Hello everybody, my name is Ian 
Applebaum, and I'm an associated 
professor of physics at the University of 
Maryland. 
Today I'll be giving a guest lecture in 
this exciting online course exploring 
quantum physics. 
I'm thrilled to be delivering this topic 
in part because my own research deals 
with experimental aspects of spin 
polarized electron transport in semi 
conductor materials. 
And I feel strongly that understanding 
the historical background I'm going to 
tell you about today is absolutely 
essential in making future progress in 
this modern field. 
I want to start our discussion of the 
discovery of electron spin by reminding 
you of what was known around the turn of 
the 19th century about atomic spectra. 
High voltage across a discharge tube 
filled with low pressure hydrogen causes 
emission of electromagnetic radiation, 
optical photons. 
And a spectrometer can be used to 
disburse the different wavelengths across 
a detector. 
The discreet wavelength scene were known 
long before quantum theory to satisfy 
Rydberg's formula. 
Which told us that the photon energy is 
proportional to the difference between 
the reciprocals of two squared integers. 
Bohr's theory, which included 
quantization of angular momentum in 
classical electron orbits, captured this 
famous result. 
But it was wrong for many reasons we now 
know, such as failure to predict the 
ground state absence of angular momentum. 
And importantly the degeneracy of each 
principal electronic level. 
From the solution of Schodringer's 
equation in spherical symmetric 
potentials, we know not only that the 
ground state has zero angular momentum. 
But also that the excited levels where 
the principal quantum number is not one 
are actually degenerate meaning that 
several states have the same eigen 
energy. 
This follows from the multiple spherical 
harmonics labeled by the polar quantum 
number, L, and the azimuth of quantum 
number, M. 
A spectroscopic notation is to label l 
equals 0 as S, l equals 1 P, l equals 2 
D, and so forth. 
Each level of principle quantum number n 
has a maximum l equal to n minus 1 and 
states with a given value for l have 
values of m given by minus l, minus l 
plus 1, in integer steps through 0 and up 
to l minus 1 and l. 
This gives a total degeneracy of 2l plus 
1. 
Therefore, each set labeled by a given 
principle quantum number n has degeneracy 
of n squared. 
A natural question to ask then is how can 
we observe this degeneracy 
experimentally? 
We need to break the degeneracy, split 
the levels and alter the photon emission 
spectrum. 
We can do this with a magnetic field. 
Here's how. 
Let's imagine that we have a classical 
electron orbit, with angular momentum, 
given by this expression. 
Now, this circulating charged particle 
comprises a current, and so the orbit 
has, necessarily, a magnetic moment. 
Classically we know that the magnetic 
moment, that absolute value of that 
magnetic moment is given by just the 
current times the area that it circulates 
around. 
Now the current in this case is one 
electron circulating around in a given 
period. 
So this is the charge of the electron, 
minus e times the frequency of its orbit. 
The area of course for circular orbit is 
just pie times the radius squared. 
Now, the frequency of the orbit is just 
given by the velocity divided by the 
circumference, 2 pie r. 
And that gives rise to a simple 
expression for the magnetic moment. 
But here we're going to play a little bit 
of a trick and multiply and divide by the 
electron mass and also planck's constant. 
And the reason why we do this, is that we 
can see mvr. 
Here is the absolute value of the total 
angular momenteum which has the same 
units as the action h bar. 
That means that this fraction out front 
carries all of the units of the magnetic 
moment. 
In fact, it has a special name. 
It's called the Bohr magneton. 
So, with this magnetic moment, we know 
that in a magnetic field, each state is 
going to acquire an energy, due to the 
interaction. 
Now, if the magnetic field is along the z 
axis. 
Then we can simply write this interaction 
energy as minus the z component of the 
magnetic moment times the magnetic field 
along z. 
So, if different states, with the same 
principle quantum number n have different 
values for the z component of the 
magnetic moment, there energy eigenvalues 
will shift differently in a magnetic 
field, and the degeneracy will be broken. 
So we need to calculate the magnetic 
moment associated with each orbital from 
the preceding discussion. 
We know the relationship between angular 
momentum and magnetic moment. 
And if the field is aligned to the z axis 
we only need to calculate the vector 
component along z. 
But this means we need to know the 
angular momentum vector component along 
z. 
So we need to calculate its expectation 
value and for that we need an operator 
representation. 
We know that the linear momentum 
operators are h bar over I times the 
derivative with respect to the conjugate 
real space variable. 
So it's easy to see that the angular 
momentum along z takes this form because 
phi, the actual total angle, is its 
concrete variable. 
The expectation value then is this matrix 
element. 
The operator sandwiched by the state. 
The only part of the wave function that 
matters here is the phi dependence, which 
we know from the spherical harmonics. 
The derivative brings down a factor of I 
times m and what's left is the original 
normalized state. 
We see, then, that our answer is integer 
units of h bar determined by the 
azimuthal quantum number m. 
This is why m is commonly called the 
magnetic quantum number. 
Now we can complete our calculation. 
The energy added due to the interaction 
of the orbit's magnetic moment with 
magnetic field along z is simply the Bohr 
magneton times the magnetic field 
strength times the magnetic quantum 
number. 
This result is pleasing enough, but some 
understanding of the scale of the effect 
is helpful. 
The Bohr magneton is small, about 60 
microelectron volts per Tesla. 
And a Tesla is a huge magnetic field, 
about 20,000 times the strength of the 
Earth's geomagnetic field. 
The largest fields in the lab created 
with super conducting coils are several 
tens of Tesla. 
The energies are then small in comparison 
to the electronic transitions, so any 
observed shifts of the spectral lines are 
going to be proportionately small. 
So here's what happens, a non-zero 
magnetic field induces a splitting 
between degenerate states, adding energy 
to states with positive magnetic quantum 
number m. 
And subtracting from states with negative 
m. 
All states with m equal 0, including the 
sole ground-state level, are un-affected. 
The splitting energy is named after 
Pieter Zeeman who won the Nobel Prize in 
1902 for observing the spectral line 
splitting when gas tubes were placed in a 
magnetic field. 
Now we have to remember that this energy 
spectrum we calculate is not the same as 
the optical spectrum of emitted photos. 
Which are only due to transitions between 
the levels and not all transitions are 
allowed. 
The expressions which determine when the 
transitions are allowed are known as 
selection rules. 
We're going to calculate them next. 
During an electronic transition, the 
electron wave function forms a 
superposition of the initial state and 
the final state with lower energy. 
Each component evolves differently in 
time. 
So the electron probability distribution 
centre of mass can move. 
This oscillating dipole is what radiates 
electromagnetic energy. 
A simple calculation of the expectation 
value of position yields this expression. 
Where we can see the first two direct 
terms from the interproduct of the wave 
function are symmetric. 
Whereas r, the radio, radial variable, is 
antisymmetric, so integration over them 
benefits. 
The cross terms which oscillate at the 
frequency determined by the energy 
difference of initial and final states 
may survive. 
Now, though our wave functions have 
previously been written in spherical 
coordinates. 
We want to convert this integral into 
Cartesian coordinates, so we may see 
along which axis we might get charge 
oscillation and electromagnetic 
radiation. 
Here's the transformation we need in 
terms of the polar and azimuthal angles, 
theta and phi. 
It allows us to split the vector integral 
from the previous slide into individual 
Cartesian components. 
First lets consider the x and y 
components of the integral over azimuthal 
angle. 
We'll have to evaluate something like 
this. 
Two complex exponents containing phi and 
a sign of cosine over phi. 
Using the euler formulas you see that 
this will involve integrals over integer 
periods of the osci-, oscillating complex 
exponential. 
Which is identically equal to 0, except 
when the exponent is 0. 
Since the z component does not depend on 
phi. 
It's even easier to see that in this 
case, the integral will be equal to 0 
only if the magnetic quantum numbers for 
initial and final states are the same. 
Taking together we see that there is no 
dipole transition and no corresponding 
emission of photons. 
Unless the change in magnetic quantum 
number is plus or minus 1 or 0. 
This is our selection rule. 
By considering these kinds of arguments 
about periodic symmetry of the intergrand 
and the integral over the polar angel 
theta. 
We can likewise derive another selection 
rule for transitions such that the change 
in orbital quantum number must be plus or 
minus 1. 
So which transitions are allowed. 
l must change by 1 and m can change at 
most by 1. 
The 2P states can decay to the 1S ground 
state. 
And in a magnetic field, all 3 of these 
transitions from m equals 1, 0 minus 1, 
have different energies giving different 
emitted photon wavelengths that we can 
analyze with a spectrometer. 
The 3S state decays only to the 32P 
states, and 3P decays to 2S, also with 3 
distinct transition energies. 
For higher values of principal quantum 
number n, we can have transitions from 
the l equals 2d states with a broken 
degeneracy of 5. 
Which can make transitions the lower p 
states, this involves many possible 
transitions that satisfy the selection 
rules, however there are always only 3 
transition energies. 
Delta m equals minus 1, 0, and plus 1. 
So each spectral line splits into a 
triplet in a magnetic field. 
Now, I want to point out something 
extremely improtant in atomic 
spectroscopy. 
Recall the energy time uncertainty 
principle. 
It says that there's a reciprocal 
relationship between the lifetime of a 
state and the resulting spectral line 
width, which limits resolution of high 
precision measurements. 
If a transition is forbidden by dipole 
selection rule, other processes may be 
allowed. 
But are typically far less efficient, and 
result in exceptionally long lifetimes. 
Here, we see that the transition to the 
ground state, from the 2S state is 
forbidden by the selection rules. 
And other processes yield a lifetime, of 
over 100 milliseconds in comparison to 
the nearly equal transition. 
From 2p to 1s, in about 1 ns. 
The line width of this transition is 
therefore extremely narrow, allowing very 
high precision measurements of exquisite 
quantum effects, such as the lamb shift. 
Due to small corrections of energy levels 
from quantum electrodynamical effects of 
the electron interacting with short-lived 
excitations in the vacuum. 
In addition to the selection rules, we 
can use the different forms of the dipole 
vector components to explain another 
feature of Zeeman split spectral lines. 
They're optical polarizations and 
directional dependents. 
For instance, we know that the transition 
corresponding to delta N is 0. 
No change in the magnetic quantum number 
during the transition is called by a 
dipole along the magnetic field axis Z. 
However we also know from classical 
electrodynamics that an oscillating 
dipole does not radiate along its axis. 
Therefore this spectral line is absent 
when observed along this orientation. 
The other two are present and do have the 
dipoles oscillating 90 degrees out of 
phase along X and Y, yielding left and 
right handed circularly polarized light. 
If we observe from a direction 
perpendicular to the magnetic field axis. 
Then, the delta m equal to zero line in 
the middle can be seen, and has a linear 
polarization along z. 
The other two are polarized perpendicular 
to the field axis because they are, 
again, due to dipoles along x and y. 
Hendrik Lorentz won the Nobel Prize in 
1902 along with Zeeman for explaining 
this polarization dependence. 
He used only classical physics, a theory 
which we now know is wrong. 
Despite this perceived success in 
explaining the Zeeman effect, a serious 
problem remained, some special lines 
split into triple lines as predicted. 
But others look into a multiplex, four, 
six, etc. 
Here's a few examples. 
Now you haven't made a silly mistake. 
We just need to re-examine the 
ingredients of our theory, namely the 
Schrodinger equation. 
Converting the classical kinetic energy 
into an operator is correct in the 
absolutely non-relativistic case, as 
we've done here with construction of the 
Schrodinger equation. 
But we're clearly leaving out an 
essential piece of physics. 
We're not even using the relativistically 
covariant expression. 
Now, can we fix the problem by starting 
from scratch, constructing a wave 
equation, by starting with the 
relativistic expression for kinetic 
energy that's given here. 
That's what we're going to see in the 
next slides. 
By the way, those of you who haven't seen 
this expression before, might want to see 
that It's asymptotically equivalent to 
the classical expression in the limit 
that the momentum p is small. 
Then we can expand the square root in a 
Taylor series and see that the dominant 
terms are the familiar mc squared, rest 
mass energy, and the classical kinetic 
energy. 
Everything else is small, although not 
negligible, as we will see. 
If we take the relativistic expression, 
and try to use it as an operator on a 
wave function. 
We immediately encounter a problem. 
Our momentum operators are within the 
square root. 
And it's not clear at all whether this 
makes any mathematical sense. 
The problem disappears if the expression 
inside the square root is, itself, a 
perfect square. 
If we write the rest mass energy and 
kinetic energy components in x, y, and z 
here, 1, 2, and 3 with arbitrary 
coefficients. 
Then we can make this a perfect square, 
if these coefficients satisfy what 
appears, at first, to be an unusual 
constraint. 
They give unity when squared, but they 
anti commute with each other. 
These coefficients clearly are not scalar 
values, however this fact didn't frighten 
Paul Dirac from wriiting down this 
relativisticly and variant equation in 
1928. 
The alpha coefficients are clearly not 
scalar values. 
But matrices which satisfy the anti 
commutation relations. 
And form a so called Clifford Algebra. 
Now, we can write down many matrices 
which satisfy the algebra. 
But it makes sense to first look at the 
simplest case with the smallest dimension 
matrices. 
It turns out this can be done with 
matrices as small as four by four. 
Here's one choice of basis. 
I2 is a 2 by 2 identity, and o2 is a 2 by 
2 matrix of all zeroes. 
The 2 by 2 Pauli matrices, the sigmas, 
form the off diagonal blocks of the alpha 
1, 2, and 3, 4 by 4 matrices. 
Now importantly this converts the wave 
equation into a four by four matrix 
equation. 
And the wave function into a four 
component vector. 
Two of these correspond to a rest mass 
energy of mc squared, when the momentum p 
is equal to 0, as we expect for an 
electron. 
But an actual question to ask is why two 
values? 
Degeneracies like this are a signature of 
symmetry, but which one? 
Which degree of freedom do these two 
values correspond to? 
We're going to look at an experiment for 
a clue