Lecture I

Reading: Lecture Notes + Sakurai: Chapter 1: sections 1, 2 and 3 )

I. INTRODUCTION

When we say Quantum Physics, we mean non-relativistic QM, its relativistic generalization

and also Quantum-Electrodynnamics (QED)

Quantum Weirdness & Quantum-Mechanical Way of Thinking

Quantum physics is the most successful theory and there has been NO violation of the

theory..That is, there are NO experiments that contradict quantum physics. Now, let us recall that

Newton’s theory survived more than 200 years before violations were found. Quantum Physics is

still less than 100 years old.

Is Quantum Mechanics the ultimate Theory..... NO.. We do not know how to describe gravity

in quantum mechanics... Quantizing gravity remains an open challenge ??

String Theory.... Quantizing Einstein’s equations of General Relativity...... possible alternative

theories in working...

Quantum mechanics is weird.. non-intuitive..why ???

Probabilistic Theory and EPR paradox ( quantum entanglement... )

Hidden variable Theory... Multi-universe theory...???

To cultivate QM way of thinking, we will not start with the historical approach, but instead

begin with one of the most important experiment, that sends a shock wave to those not used to QM..

Aim is to start with quantum spin: two reasons:

(1) Simplest way to introduce quantum mechanics: that is all fundamental aspects of

quantum can be introduced using spin which is purely quantum mechanical. We do not start with

Scrounger equation–that is not the most fundamental thing about quantum description, as we will

1

see shortly.

(2) There is another reason: The story of spin is one of the most beautiful story..

Story of spin is one of the most fascinating story in the history of science...

Where did it begin ???

I : Discovery of Spin in physics: Pauli & The Postulate of Kronig, Uhlenbeck and Goudsmit

• The story of spin begins with the discovery of the multiplicity of spectral terms and

anomalies of Zeeman effect[1].

• Around 1924, the idea that origin of multiplicity of specral lines is the electron began

to crystallize in Pauli’s mind. He proposed that the multiplicity is caused by a strange

two-valuedness of the quantum mechanical properties of the electron which cannot be

described classically.

• In January 1925, a young person physicist Ralph Kronig wanted to propose that electron is

self -rotating, as an extension of Pauli’s idea of two-valuedness of electron. This explained

the multiplicity, except for a factor of two in the level spacing. He discussed this with Pauli

who showed no enthusiasm for it. Factor of half discrepancy and lack of support from Pauli

and also Copenhagen group and idea of self-rotating electron presents difficulties when

examined within the framework of classical theory. For example, if the size of the electron

is e2/mc2, as H.A. Lorentz has considered, then to produce angular momentum of 1/2

requires electron rotating with speed that is 10 times the speed of light. All this prevented

Kronig to publish his idea.

• The rotation of an electron about its own axis, i.e., the electron spin, was first proposed by

Uhlenbeck and Goudsmit in Fall of 1925. Luckily, they did not discuss this with Pauli and

also were not aware of the factor of two discrepancy. After their paper was submitted, they

found out that their spinning electron model violates relativity. They wanted to withdraw

their paper. But their advisor Ehrenfest loved the idea and saw some merit in it even if it

turned out to be wrong.

2

• Later on L. H. Thomas resolved the discrepancy of this factor of two. Thomas used classical

relativity, and the classical theory of tops, supplemented by the correspondence principle to

describe the self-rotation of the electron and the discrepancy between theory and experiment

on energy spacing disappeared. ( This factor of two was predicted in Dirac theory...). After

this, the idea of self-rotating electron won Pauli’s approval.

• Pauli presented the first quantum-mechanical formulation of spin using his 2 × 2 matrices

still based on ad hoc assumptions. This led to introduction of spinors to describe the state

of the electron, in contrast to the scalar wave function of the Schodinger theory to describe

the state of the electron.

• II : Dirac Equation

In 1929, Paul Adrian Dirac derives relativistic equation which turns out to be the “ultimate”

theory of electron spin.

“I cannot adequately convey the admiration I feel for Dirac in making this discovery,

nor my gratitude to God for having lifted the scales from mankind’s eyes and giving us

this fantastic gift. Stephen Hawking has stated that if Dirac had somehow been able to

patent his discovery, the royalties he would have received from the worlds electronics

manufacturers alone would have made him a multibillionaire. As it was, in 1933 Dirac

settled for a few thousand dollars and the Nobel Prize in Physics!”...William O. Straub

• III Relation between Spin and Statistics

“ .... the connection between spin and statistics is one of the most important applications of

the special theory of relativity ”.. Pauli, 1940

In 1940, Pauli published his theory that only particles with integral spins are bosons and

only ones with half-integral spin are fermions.

• IV: The g-factor & QED

3

Magnetic property of the spin provides, with remarkable precision, the validity of quantum

electrodynamics or QED.

– 2.00244 (experimental) ( 1932)

– 2.08 (theoretical)

– 2.00231930482 ( experimental) (1983)

– 2.00231930476 ( theoretical)

– 2.00231930436182 ( experimental) ( 2017)

– 0.0000000000005 ( uncertanity)

• V : Quaternions & Spin

Spin was actually discovered in 1840’s in mathematics, disguised as ‘quaternions”. See

Appendix II.

II. MAGNETIC MOMENT OF AN ELECTRON

Electron is a charged particles that goes around the nucleus just the way planets go around

the sun. Moving charge is equivalent to a current I = e/T = ev/(2πr.

Current I in a loop of area A, acts like a magnet of dipole moment µl:

µl = IA

= ev(πr2)/(2πr)

= evr/2

=e

2mL

= gl(e~2m

)(L/~)

= glµb√l(l + 1)

µzl = glµbl

4

µl is the dipole moment of the electron due to its orbital motion.

Behavior of a Magnetic Dipole Moment in a Magnetic Field

A stationary dipole subjected to magnetic field B precesses about the field, with precession

frequency ω = glµb~ . ( Torque is perpendicular to the angular momentum and hence can only

change the direction of the angular momentum ).

E = −µl ·B

τ =dL

dt

τ = µl ×B

= −glµb~L×B

= −(glµb~B)L sin θ

= ωL sin θ

Effect of Inhomogeneous Field on a dipole µ

If the magnetic field is not uniform, the dipole will experience a force, in addition to torque:

Let B(x, y, z) = −axi + (B0 + bz)k, where B0 is a strong uniform field and the constants a, b

describe the non-uniform part of the field. Note, ∇ · B = 0. Due to Larmor precession, µxloscillates rapidly and averages to zero and the net force is in the z-direction

5

B(x, y, z) = −axi+ (B0 + bz)k

F = ∇(µ ·B)

Fz ≈ µz∂zBz

III. STERN- GERLACH EXPERIMENT...

Read section 1.1 of Sakurai. Problem: Compute the deflection

A beam of H-atoms at T = 400 K is sent through SG magnet of length L = 1 m. If the

atoms experience a gradient of 10 Tesla/m, calculate the transverse deflection Z of the atom, due

to the force on its spin dipole moment

1/2mv2x = 2KT

vx =√4KT/m

t = L/vx

Z = 1/2Fzmt2

Fz = ∂zBµ

Ans: Z = ±2.1X10−3m. Introducing Electron Spin

NOTE: (1) In Classical mechanics, a rigid body can have two types of angular momentum,

orbital: L = r × p, associated with the motion of the center of mass and spin: S = Iω, associated

with the motion about the center of mass. But in QM, distinction between these two types is very

fundamental as spin has nothing to do with motion in space-time.

6

NOTE (2) Schrodinger QM is completely compatible with the existence of electron spin;

but it does not predict it, so spin must be introduced as a separate postulate.

NOTE (3) Spin operators obey the same algebra as the angular momentum operators. However,

the g factor that relates spin magnetic moment and spin angular momentum, µs = gsµbS/~ is the

gs = 2.

IV. IMPACT OF SG EXPERIMENT

The SG experiment had one of the biggest impacts on modern physics. Firstly, it proves

the existence of spin. Practically all current textbooks describe the Stern-Gerlach splitting as

demonstrating electron spin, without pointing out that the intrepid experimenters had no idea it

was spin that they had discovered.

From Physics Today 2003:

Descendants of the SternGerlach experiment (SGE) and its key concept of sorting quantum

states via space quantization are legion. Among them are the prototypes for nuclear magnetic

resonance, optical pumping, the laser, and atomic clocks, as well as incisive discoveries such

as the Lamb shift and the anomalous increment in the magnetic moment of the electron, which

launched quantum electrodynamics. The means to probe nuclei, pro- teins, and galaxies; image

bodies and brains; perform eye surgery; read music or data from compact disks; and scan bar

codes on grocery packages or DNA base pairs in the human genome all stem from exploiting

transitions be- tween space-quantized quantum states.

(1) In the decade that followed, scientists showed using similar techniques, that the nuclei of

some atoms also have quantized angular momentum. It is the interaction of this nuclear angular

momentum with the spin of the electron that is responsible for the hyperfine structure of the

spectroscopic lines.

7

(2) In the thirties, using an extended version of the SG apparatus, Isidor Rabi and colleagues

showed that by using a varying magnetic field, one can force the magnetic momentum to go

from one state to the other. The series of experiments culminated in 1937 when they discovered

that state transitions could be induced using time varying fields or RF fields. The so called Rabi

oscillation is the working mechanism for the Magnetic Resonance Imaging equipment found in

hospitals.

(3) Norman F. Ramsey later modified the Rabi apparatus to increase the interaction time with

the field. The extreme sensitivity due to the frequency of the radiation makes this very useful for

keeping accurate time, and it is still used today in atomic clocks.

(4) In the early sixties, Ramsey and Daniel Kleppner used a SG system to produce a beam of

polarized hydrogen as the source of energy for the hydrogen Maser, which is still one of the most

popular atomic clocks.

(5) The direct observation of the spin is the most direct evidence of quantization in quantum

mechanics.

————————————————————————–

8

Stern-Gerlach Experiment & Vector Space

If we ignore nuclear angular magnetic moment ( which is 2000 times smaller than the

electronic moment ) and since angular momentum quantum number l = 0, the magnetic moment

of the Ag atoms is solely due to the intrinsic spin of the electron. This spin cannot be described

by laws of classical mechanics

Note, in SG experiment, only thing we care about is the spin of the atom, which way it is

pointing... We do not care about the spatial coordinates of the atom.. Physical state, like Ag atom

with a definite spin orientation...

Figure 1.3 suggests that this state ( ie, state with the definite spin ) is acting like a vector

in 2D space, as it has only two possible orientations.

• 2D has nothing to do with the x-y coordinates; it is a kind of abstract 2D vector space, where

if we know two vectors, we can find all other vectors:.

• Why states are represented as vectors ???

The general principle of superposition of quantum mechanics applies to state. This is why

sometimes quantum mechanics is also called wave mechanics...

A state can be regarded as a result of a kind of superposition of two or more new states, in

a wave that cannot be conceived on classical ideas. The procedure of expressing a state as a

result of superposition of other states is a mathematical procedure that is always permissible,

independent of any physical conditions, like the procedure of resolving a wave into Fourier

components. Its usefulness depends upon the special physical conditions of the problem

under consideration.

• Non-classical nature of superposition emerges from the fact that coefficients of the

superposition give the probability for determining the system in those superimposed states.

• Superposition process is an additive process that implies that states can be added to give

9

other quantities ( that is other states ). The most obvious of such quantities are vectors.

• As we will see, finite number of dimensions may not be enough to describe a state. We have

to make a generalization to space of infinite dimension.

• Following Dirac, there is a special name for describing vectors which are connected with the

states of a system in quantum mechanics, called the ket vectors , or simply kets and denote

them by a special symbol | 〉 . If we want to specify them with a certain label say “x”, we

write | x 〉. In general,

| x 〉 =∑

Ai| xi 〉 , or, | x 〉 =

∫a(x′)| x′ 〉 dx′ (1)

• In SG experiment, Possible states of the system are:

| Sx,+〉, | Sx,−〉, | Sy,+〉, | Sy,−〉, | Sz,+〉 and | Sz,−〉

But, they are not independent: given any two, others can be determined, as suggested by

Fig 1.3

Suppose, we are given |Sz,+ > and |Sz,− >

How can we be express all other states in terms of these two alone ??? The simplest

possibility is,

| Sx,+〉 = A1| Sz,+〉+ A2| Sz,−〉

| Sx,−〉 = A1| Sz,+〉 − A2| Sz,−〉

where it is obvious that A1 = A2 = A = 1/√2

Now, what about | Sy,±〉??

10

The only way to make | Sy,±〉 independent of | Sx,±〉 is to use A1 = ±iA2.

In other words, the vector space is complex

This is amazing – the outcome of this SG expt tell us that QM states can be represented by

vectors in an abstract complex vector space.

V. BASIC MATHEMATICS OF VECTOR SPACE USED IN QM

Postulates of QM

• Each state of a system at a particular time corresponds to a ket vector.

Note that superposition of a state with itself does not lead to a new state. In other words,

| x 〉 and c| x 〉 represent same state.

Also zero ket vector corresponds to no state.

• Bra and ket vectors.

To every ket vector | x 〉, there exists its dual, call the bra, denoted as 〈 x |.

The bra dual to c| x 〉 is c∗〈 x |

• We can define scaler product 〈 x | y 〉.

• Two kets | x 〉 and | y 〉 are orthogonal if 〈 x | y 〉 = 0

• 〈 x | x 〉 > 0 except when | x 〉 = 0

• 〈 x | y 〉 = (〈 y | x 〉)∗

• Normalized ket is 1√〈 x | x 〉

| x 〉

• Where are the wave functions ???? They will appear soon – just wait–

VI. LINEAR OPERATORS

Let us consider a ket that is a linear function of another ket. That is, to each ket say | α 〉,

there corresponds one ket | β 〉 and suppose that the function is linear one which means that we

can define an operator X such that if

11

X| α 〉 = | β 〉 (2)

Xc| α 〉 = cX| α 〉 (3)

X(c1| α1 〉+ c2| α2 〉) = c1X| α1 〉+ c2X| α2 〉 (4)

NOTE:

• X| α 〉 = | β 〉 ↔ 〈 α |X† = 〈 β |.

The operator X† is called the adjoint of X . Operator is called HermitianX† = X . In this

case,

〈 α | β 〉 = (〈 β | α 〉)∗

That is for the ket the operator acts on the left while for the bra, the operator acts on the right.

( Give an example of a non-linear operator ...)

READ section 1.2 of Sakurai.

12

VII. APPENDIX I : WHAT IS QED

What is QED ???

Firstly, trying to picture electron as a wave causes lot of headaches... These waves are clearly

not ordinary waves – they are not like ripples on the surface of water...

Despite all this, Sch”odinger wave theory has been very successful. However, at end of 1920,

physicist realized that it told only part of the truth... It cannot handle physics when particles are

created and destroyed. Like, when electron and positron annihilate each other, creating photons....

Dirac equation also pointed towards such problems...

This led to the notion of quantum fields.

In this description, instead of talking about particles, one talks about quantum fields. Creation

and destruction of particles is adding or removing quanta from the field. Earlier versions of

quantum field theory had lot of problems... like infinity occurs due to electrons interacting with its

own electromagnetic field...

Then physicists realized that this problem can be solved and theory provides a new way to

understand forces between two particles... like repulsion between two electrons.

In 1932, Bethe and Fermi suggested that repulsion between two electrons is due to exchange

of photons between two electrons. These photon cannot be observed and are called virtual photons.

This is wonderful... photon is not only simply the quantum particle of light or electromagnetic

waves, it is the ”carrier” of the electromagnetic force.

Three physicists independently developed the complete theory of QED – the quantum

electrodynamics. They are two American physicists Richard Feynman and Julian Schwinger and

Japanese physicist Sin-Itiro Tomonaga. Interestingly, these theories that on the surface looked

very different were shown to be equivalent by Freeman Dyson.

VIII. APPENDIX II: QUATERNION

<https://arxiv.org/ftp/arxiv/papers/1504/1504.04885.pdf>

Complex numbers are necessary to solve certain algebraic equations such as x2 + 1 = 0.

Furthermore, complex numbers may be represented as two-dimensional vectors lying in the

so-called Argand plane, with the x-axis representing the real numbers and the y-axis representing

13

14

the pure-imaginary numbers. The geometric interpretation of complex numbers is supported by

the observation that the addition of two complex numbers represents the addition of two vectors,

and the product of two unit-length complex numbers represents a sequence of two rotations.

Upon seeing the connection between complex numbers and two-dimensional geometry, the

curious mind is apt to wonder whether there is an extension applicable to three-dimensional

geometry. During the nineteenth century, this germ of an idea was lodged in the minds of many

people including William Rowan Hamilton. In 1843 Hamilton discovered an algebra of three-

dimensional rotations that was based on a new set of objects he called quater- nions (van der

Waerden, 1976 provides a good history). So impressed was he with these new objects that he

spent the remaining 22 years of his life working out their properties.

This discovery of sets of four numbers – known as quaternions in scientific literature are

disguised as what quantum physicist call spin-1/2 Excellent reference is the following paper...

<https://arxiv.org/ftp/arxiv/papers/1504/1504.04885.pdf>

[1] NOTE: Compton anticipated by five years the spectroscopic discovery by Goudsmit and Uhlenbeck of

the electron spin. Compton concluded that the electron must be “ spinning like a tiny gyroscope” and

have “angular momentum ~..... offering an explanation for the large value of e/m observed......“ about

one-third the magnetic moment of the iron atom” in order to generate the magnetic properties of matter.

See Arther H. Compton, Journal of the Franklin Institute, 192-2, 145 (1921)

15