All language subtitles for s9.1

hello everybody

my name is Ian Applebaum

and I'm an associate 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

and semiconductor materials

and I feel strongly that understanding the

historical background I'm gonna tell you about today

is absolutely essential

in making future progress in this modern field

I wanna 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 the emission of electromagnetic radiation

optical photons

and a spectrometer can be used to disperse

the different wavelengths across a detector

the discreet waveleng scene

were known long before quantum theory

to satisfy Ridberg's formula

which told us that the photon energy

is proportional to the difference

between the reciprocals of two squares of integers

Boar's theory

which included quanization of angular momentum

in classical electronic 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 Schrodinger's equation

in spherically 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 1

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 asthmath 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 principal quantum number n

has a maximum L equal to n minus 1

and stage with a given value for L have values of m

given by minus l minus l plus 1

in inter steps through 0 and up to l minus 1 and L

this gives a total degeneracy of 2 L+ 1 therefore

each set labeled by given principal 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 mission spectrum

we can do this with a magnetic field here's how

let's imagine that we have a classical electron orbit

with total angular momentum given by this expression

now this circulating charge 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

and the area

of course for circular orbit

is just pi times the radius squared

now the frequency

of the orbit is just given by the velocity

divided by the circumference to pi r

and that gives rise to a simple expression

for the magnetic moment

but here we're gonna play a little bit of a trick

and multiply and divide by the electron mass

and also planks constant

and the reason why we do this is that we can see MVR

here is the absolute value of

the total angular momentum

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 boar 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

their energy

igan values 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 this 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 asthma with all angle

is its conjugate variable

the expectation value then

is this matrix element

the operator sandwich

by the state

the only part of the way function that matters here

is the fi 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 energy units of h bar

determined by the asmothal 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 orbits

magnetic moment with magnetic field long z

is simply the boar 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 bore magneton is small

about 60 micro electron volts per Tesla

and a Tesla is a huge magnetic field

about 20,000 times the strength

of the Earth's geomagnetic field

largest fields in the lab

created with superconducting coils

are several tens of Tesla

the energies are then small

in comparison to the electronic transitions

so any observe 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 equals 0

including the sole ground state level

are unaffected

the splitting energy is named after Peter Zeman

who won the Nobel Prize in 19:02

for observing the spectral line splitting

when gas discharge tubes

were placed in the magnetic field

now we have to remember

that this energy spectrum we calculate

is not the same as the optical spectrum

of emitted photons

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 gonna calculate them next

during an electronic transition

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

center 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 inner product of the way function

are symmetric whereas r

the radio radio variable

is antisemmetric

so integration over them vanishes

the cross terms

which oscillate at the frequency determined by

the energy difference of initial and final states

may survive

now though away functions

have previously been written in spherical coordinates

we wanna convert this integral

into Cartesian coordinates

so we may see along which axis we might get it

charge oscillation

and electromagnetic radiation

here's the transformation we need

in terms of the polar and asmothal angles

theta and fi

it allows us to split

the vector integral from the previous slide

into individual Cartesian components first

let's consider the x and y components of the integral

over asmothalangal

we'll have to evaluate something like this

two complex exponents containing phi

and a sign of cosine over phi

using the oiler formulas

we see that this will involve integrals over

inager periods of the oscillating complex exponential

which is identically equal to 0

except when the exponent is zero

since the z component does not depend on Fi

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

taken together

we see that there is no dipole transition

and no corresponding emission of photons

unless the change in magnetic one number

is plus or minus 1 or 0

this is our selection rule

by considering these kinds of arguments

about periodic symmetry of the integrand

and the integral over the polar angle theta

we can likewise derive another selection rule

for transitions

such that the change in orbital quantum number

must be plush minus 1

so which transitions are allowed

l must change by 1

and m can change at most by one

the two piece states can decay to the 1 s ground state

and in a magnetic field

all three of these transitions from m equals 1 0MINUS1

have different energies

giving different emitted photon wavelengths

that we can analyze with the spectrometer

3 s state decays only to the three 2P states

and 3P decays to 2S

also with three distinct transition energies

for higher values of principal quantum number n

we can have transitions from the L equals to d states

with a broken degeneracy of 5

which can make transitions to the lower p states

this involves many possible transitions that satisfy

the selection rules however

there are always only three different 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 wanna point out something extremely important

in atomic spectroscopy

recall the energy time uncertainty principle

it says that there is a reciprocal relationship

between the lifetime of a state

and the resulting spectral line with

which limits resolution of high precision measurements

if a transition is forbidden by dipole selection rules

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 2 s 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 1 s in about 1 nanosecond

the line with 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 ex

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 Zaman split spectral lines

their optical polarization and directional dependence

for instance

we know that the transition

corresponding to delta m equal to 0

no change in the magnetic quantum number

during the transition

is caused by a dipole along the magnetic field axis z

however we also know from classical electrodynamics

that an oscillating dipol

does not radiate along its axis

therefore this spectral line is absent

when observed along this orientation

the other two are present

and due to dipoles oscillating 90 degrees out of phase

along X and y

yielding right and left 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

Hendrick Lorenz won the Nobel Prize in 19:02

along with Zaman

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 Zaman effect

a serious problem remained

some special lines split into triplets as predicted

but others split into multiplets a 4 6 etc

here's a few examples

now we haven't made a silly mistake

we just need to reexamine the ingredients of our theory

namely distruding our equation

converting the classical kinetic

energy into an operator

is correct in the absolutely non relativistic case

as we've done here

with the construction of the Shortinger equation

but we're clearly leaving

out an essential piece of physics

by not even using a relativistically

invariant expression

now can we fix the problem by starting from scratch

constructing a wave equation

by starting with the relativeistic expression

for kinetic energy that's given here

that's what we're gonna see in the next slides

by the way those of you

who haven't seen this expression before

might wanna 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 its Taylor series

and see that the dominant terms are the familiar MC

square rest

mass energy

and the classical kinetic energy

everything else is small

although not negligible

as we will see

if we take the relativeistic expression

and try to use it as an operator on a way 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 Durak from writing down this

relativistically in varying equation in 1928

the alpha coefficients are clearly not scalar values

but matrices

which satisfy the anticommutation 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 4 by 4

here's one choice of basis

I 2 is a 2 by 2 identity

and O 2 is a 2 by 2 matrix of all zeros

the 2 by 2 poly matrices

the sigmas form the off diagonal blocks of the alpha 1

2 and three 4 by 4 matrices

now importantly

this converts the wave equation into

a 4 by 4 matrix equation

and the wave function into a four component vector

two of these correspond to arrest mass energy of

MC squared when the momentum p is equal to zero

as we expect for an electron

but a natural question to ask is why two values

degeneracies like this are 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