Week 1 ( Aug 30 & Sept 1 2016)

Exotic Aspects of Quantum Physics... some examples

Double Slit Experiment

Quantization of energy

Uncertainty principle

Zero point Energy – Even at T=0, particles have finite energy

I. SOME QUANTUM THEORY – FROM PLANCK TO SCHRODINGER EQUATION (SE)

• To explain radiations from hot objects, Planck introduced E = hf , that is, energy is related

to frequency. Also a new fundamental constant h emerged, that cannot be expressed in terms

of any known constants from classical physics.

• To explain sharp spectral lines of light from H-atom, and also the instability of atom, Bohr

proposed planetary model where electrons have quantized angular momentum. This led

to quantization of energy and “space” quantization, that is only certain orbits are allowed.

In Bohe theory, En = −13.6..Z2

n2 = −Z2

n212(mec

2α2) where α is the fine structure constant,

α ≈ 1137

. It is a non-relativistic theory, that is, v/c << 1 or equivalently E << mec2.

• de-Broglie proposed wave particle duality: every particle is associated wave-like properties

characterized by a wave length λ and a frequency f . Also, radiations have particle character

as they are associated with energy E and momentum p. In both cases, λ, f, E, p are given

by,

λ =h

p, f =

E

h

p =h

λ, E = hf

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• Schrodinger proposed wave equation to describe quantum behavior of particles ( replacing

Newton’s equation),

Hψ(x, t) = i~∂

∂tψ(x, t), H =

p2

2m+ V (x) = − h2

2m∇2 + V (x) (1)

Note the presence of ~ = h/(2π) and the complex number i. The solutions ψ(x) are

complex quantities and have no classical analog. Also the the momentum p is an operator

and in SE theory is given by p = ~i∇.

Magically, SE gave same solution for the energy En as the Bohr model.

In SE theory, three quantum numbers (n, l,ml) emerged. But energy depends only on n.

• Bohm provided physical interpretation of ψ(x) as |ψ(x)|2 gives the probability of finding

the particle at x.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

SPIN – How spin “spins”

Our exotic quantum world in this class will begin with the story of spin of elementary

particles. Few of our lectures will revolve around “spin”, however, this story of spin is also the

story of development of quantum mechanics as the history of the ripening of quantum mechanics

will emerge from it. As we will see later, there are many “siblings” of spin in nuclear and

elementary particle physics.. This includes the idea of isospin, color, strangeness.... etc.. It turns

out that the idea of spin is as abstract as the ideas of isospin, color, strangeness that some of you

may have heard of. It is the name “spin” that initially allows us to embrace it as it is something

familiar in classical physics.

Unlike what the name “spin” implies, the quantum particles do not spin like classical

particles. These particles are associated with “spin”, even in their rest frame. It is purely a

quantum mechanical abstract concept that has no analog in the classical physics.

According to Landau and Lifshitz, spin character of such particles is peculiar to quantum

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theory... and has no classical interpretation. It would be wholly meaningless to imagine the

“intrinsic” angular momentum of an elementary particle as being the result of its rotation “about

its own axis”. ( See page 198 of Landau Lifshitz, third edition)

“Spin is a mysterious beast, and yet its practical effects prevails over the whole of science.

The existence of spin, and the statistics associated with it, is the most subtle and ingenious design

of nature– without it the whole universe would collapse.”... Sin-itiro Tomonaga

Another weird aspect of spin is that although spin character of particles and relation between

spin and statistics exists in non-relativistic quantum mechanics, theory of relativistic quantum

mechanics is essential in understanding it. According to Richard Feynman, “It appears to be one

of the few places in physics where there is a rule which can be stated very simply, but for which

no one has found a simple and easy explanation. The explanation is down deep in relativistic

quantum mechanics. This probably means that we do not have a complete understanding of the

fundamental principle involved”.

Also, there is something more about the spin, more recent than what was known in the

early development of quantum mechanics. This relates to the idea of “geometric” or ” Berry

phase” discovered about 30 years ago, that we will discuss later in this course. Yes, the idea of

geometric phase has precursors in the story of spin.

II. STORY OF SPIN – HOW THE IDEA OF SPIN WAS BORN

How the idea of spin was born and under what circumstances..

It emerged independently in four different settings.. consisting of three different groups trying

to resolve experimental results–and a solo theoretician trying to generalize Schrodinger equation

to relativistic domain. Here is the summary...

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• (1) Germany –Between 1922-1925, Sommerfeld, Lande and Pauli were in hot competition to

explain the discovery of multiplicity of the spectral lines ( an energy level consists of many

closely spaced levels – known as the fine structure of the energy levels ) and also splitting

of energy levels in the presence of magnetic field, as seen in the experimental observations

–described below.

• (2) Netherlands – Graduate students Uhlenbeck and Goudsmit working in Leiden with Paul

Ehrenfest were trying to explain experimental observation of energy spectrum in Paschen’s

lab. As described by Goudsmit –One of the things which stuck to me is that in Paschen’s

experiments on the helium line, its fine structure and the relativistic explanation, there was

a forbidden component which was obviously present.

Read the article –” Discovery of the electron spin by Goudsmit”

.

• (3) England (Dirac ) – To develop relativistic equation for electrons.. That is to generalize

Schrodinger theory to relativistic particles. Dirac’s theory will be discussed in detail in

coming weeks.

• (4) Stern Gerlach Experiment – will not be discussed...

The SternGerlach experiment was performed in Frankfurt, Germany in 1922 by Otto Stern

and Walther Gerlach. At the time, Stern was an assistant to Max Born at the University

of Frankfurt’s Institute for Theoretical Physics, and Gerlach was an assistant at the same

university’s Institute for Experimental Physics.

At the time of the experiment, the most prevalent model for describing the atom was the

Bohr model, which described electrons as going around the positively charged nucleus only

in certain discrete atomic orbitals or energy levels. Since the electron was quantized to

be only in certain positions in space, the separation into distinct orbits was referred to as

space quantization. The Stern-Gerlach experiment was meant to test the Bohr-Sommerfeld

hypothesis that the direction of the angular momentum of a silver atom is quantized.

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Note that the experiment was performed several years before Uhlenbeck and Goudsmit

formulated their hypothesis of the existence of the electron spin. Even though the result

of the Stern?Gerlach experiment has later turned out to be in agreement with the predictions

of quantum mechanics for a spin-1/2 particle, the experiments were seen as a corroboration

of the BohrSommerfeld theory. We will not discuss this here further.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Germany

Netherlands

England

In Schrodinger theory for H-atoms, there are three quantum numbers (n, l,ml) where n ≥ l.

The quantum number n determines the energies, the radius of the orbit. The quantum number l

determines the angular momentum in units of ~. The quantum number −ml ≤ l ≤ ml.

With improvement in spectroscopic measurement, it was found that for Alkaline metals, with

the exception of S electron orbit ( l = 0), other levels have multiplet structure. This was explained

by Sommerfeld theory of relativistic correction to Bohr model.

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Brief Review of Bohr Model and Sommerfeld Correction

• From de-Broglie relation λ = hp

, and L = mvr = n~, we obtain the key result of Bohr

quantization – “space quantization” — 2πr = nλ.

• En = −13.6..Z2

n2 = −Z2

n212(mec

2α2) where α is the fine structure constant, α ≈ 1137

. Using

relativistic theory, show that upto leading power in α, E = En[1 + (αZ2n

)2]

For elliptical orbits, Sommerfeld, there are two quantum numbers n ≡ nφ and nr ( which is

zero for circular orbits). And the expression for energy becomes,

E = En[1 + (αZ

2n)2(

n

nφ− 3

4)] n = nr + nφ (2)

For nr = 0 it reduces to the formula for circular orbit. This shows that an electron in a

H-atom is non-relativistic ( E << mec2 ) . It has v

c≈ 10−2. Therefore, relativistic corrections

to energy will be of the order of 10−4. It turns out that this is just the order of magnitude of the

spitting that was observed in the experiments. Therefore, relativistic treatment of Bohr model will

explain the fine structure of the energy spectrum observed in the experiments.

Experiments showed that the S shell, that is l = 0 quantum state remained an exception

and did not split and this is consistent with Sommerfeld correction term.

However, when we apply magnetic field, it was found to split into two levels.. This is really

bizarre !!!

To understand why, let us review two central concepts about dipole moments of an orbiting

electron and its response to magnetic field.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

III. BACKGROUND – DIPOLE MOMENTS AND MAGNETIC FIELD

Recall that as electrons revolves in an orbit, it generates currents and hence magnetic field

and acts like a little magnet –particle with a dipole moment.

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A. Relating Dipole moment to the orbital angular momentum of Bohr model

Consider a particle of charge e in a circular orbit of radius r. If T is the period for one

revolution, the current I , orbital magnetic moment µl and the orbital angular momentum L = mvr

are related by the following equations

I = e/T = ev/(2πr)

µl = I(πr2)

µlL

= (e/2m) ≡ glµb/~

~µl = glµb~L

~

Here gl = 1 is the Lande g-factor. Its importance will emerge later when we talk about spin.

The quantity µb forms a natural unit for atomic magnetic dipole moments and is called Bohr

magneton.

µB =e~2m

(3)

Note that the ratio µlL

does not depend upon the size of the orbit or the period T . It turns out

that this result is also valid for elliptical orbits.

We will next discuss what happens to a dipole in a magnetic field – the classical physics

problem.

B. Dipole in a Magnetic Field

Dipole experiences a torque τ = µl × B that tends to align the dipole with the field . The

potential energy is given by,

∆E = −µl ·B = glµb~L · ~B~

(4)

This shows that when l = 0, the energy levels will not split in a magnetic field. This is in

contradiction to what was observed in experiments

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

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Three brilliant physicists, Sommerfeld, Lande and Pauli grope for understanding multiplicity

of levels and Zeeman effect. They felt the need to introduce a new degree of freedom in addition

to orbital angular momentum.

In 1924, Pauli, the youngest of the three, imaginatively assigns this new degree of

freedom to the electron. His new idea was that all the electrons have “classically indescribable

two-valuedness”.

Pauli Exclusion Principle: Pauli knew that the understanding of his new concept of

two-valuedness of electron is not complete. However, if we accept this, we can explain the

remarkable fact, well known experimentally, that each orbit in an atom has a certain capacity

and observation that an orbit is closed when its capacity is filled, once we specify the values

of (n, l,ml,ms) where ms = ±12. No two electrons in an atom can have same set of quantum

numbers. In other words, once an electron take these values, it prevents any other electron to

take the same value. This is called Pauli exclusion principle. This rule explained the periodic table.

Kronig came from United States to Lande’s lab and when he saw Pauli’s letter about new

degree of freedom, he immediately thought of a self-rotating electron. That is, electron is rotating

about its own axis with an angular momentum ( which Bohr later called spin ) equal to 12~ and a

g factor, will be written as gs of 2. That is, in the rest frame of electron, it possess dipole moment

µs,

~µs = ~µl = gsµb~S

~(5)

This empirical fact explained the splitting of S-levels into a doublet by Zeeman effect, except

for a factor of two!!!.

Pauli’s sanctions – Pauli comes in the way!!! ???

Pauli visited Landes lab and met Kronig , who was only 20 years old, who explained his

theory of self-rotating electron and this spinning is what gives rise to a new degree of freedom in

Paulis theory.

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However, Pauli rejected this idea . Recall, Pauli had proposed “two-valuedness” idea, but he

was not willing to think it in terms of spinning of the electron. Why ???

To calculate the splitting of levels due to spinning electron, one goes back to relativity theory.

All was well except a factor of two discrepancy... Kronig discussed his results with Pauli. Pauli

showed no interest in Kronig’s idea about self rotating electron. Kronig went to Copenhagen and

also discussed his theory, but could not convince the physicists there.

He himself was not confident because of this factor of 2 difference between the relativistic

theory using spinning electron and the experimental result. Also, there was another problem. Idea

of self rotating electron was problematic within the framework of classical theory. H. A. Lorentz

has considered the size of the electron to be e2/mc2. In this case a very fast rotation is needed ( 10

times the speed of light ) to attain the spin angular momentum equal to 1/2~. For these reasons,

Kronig decided not to publish his results.

IV. LEIDEN – NETHERLANDS WITH PAUL EHRENBERG

In the Fall of 1925, graduate students Uhlenbeck and Goudsmit published precisely the same

idea...

This remarkable story is beautifully narrated by Goudsmit in his 1971paper ( see the

“Discovery of Spin” on the course web site... )

They were well aware of all the problems with the self-rotating electrons and in fact wanted

to withdraw their paper. But their advisor said that since they have no reputation, they have

nothing to loose and submitted their paper anyway. Actually, when they asked Ehrenfest, he

answered

“ I have already sent your letter in long ago; you are both young enough to allow yourselves

some foolishness!!”.

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While the proposition of self rotating electron was in turmoil, famous work of L. H. Thomas

appeared. He showed that factor of two discrepancy was due to incorrect definition of the electron

mass.

When Pauli saw this, he embraced the self rotating electron model, clearing the Fog of

1923-1924..... It was fortunate that Pauli had not seen this work, this was before the “factor of

two” problem was resolved and Pauli had embraced the self-rotating electron ( or spin of the

electron ) model.

V. HOME WORK FOR WEEK 1: DUE SEPT 8

(1) Read the article –” Discovery of the electron spin by Goudsmit” ( It is on the class web

site )

(2) (a)Using relativistic theory, show that upto leading power in α, E = En[1 + (αZ2n

)2]

(b) For the elliptic orbits, the Sommerfeld formula E = En[1 + (αZ2n

)2( nnφ− 3

4)] n = nr + nφ

reduces correctly to the formula for the circular orbits.

See <http://scitation.aip.org/docserver/fulltext/aapt/journal/

ajp/42/10/1.1987875.pdf?expires=1472558003&id=id&accname=2120773&

checksum=B3D21180E16C828EC0925150C3D8E8DD> .

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