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