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Hello everybody, my name is Ian 
Applebaum, and I'm an associated 

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professor of physics at the University of 
Maryland. 

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Today I'll be giving a guest lecture in 
this exciting online course exploring 

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quantum physics. 
I'm thrilled to be delivering this topic 

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in part because my own research deals 
with experimental aspects of spin 

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polarized electron transport in semi 
conductor materials. 

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And I feel strongly that understanding 
the historical background I'm going to 

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tell you about today is absolutely 
essential in making future progress in 

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this modern field. 
I want to start our discussion of the 

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discovery of electron spin by reminding 
you of what was known around the turn of 

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the 19th century about atomic spectra. 
High voltage across a discharge tube 

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filled with low pressure hydrogen causes 
emission of electromagnetic radiation, 

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optical photons. 
And a spectrometer can be used to 

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disburse the different wavelengths across 
a detector. 

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The discreet wavelength scene were known 
long before quantum theory to satisfy 

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Rydberg's formula. 
Which told us that the photon energy is 

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proportional to the difference between 
the reciprocals of two squared integers. 

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Bohr's theory, which included 
quantization of angular momentum in 

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classical electron orbits, captured this 
famous result. 

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But it was wrong for many reasons we now 
know, such as failure to predict the 

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ground state absence of angular momentum. 
And importantly the degeneracy of each 

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principal electronic level. 
From the solution of Schodringer's 

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equation in spherical symmetric 
potentials, we know not only that the 

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ground state has zero angular momentum. 
But also that the excited levels where 

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the principal quantum number is not one 
are actually degenerate meaning that 

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several states have the same eigen 
energy. 

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This follows from the multiple spherical 
harmonics labeled by the polar quantum 

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number, L, and the azimuth of quantum 
number, M. 

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A spectroscopic notation is to label l 
equals 0 as S, l equals 1 P, l equals 2 

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D, and so forth. 
Each level of principle quantum number n 

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has a maximum l equal to n minus 1 and 
states with a given value for l have 

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values of m given by minus l, minus l 
plus 1, in integer steps through 0 and up 

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to l minus 1 and l. 
This gives a total degeneracy of 2l plus 

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1. 
Therefore, each set labeled by a given 

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principle quantum number n has degeneracy 
of n squared. 

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A natural question to ask then is how can 
we observe this degeneracy 

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experimentally? 
We need to break the degeneracy, split 

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the levels and alter the photon emission 
spectrum. 

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We can do this with a magnetic field. 
Here's how. 

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Let's imagine that we have a classical 
electron orbit, with angular momentum, 

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given by this expression. 
Now, this circulating charged particle 

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comprises a current, and so the orbit 
has, necessarily, a magnetic moment. 

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Classically we know that the magnetic 
moment, that absolute value of that 

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magnetic moment is given by just the 
current times the area that it circulates 

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around. 
Now the current in this case is one 

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electron circulating around in a given 
period. 

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So this is the charge of the electron, 
minus e times the frequency of its orbit. 

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The area of course for circular orbit is 
just pie times the radius squared. 

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Now, the frequency of the orbit is just 
given by the velocity divided by the 

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circumference, 2 pie r. 
And that gives rise to a simple 

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expression for the magnetic moment. 
But here we're going to play a little bit 

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of a trick and multiply and divide by the 
electron mass and also planck's constant. 

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And the reason why we do this, is that we 
can see mvr. 

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Here is the absolute value of the total 
angular momenteum which has the same 

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units as the action h bar. 
That means that this fraction out front 

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carries all of the units of the magnetic 
moment. 

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In fact, it has a special name. 
It's called the Bohr magneton. 

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So, with this magnetic moment, we know 
that in a magnetic field, each state is 

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going to acquire an energy, due to the 
interaction. 

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Now, if the magnetic field is along the z 
axis. 

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Then we can simply write this interaction 
energy as minus the z component of the 

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magnetic moment times the magnetic field 
along z. 

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So, if different states, with the same 
principle quantum number n have different 

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values for the z component of the 
magnetic moment, there energy eigenvalues 

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will shift differently in a magnetic 
field, and the degeneracy will be broken. 

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So we need to calculate the magnetic 
moment associated with each orbital from 

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the preceding discussion. 
We know the relationship between angular 

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momentum and magnetic moment. 
And if the field is aligned to the z axis 

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we only need to calculate the vector 
component along z. 

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But this means we need to know the 
angular momentum vector component along 

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z. 
So we need to calculate its expectation 

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value and for that we need an operator 
representation. 

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We know that the linear momentum 
operators are h bar over I times the 

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derivative with respect to the conjugate 
real space variable. 

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So it's easy to see that the angular 
momentum along z takes this form because 

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phi, the actual total angle, is its 
concrete variable. 

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The expectation value then is this matrix 
element. 

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The operator sandwiched by the state. 
The only part of the wave function that 

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matters here is the phi dependence, which 
we know from the spherical harmonics. 

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The derivative brings down a factor of I 
times m and what's left is the original 

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normalized state. 
We see, then, that our answer is integer 

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units of h bar determined by the 
azimuthal quantum number m. 

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This is why m is commonly called the 
magnetic quantum number. 

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Now we can complete our calculation. 
The energy added due to the interaction 

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of the orbit's magnetic moment with 
magnetic field along z is simply the Bohr 

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magneton times the magnetic field 
strength times the magnetic quantum 

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number. 
This result is pleasing enough, but some 

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understanding of the scale of the effect 
is helpful. 

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The Bohr magneton is small, about 60 
microelectron volts per Tesla. 

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And a Tesla is a huge magnetic field, 
about 20,000 times the strength of the 

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Earth's geomagnetic field. 
The largest fields in the lab created 

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with super conducting coils are several 
tens of Tesla. 

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The energies are then small in comparison 
to the electronic transitions, so any 

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observed shifts of the spectral lines are 
going to be proportionately small. 

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So here's what happens, a non-zero 
magnetic field induces a splitting 

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between degenerate states, adding energy 
to states with positive magnetic quantum 

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number m. 
And subtracting from states with negative 

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m. 
All states with m equal 0, including the 

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sole ground-state level, are un-affected. 
The splitting energy is named after 

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Pieter Zeeman who won the Nobel Prize in 
1902 for observing the spectral line 

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splitting when gas tubes were placed in a 
magnetic field. 

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Now we have to remember that this energy 
spectrum we calculate is not the same as 

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the optical spectrum of emitted photos. 
Which are only due to transitions between 

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the levels and not all transitions are 
allowed. 

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The expressions which determine when the 
transitions are allowed are known as 

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selection rules. 
We're going to calculate them next. 

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During an electronic transition, the 
electron wave function forms a 

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superposition of the initial state and 
the final state with lower energy. 

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Each component evolves differently in 
time. 

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So the electron probability distribution 
centre of mass can move. 

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This oscillating dipole is what radiates 
electromagnetic energy. 

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A simple calculation of the expectation 
value of position yields this expression. 

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Where we can see the first two direct 
terms from the interproduct of the wave 

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function are symmetric. 
Whereas r, the radio, radial variable, is 

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antisymmetric, so integration over them 
benefits. 

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The cross terms which oscillate at the 
frequency determined by the energy 

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difference of initial and final states 
may survive. 

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Now, though our wave functions have 
previously been written in spherical 

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coordinates. 
We want to convert this integral into 

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Cartesian coordinates, so we may see 
along which axis we might get charge 

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oscillation and electromagnetic 
radiation. 

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Here's the transformation we need in 
terms of the polar and azimuthal angles, 

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theta and phi. 
It allows us to split the vector integral 

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from the previous slide into individual 
Cartesian components. 

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First lets consider the x and y 
components of the integral over azimuthal 

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angle. 
We'll have to evaluate something like 

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this. 
Two complex exponents containing phi and 

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a sign of cosine over phi. 
Using the euler formulas you see that 

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this will involve integrals over integer 
periods of the osci-, oscillating complex 

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exponential. 
Which is identically equal to 0, except 

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when the exponent is 0. 
Since the z component does not depend on 

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phi. 
It's even easier to see that in this 

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case, the integral will be equal to 0 
only if the magnetic quantum numbers for 

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initial and final states are the same. 
Taking together we see that there is no 

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dipole transition and no corresponding 
emission of photons. 

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Unless the change in magnetic quantum 
number is plus or minus 1 or 0. 

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This is our selection rule. 
By considering these kinds of arguments 

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about periodic symmetry of the intergrand 
and the integral over the polar angel 

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theta. 
We can likewise derive another selection 

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rule for transitions such that the change 
in orbital quantum number must be plus or 

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minus 1. 
So which transitions are allowed. 

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l must change by 1 and m can change at 
most by 1. 

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The 2P states can decay to the 1S ground 
state. 

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And in a magnetic field, all 3 of these 
transitions from m equals 1, 0 minus 1, 

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have different energies giving different 
emitted photon wavelengths that we can 

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analyze with a spectrometer. 
The 3S state decays only to the 32P 

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states, and 3P decays to 2S, also with 3 
distinct transition energies. 

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00:11:04,175 --> 00:11:08,790
For higher values of principal quantum 
number n, we can have transitions from 

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the l equals 2d states with a broken 
degeneracy of 5. 

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00:11:13,495 --> 00:11:17,968
Which can make transitions the lower p 
states, this involves many possible 

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transitions that satisfy the selection 
rules, however there are always only 3 

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transition energies. 
Delta m equals minus 1, 0, and plus 1. 

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So each spectral line splits into a 
triplet in a magnetic field. 

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Now, I want to point out something 
extremely improtant in atomic 

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spectroscopy. 
Recall the energy time uncertainty 

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principle. 
It says that there's a reciprocal 

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relationship between the lifetime of a 
state and the resulting spectral line 

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width, which limits resolution of high 
precision measurements. 

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If a transition is forbidden by dipole 
selection rule, other processes may be 

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allowed. 
But are typically far less efficient, and 

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result in exceptionally long lifetimes. 
Here, we see that the transition to the 

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ground state, from the 2S state is 
forbidden by the selection rules. 

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And other processes yield a lifetime, of 
over 100 milliseconds in comparison to 

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the nearly equal transition. 
From 2p to 1s, in about 1 ns. 

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The line width of this transition is 
therefore extremely narrow, allowing very 

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high precision measurements of exquisite 
quantum effects, such as the lamb shift. 

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Due to small corrections of energy levels 
from quantum electrodynamical effects of 

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the electron interacting with short-lived 
excitations in the vacuum. 

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In addition to the selection rules, we 
can use the different forms of the dipole 

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vector components to explain another 
feature of Zeeman split spectral lines. 

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They're optical polarizations and 
directional dependents. 

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00:12:54,350 --> 00:12:58,900
For instance, we know that the transition 
corresponding to delta N is 0. 

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No change in the magnetic quantum number 
during the transition is called by a 

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00:13:02,676 --> 00:13:08,670
dipole along the magnetic field axis Z. 
However we also know from classical 

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electrodynamics that an oscillating 
dipole does not radiate along its axis. 

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00:13:13,978 --> 00:13:18,610
Therefore this spectral line is absent 
when observed along this orientation. 

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00:13:19,750 --> 00:13:23,715
The other two are present and do have the 
dipoles oscillating 90 degrees out of 

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phase along X and Y, yielding left and 
right handed circularly polarized light. 

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00:13:29,550 --> 00:13:35,055
If we observe from a direction 
perpendicular to the magnetic field axis. 

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00:13:35,055 --> 00:13:39,402
Then, the delta m equal to zero line in 
the middle can be seen, and has a linear 

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00:13:39,402 --> 00:13:45,220
polarization along z. 
The other two are polarized perpendicular 

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00:13:45,220 --> 00:13:49,540
to the field axis because they are, 
again, due to dipoles along x and y. 

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00:13:51,540 --> 00:13:55,892
Hendrik Lorentz won the Nobel Prize in 
1902 along with Zeeman for explaining 

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00:13:55,892 --> 00:14:01,065
this polarization dependence. 
He used only classical physics, a theory 

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00:14:01,065 --> 00:14:05,522
which we now know is wrong. 
Despite this perceived success in 

186
00:14:05,522 --> 00:14:10,340
explaining the Zeeman effect, a serious 
problem remained, some special lines 

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00:14:10,340 --> 00:14:18,645
split into triple lines as predicted. 
But others look into a multiplex, four, 

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00:14:18,645 --> 00:14:23,485
six, etc. 
Here's a few examples. 

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00:14:23,485 --> 00:14:28,785
Now you haven't made a silly mistake. 
We just need to re-examine the 

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00:14:28,785 --> 00:14:32,886
ingredients of our theory, namely the 
Schrodinger equation. 

191
00:14:34,610 --> 00:14:37,823
Converting the classical kinetic energy 
into an operator is correct in the 

192
00:14:37,823 --> 00:14:41,240
absolutely non-relativistic case, as 
we've done here with construction of the 

193
00:14:41,240 --> 00:14:45,593
Schrodinger equation. 
But we're clearly leaving out an 

194
00:14:45,593 --> 00:14:48,932
essential piece of physics. 
We're not even using the relativistically 

195
00:14:48,932 --> 00:14:54,342
covariant expression. 
Now, can we fix the problem by starting 

196
00:14:54,342 --> 00:14:57,422
from scratch, constructing a wave 
equation, by starting with the 

197
00:14:57,422 --> 00:15:01,980
relativistic expression for kinetic 
energy that's given here. 

198
00:15:03,680 --> 00:15:05,230
That's what we're going to see in the 
next slides. 

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00:15:07,180 --> 00:15:10,597
By the way, those of you who haven't seen 
this expression before, might want to see 

200
00:15:10,597 --> 00:15:13,963
that It's asymptotically equivalent to 
the classical expression in the limit 

201
00:15:13,963 --> 00:15:19,723
that the momentum p is small. 
Then we can expand the square root in a 

202
00:15:19,723 --> 00:15:23,221
Taylor series and see that the dominant 
terms are the familiar mc squared, rest 

203
00:15:23,221 --> 00:15:27,000
mass energy, and the classical kinetic 
energy. 

204
00:15:28,110 --> 00:15:31,230
Everything else is small, although not 
negligible, as we will see. 

205
00:15:31,230 --> 00:15:35,632
If we take the relativistic expression, 
and try to use it as an operator on a 

206
00:15:35,632 --> 00:15:40,600
wave function. 
We immediately encounter a problem. 

207
00:15:40,600 --> 00:15:42,870
Our momentum operators are within the 
square root. 

208
00:15:42,870 --> 00:15:45,979
And it's not clear at all whether this 
makes any mathematical sense. 

209
00:15:47,110 --> 00:15:50,230
The problem disappears if the expression 
inside the square root is, itself, a 

210
00:15:50,230 --> 00:15:54,509
perfect square. 
If we write the rest mass energy and 

211
00:15:54,509 --> 00:15:58,108
kinetic energy components in x, y, and z 
here, 1, 2, and 3 with arbitrary 

212
00:15:58,108 --> 00:16:02,368
coefficients. 
Then we can make this a perfect square, 

213
00:16:02,368 --> 00:16:06,085
if these coefficients satisfy what 
appears, at first, to be an unusual 

214
00:16:06,085 --> 00:16:10,529
constraint. 
They give unity when squared, but they 

215
00:16:10,529 --> 00:16:16,430
anti commute with each other. 
These coefficients clearly are not scalar 

216
00:16:16,430 --> 00:16:21,955
values, however this fact didn't frighten 
Paul Dirac from wriiting down this 

217
00:16:21,955 --> 00:16:27,410
relativisticly and variant equation in 
1928. 

218
00:16:27,410 --> 00:16:30,580
The alpha coefficients are clearly not 
scalar values. 

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00:16:30,580 --> 00:16:34,420
But matrices which satisfy the anti 
commutation relations. 

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00:16:34,420 --> 00:16:38,600
And form a so called Clifford Algebra. 
Now, we can write down many matrices 

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00:16:38,600 --> 00:16:41,942
which satisfy the algebra. 
But it makes sense to first look at the 

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00:16:41,942 --> 00:16:45,160
simplest case with the smallest dimension 
matrices. 

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00:16:45,160 --> 00:16:49,260
It turns out this can be done with 
matrices as small as four by four. 

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00:16:49,260 --> 00:16:53,541
Here's one choice of basis. 
I2 is a 2 by 2 identity, and o2 is a 2 by 

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00:16:53,541 --> 00:17:00,132
2 matrix of all zeroes. 
The 2 by 2 Pauli matrices, the sigmas, 

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00:17:00,132 --> 00:17:07,835
form the off diagonal blocks of the alpha 
1, 2, and 3, 4 by 4 matrices. 

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00:17:07,835 --> 00:17:12,671
Now importantly this converts the wave 
equation into a four by four matrix 

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00:17:12,671 --> 00:17:17,610
equation. 
And the wave function into a four 

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00:17:17,610 --> 00:17:21,970
component vector. 
Two of these correspond to a rest mass 

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00:17:21,970 --> 00:17:26,170
energy of mc squared, when the momentum p 
is equal to 0, as we expect for an 

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00:17:26,170 --> 00:17:31,422
electron. 
But an actual question to ask is why two 

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00:17:31,422 --> 00:17:35,395
values? 
Degeneracies like this are a signature of 

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00:17:35,395 --> 00:17:39,830
symmetry, but which one? 
Which degree of freedom do these two 

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00:17:39,830 --> 00:17:43,330
values correspond to? 
We're going to look at an experiment for 

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00:17:43,330 --> 00:17:44,070
a clue