and y as the initial beam-axis.]
In practise the atomic beam travels through a fairly good vacuum, so that collisions between the atoms in the beam and
air molecules are infrequent during the 1-2 m the beam must travel. The magnets are about 10 cm long and the beams
travel for about a further 50 cm upon leaving the magnetic field, before striking the detector. The actual detector may
consist of a glowing hot (incandescent) niobium wire connected to an electrometer, which functions as a very sensitive
ammeter (galvanometer) which can measure currents as small as one nanao-Amp (10E-12 A). Caesium atoms become
ionised when they strike the hot wire and can then be drawn off by another (negative) electrode and so generate a
current by completing the circuit. The wire can be scanned across to build up a complete 2D picture.

  • The Stern-Gerlach experiment demonstrates that the angular momentum of atoms is quantised.

  • It actually measures the spin angular momentum of a single electron when alkali metals are used, or
    other atoms with a single unpaired valence s-electron, such as silver (Ag).

The electron configurations of caesium (Cs) and silver are given below:
Many particles, including electrons in atoms, have
angular momentum. This implies rotational
motion, and although it is not quite correct to think
of an electron rotating, it is still helpful to think in
these terms.

An electron, as shown on the left, can be thought
of as a spinning ball. We can then represent its
intrinsic angular momentum, or spin, as a
vector, the length of which equals the magnitude
of the spin and the direction of which is given by
standard definition. We could define the direction
of rotation to be clockwise when looking along the
arrow towards its tip, as shown here, or
anticlockwise.

In the case of an electron only two possible
values of the spin (s) are possible: s = +1/2 and s
= -1/2. We can define the plus to indicate spin-up
and the minus spin-down. The direction we use
(the axis, chosen to be vertical in this diagram) is
arbitrary but becomes important in a magnetic
field in which the spins all aline. The fact that only
two values are allowed is because the angular
momentum is
quantised.
Background

Ordinary classical linear momentum is very familiar from everyday life. A bullet has momentum, which is the
product of its velocity and mass, so a heavy bullet moving at the same speed has a greater momentum, as does a
faster bullet of the same mass.
Momentum is a vector, it has both magnitude (value) and direction. Momentum is
conserved, so when the chemical reaction in the gun barrel imparts momentum to a firing bullet, it also imparts a
momentum of the same magnitude, but acting in the opposite direction to the gun, causing recoil. This happens
because initially before the gun was fired the momentum was zero and so after the gun is fired the total momentum
must also be zero, and so a negative momentum is produced in recoil so that the total momentum of gun + bullet is
still zero. Similarly, when the bullet hits its target it imparts the same momentum to the target (if the target is much
heavier than the bullet then the velocity gained by the target will be small).

      Classical linear momentum = mass x velocity = m
v.

Remember that velocity is a vector with magnitude equal to the speed and with direction pointing in the direction of
motion.
Classical angular momentum is a similar concept. A rotating object possesses angular momentum. If it
rotates on the spot then it only has angular momentum - momentum due to rotary motion. If its is also flying through
the air then it also has linear momentum. A spin-bowler imparts spin to the ball, giving it both angular and linear
momentum. Classically, linear and angular momenta can possess any values - they are continuous variables and not
quantised. We shall now look at an experiment that demonstrates the quantisation of angular momentum in the world
of the very small - the world of atoms and electrons.

        Classical angular momentum =
r x mv

Where, r is the position (a vector) of the particle from the origin (point of rotation).
Above: electron spin.
More details on the quantum numbers, and their values, which are used to describe the spin, orbital and total angular
momenta of electrons and atoms are given in the two pages below (click pages to enlarge):
1s hydrogen atomic orbital
a 2p hydrogen atomic orbital
a 3d hydrogen atomic orbital
By spin, we mean the intrinsic rotational angular momentum of the electron, which is quantised (unlike the rotation of a
football). Electrons can also have orbital angular momentum. If the orbital of the electron around the nucleus is not
spherically-symmetric (that is the likelihood of finding the electron is the same in all directions from the nucleus) then it
has zero orbital angular momentum. This is the case with an s-electron. Recall (see atomic orbitals) that s-orbitals are
spherical, and so have zero orbital angular momentum. Thus, the unpaired valence electron in the Cs or Ag atom has
only spin angular momentum. The total angular momentum of an electron (j) is equal to the spin angular momentum
(s) plus the orbital angular momentum (
l).
So, we have seen that an electron has an intrinsic angular momentum, called spin, which we can think of as the
rotational angular momentum of the electron. If an atom has one unpaired electron then this spin is imparted to the
whole atom (if there are more than one unpaired electron then all contribute to the total atomic spin in various ways).
However, we should stress that electrons are not thought to actually spin as such, rather we have a quantum
mechanical phenomenon, which gives a measurable or observable value, which behaves in some ways like classical
rotational angular momentum, but in truth nobody knows what 'spin' really is in the world of particles. It can have
strange effects which are quite unlike classical rotation as we perceive it.

Why does the electron spin cause the beam to split in two in the magnetic field?

The electron carries electric charge and we can think of it as spinning. Rotation is a form of acceleration (even though
the rotational speed may remain constant, since velocity is a vector, acceleration can be a change in the speed, or
magnitude of the velocity vector, or a change in its direction, as in rotation). An accelerating electric charge generates
a magnetic field. We can think of a spinning electron as a little bar magnet, with a magnetic North pole and a magnetic
South pole and magnetic field-lines connecting these poles (and running from magnetic North to South).
The orbital angular momentum of an electron is
represented by the orbital quantum number,
l.  For an
s-electron (top left)
l = 0, that is there is no orbital angular
momentum. After all, rotating an s-orbital causes no
observable change (recall that electrons are elementary
particles and not made up of smaller parts).

For a p-electron (top right)
l = 1. The distribution of
charge (strictly the probability of finding the electron) is no
longer spherically-symmetric. A d-electron (bottom left)
has more orbital angular momentum with
l  = 2. (A value of
l = 3 corresponds to f-orbitals). The value of l equates to
the number of nodal planes in the electron charge
distribution/cloud - that is a plane where the electron is
never found (probability of finding the electron here = 0).

Note: If we consider the orbitals to be circular, as in the
Bohr model of the hydrogen atom, then they also have
angular momentum. However, we need to use
Schrodinger's model, as shown here.
s-orbital (1s)
p-orbital (2p)
d-orbital (3d)
Spin-Orbit (L-S) Coupling

The fact that electrons are magnetic dipoles has another interesting consequence. The atom generates its own
magnetic field which interacts with the electrons. Classically we would say that the orbiting electrons constitute a
circulating current, that is an accelerating electric charge. Accelerating electric charges generate magnetic fields. In
other words, just as the rotation of the electron generated a magnetic field, so does the orbital motion of the electron
about the nucleus.

[Note: electrons in s, p, d and f-orbitals are NOT expected to orbit the nucleus, since these are stationary states, but
in-between measurements the electron can exist in a superposition state, having some s character and some p and d
character, for example, and such superposition states appear capable of motion, sop the electron in such a state might
move around the nucleus, though not in a precise trajectory, since the Uncertainty principle forbids this!]

These orbital and spin magnetic fields interact (both are non-uniform). For example, the apparent orbital motion of the
electron in a hydrogen atom generates a magnetic field of about 2 tesla (20 000 gauss). This interaction, called
spin-orbit coupling or L-S coupling, with L representing the orbital angular momentum and S the spin angular
momentum. This spin-orbit coupling shifts the energy of the electron slightly, splitting each energy level into a pair of
closely-spaced energy levels depending again on the direction of spin of the electron. One such split-level in each pair
is slightly higher in energy, and this corresponds to  the spin of the electron aligning parallel to the magnetic field of the
atom (which is due to orbital motion) whilst the other level with a slightly lower energy occurs for atoms in which the spin
is antiparallel to the magnetic field. In any population of atoms some atoms will be in the higher-energy state and some
in the lower-energy state at any one moment in time. The s-orbital energy is NOT split since this state has no orbital
angular momentum (L = 0) and thus zero L-S coupling, but it happens for p, d and f-orbitals.

This spin-orbit coupling induced splitting of the energy levels results in splitting of the observed spectral lines. For
example, the pair of orange D lines of sodium, observable in a spectroscope are one example. Typically, however, the
splittings are very small and can only be detected with very precise high-resolution spectroscopes. Such slight splitting
is called fine-splitting and the resultant high-resolution spectrum is called the
fine structure of the hydrogen atom.
Spin-orbit coupling is one contribution to this fine structure, two other contributions are due to relativistic effects, due to
the relativistic speeds of the electrons. The energy shifts induced in the spectral lines, which causes them to split, are
only of the order of about 0.1 % of the spectral line energies and so only observed in high-resolution measurements
and so are ignored in the lower-resolution standard hydrogen-atomic spectrum usually depicted in illustrations.

Changing Spin States

In quantum mechanics, measurements are usually what we call strong measurements. Strong measurements perturb
the system being measured irreversibly. This is hardly surprising as we measuring particles which are extremely tiny
and we measure them by causing them to interact with measurement apparatus, that is with other particles, and so it is
hardly surprising that the system gets disturbed.
Weak measurements, in which the system is perturbed only very
slightly and reversible are possible, but generally when we measure the system we change it and the result of our
measurement is actually the state of the system after the measurement-interaction (this is sometimes called
post-selection). [See
quantum measurements for more details.]

Before the measurement, a quantum system may be in a pure state, that is with all the particles in the same stable state
(a stationary or
eigenstate) such as with all their spins pointing up relative to the z-axis; or it may be in a mixed state in
which individual particles have a mixed state, that is with some spin-up character and some spin-down character. This
odd state of affairs is possible because particles are also waves and several waves can be added together
(
superposition). A system that is initially in a pure eigenstate may remain unchanged by the measurement, for
example if z-spin is up and we measure z-spin, then we shall find that it is still up. Measuring the degree of x-spin will
change the eigenstate, however, and  result in a mixed state with some spin-up and some spin-down relative to the
z-axis, which will also be a pure state with spin-up or spin-down relative to the x-axis! We generally select what we
measure for! If we measure spin in the x-direction then we change the system to have spin in the x-direction!

We can prepare a pure spin-state by using a
spin-preparer. If we return to the Stern-Gerlach apparatus and begin
with a beam of atoms which will be in some superposition state initially (with degrees of spin in the x, y and z-directions)
and then perform a measurement on the z-component of spin, which we do by having our magnets aligned along the
z-axis (which we define) as in our diagram of the apparatus above, then we obtain two beams, one with z-spin up, the
other with z-spin down. If we use a shutter to block the down-spin state, then we left with a beam of atoms in a pure
z-spin up state.

We can then perform a measurement by passing this z-spin up beam through another set of identical magnets, a
spin-analyzer, but rotate these magnets by theta-degrees around the beam axis (y-axis in our diagram). For example
if we rotate by theta = zero degrees (i.e. no rotation) then we have another pair of z-magnets and are measuring the
z-component of spin (again) and we will find it still to be z-spin up. The pure state is an eigenstate or
stationary state
and does not change over time. If, however, we rotate the magnets by 90 degrees clockwise (looking in the +y
direction) around the y-axis (beam-axis) then the magnets will lie along the x-axis and we will be performing a
measurement of the x-component of spin. This will generally produce two new beams, one with x-spin up, the other with
x-spin down. In terms of z-spin these particles now have some degree of both z-spin up and z-spin down, the z-state
has become mixed. This odd state of affairs takes some getting used to! When we measure one component of spin we
collapse the initial state of the particles into a pure z-spin state. If we then measure their x-component of spin, say, then
we obtain pure x-spin states and destroy the initial z-spin state. These are strong measurements!

A mathematical description of how measurements work in quantum mechanics, using spin

Our articles are not meant to be mathematically comprehensive, that would take hundreds of pages and reproduce
what can be found in many textbooks. Instead, we study the maths and try to tease out the key ideas and put these into
words. It is easy for students to get lost in pages of mathematical proofs, understanding perhaps perfectly every step in
the algebra and calculus, but overlooking the important physics! To help give some pointers, I will summarise some of
the mathematical concepts in this section. This requires prior knowledge of the basics of vectors and matrices.

The purpose of quantum mechanics is not just to describe physical systems, but to construct mathematical models that
accurately predict the behaviour of systems. The behaviour is what we observe, for example a measurement of energy.
Observables, like energy, position, momentum, velocity, etc. describe the state and tell us all we can know about it.
Particles are also waves (
wave-particle duality) and in quantum mechanics the state of the system under study is
described by a
wave equation. For example, when obtaining the shapes of the hydrogen atomic orbitals, plotted
above, we used wave functions (eigenfunctions) that satisfy
Schrodinger's wave equation. Schrodinger's wave
equation describes the system and its behaviour. We have to feed the wave functions into Schrodinger's equation to
get the stationary states (eigenstates) corresponding to what we observe in strong measurements and the values
(eigenvalues) we want, e.g. the values of energy that we would measure or observe (energy eigenvalues)
corresponding to each stationary state. We do this by using mathematical operators to extract the information from the
wave function. For example, Schrodinger's wave equation uses the Hamiltonian operator to obtain the possible values
of energy that we might observe upon measuring the energy of an initial state. Quantum mechanics is inherently
probabilistic (stochastic) and we can rarely predict the outcome of a measurement with certainty (probability = 1) but
more generally we obtain all the possible values that could be observed and the probabilities that they will be observed.
The electron in a hydrogen atom, for example, could be any state, but we can obtain the probability of finding it in a
given region of space around the nucleus.

Mixed states are also mixtures of these stationary eigenstates, and so the eigenstates form a basis for describing any
state of the system. Schrodinger's wave equation, however, is non-relativistic and does not incorporate spin.
Nevertheless, the treatment for spin is analogous. We begin with an initial state, simulate a measurement by acting on
the initial state with a mathematical operator, which corresponds to the observable we are measuring, and obtain the
final state(s) which we can observe and measure.

Mathematically, we can represent any state as a
vector. For spin a 2D vector is sufficient, as we have only spin-up
and spin-down as the only two possible outcomes of measurement. More generally we can use a vector in
n-dimensions when there are n-possible outcomes. We use a set of vectors, with values representing each possible
outcome of measurement. Such a set of vectors occupies a mathematical n-dimensional space called a
Hilbert space.
We use a minimum set of vectors as
basis vectors (an eigenbasis). An analogy would be when plotting points on a
graph, you have just two axes, the x-axis and the y-axis, which can be represented as unit vectors (vectors of unit
length) which are orthogonal (at 90-degrees) to one-another. The x-axis can then be represented by the unit vector
(1,0) and the y-axis by (0,1). Any point can then be represented as a linear combination of these two basis vectors, so
the point x = 7, y = 4 = (7,4) = 7*(1,0) + 4*(0,1). Similarly with spin, we have two unit basis vectors, called
spinors, and
any spin can then be represented as a linear combination of these.

Our operators are then represented by
matrices (with n by n dimensions, in the case of spin we have 2 by 2 matrices).
Matrices can operate on a vector, by multiplying it, to produce another vector. This is used in geometry, for example to
translate or rotate a point on a graph, the point, represented by a vector, is multiplied by the translation or rotation
matrix. The elements of the matrix may be complex or imaginary numbers (involving the square-root of -1 or i). Complex
numbers crop up a lot in quantum mechanics, but the mathematics always resolves itself to give measurements that are
real numbers. The basis vectors represent stationary states (eigenstates) of the matrix and so are eigenvectors. An
eigenvector of a matrix is a vector which when multiplied by its corresponding matrix becomes a scalar multiple of
itself, that is itself times a number (a scalar). The number, or eigenvalue, represents an observable value, the result of
a measurement. For example, the z-spin up spinor is an eigenvector of the matrix corresponding to a measurement of
the z-component of spin. If we make a measurement on a pure z-spin up state represented by this spinor, then we
multiply it by the z-spin operator matrix and the result is the z-spin up vector times the eigenvalue which is h-bar/2
(h-bar is Planck's constant h divided by 2 x pi). This means that when we measure the z-spin of a pure z-spin up beam
of Cs atoms, we do not change their state, as this is a stationary eigenstate, and we obtain the value +(h-bar/2) which
is the magnitude of the spin for a spin-up Cs atom! A general state might not be a basis or eigenvector, but it can be
resolved into linear multiples of the basis vectors, just like the point on our graph.

Note: our basis vectors must also be orthogonal (this may be difficult to visualise when there are more than three
eigenvectors, as there are in other systems, but mathematically the equivalent condition can be enforced and which is
why we say orthogonal rather than perpendicular, as perpendicular vectors in 2D and 3D are just examples of the more
general principle of orthogonality which holds in any number of dimensions).

Eigenvectors and eigenvalues crop up a lot in physics and mathematics as the solutions to physical problems. We are
indeed fortunate that mathematics works so well! The arguments used for spin apply to other systems, but spin is the
simplest as it only requires 2D vectors and 2 by 2 matrices. The choice of basis vectors is not unique, we chose to use
the spinors for z-spin (the eigenvectors of the z-spin operator matrix) but we can use those of the x and y operators or
some other general basis which is a mixture of these. We can switch basis when it is mathematically convenient to do so.

These arguments are summarised in the pages below (click pages to enlarge) along with worked examples:
the magnetic field around an electron modelled in Pov-Ray
The magnetic field around an electron modelled in Pov-Ray
Left: a representation of the magnetic
field around an electron. The electron
is a
magnetic dipole, that is it
possesses two magnetic poles: a
magnetic North pole and a magnetic
South pole.
Magnets of course interact with one-another, due to the presence of their magnetic fields like poles repel and unlike
pole attract and magnets will rotate in order to align in a stable configuration. Similarly, when the electron is placed in
the magnetic field of the Stern-Gerlach magnets then the electrons move, they are accelerated. It is necessary for the
magnetic field to be non-uniform in order to accelerate the electron 'magnets'. The electrons are accelerated in two
directions, depending whether their spins are up and down relative to the magnetic field. If we define the magnetic field
of the Stern-Gerlach apparatus, which extends between our pair of magnets from the magnetic North to the magnetic
South, as the z-axis (the vertical axis in our diagram) then the beam gets split into two since those electrons with
spin-up compared to the z-axis (i.e. with magnetic spin quantum number = +1/2) move in one direction and those with
spin-down relative to the z-axis (magnetic spin quantum number = -1/2) move in the opposite direction and both spins
are equally probable.

Using electron beams instead of atom beams would be complicated by the fact that electrons have electric charge and
so accelerate in a magnetic field due to magnetic-electric interactions which would swamp the magnetic-magnetic
interactions we wish to study. Why do we wish to study magnetic-magnetic interactions? Because the
magnetic
moment
of a particle is proportional to its angular momentum. Magnetic moment is essentially the turning force a
magnet experiences in a magnetic field. Thus, we are essentially measuring magnetic moment of the atoms, and from
this we can obtain their angular momentum! The constant of proportionality between the magnetic moment and
angular momentum is called the
gyromagnetic ratio:

                                magnetic moment = gyromagnetic ratio x angular momentum

This tiny magnetic moment is measured in units of the
Bohr magneton, and has a value of about 1 Bohr magneton.
For any system, the magnetic moment can be expressed as a multiple (number) of Bohr magnetons, this multiple being
called the
Lande g factor:

                                magnetic moment = Lande g factor x Bohr magneton

For an electron in an atom, modelled as a Bohr atom (with a circular orbit) the Lande g factor equals 1. For electron
spin the g factor is almost exactly 2 (though not quite). [Note: there is an acute accent on the e of Lande, so it should
be pronounced something like: 'lan-day'.]
Stern-Gerlach apparatus modelled in Pov-Ray, with labels
the mathematics of quantum spin 1
Click these two pages to enlarge them and see a
summary of the key mathematical points of the
description of spin in quantum mechanics and some
examples.
the mathematics of quantum spin 2
Magnetic Spin precession

So far we have looked at what happens to spinning dipoles, such as the electrons in Cs atoms, when they encounter a
non-uniform magnetic field - they are accelerated along the field in a direction depending on the sense of their spin
(up or down). Now we look at a phenomenon called spin precession.
diagram of the Stern-Gerlach magnets showing non-uniform magnetic field
Left: the Stern-Gerlach magnets
illustrating the non-uniform magnetic
field which accelerates dipoles by
exerting a greater force on one
magnetic pole, of the dipole, than
the other. Spinning electrons in
atoms act as such dipoles. The
direction of the force depends on
the direction (sense) of the spin.
charged particle precession in a uniform magnetic field
What happens to a spinning electric charge (such as an
electron) in a uniform magnetic field. In a uniform
magnetic field the magnetic field lines are straight,
parallel and evenly spaced. Such a field does not
translate the charge through space, as a non-uniform
magnetic field does, since in a uniform field the force
exerted on one magnetic pole, of the dipole, is
counterbalanced by the force exerted at the other pole.
Thus there is no net translation or linear acceleration of
the particle or charge through space.

Instead, the classical particle processes - its angular
momentum (spin) vector traces circles around the
magnetic field lines, as shown on the left. The particle
rotates with the Larmor frequency (though a relativistic
correction called Thomas precession may need to be
applied). This is similar in some ways to the
(non-magnetic) precession of a spinning top or
gyroscope (and of astronomical objects, such as the
Earth which precesses around its own axis once every
26 000 years).
See also: particle trajectories.

This is classical precession, but what happens quantum-mechanically?

Quantum mechanical spin precession

What we need is to let spin evolve over time in a magnetic field. To do this we can use Schrodinger's time-dependent
wave equation (or an equation which is identical to it) and replace the Hamiltonian (energy operator) with a
spin-Hamiltonian 9which incorporates Pauli's spin matrices). If we do this then it is found that something similar to spin
precession occurs. For example, if we start by preparing a beam of Cs atoms all with spin-up in the z-direction, and
then pass this beam through a uniform magnetic field in the y-direction, then we find that the expectation values for
spin change sinusoidally (i.e. cyclically) over time. An
expectation value in quantum mechanics is the predicted
value of a measurement on a population of particles, it is a bit like an average, since in quantum mechanics we can
rarely predict the behaviour of an individual particle, due to the probabilistic nature of quantum mechanics. In our
specific example, going from z-spin up to a uniform magnetic y-field, we find that the expectation value for the
y-component of spin is zero, but that the expectation values for the x and z-components vary sinuosidally in time.
Thus the spin of our particles rotates or precesses. We should caution, however, that we still do not know exactly what
spin is, on the scale of atomic and sub-atomic particles, and the uncertainty principle forbids an individual particle
from following a specific trajectory, such as a circular precession. However, when studying populations (ensembles) of
particles, something similar does indeed result.

This is one of those cases where quantum mechanics produces motion, though not like classical motion. Perhaps we
could think of the precessing particle as being in a superposition spit, having both x and z-components of spin, and
often in quantum mechanics such superpositional states produce motion.
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