IRREVERENT QUANTUM MECHANICS

Giancarlo Borgonovi

May 2004

For the purpose of this presentation the term Quantum Mechanics is equivalent to Quantum Theory Personal opinion: Quantum Mechanics is the most significant intellectual achievement of the 20th

Century. Reasons in support of this statement: QM is totally counter intuitive QM was created/invented to explain phenomena only indirectly accessible to our senses QM was created/invented to explain phenomena in the eV energy range (atomic spectra) QM has maintained its validity up to the GeV energy range (11 orders of magnitude)

QM

People

Concepts

History Applications

Results

Criticism

InterpretationImplications

MOTIVATION

What is irreverent quantum mechanics?

A discipline for OFs to keep involved with QM:

• Develop allegories/metaphors about QM • Design/build models/representations of QM effects• Investigate QM trivia• Explore connection between science and art• Write fiction around QM subjects/characters• Develop humor about QM subjects/characters• Quantum mechanical cooking?• Give presentations to other OFs.

GENERAL PRINCIPLES

Classical and quantum mechanics comparison

QuantumClassical

SystemState vector

Represented by real numbersPossible statesDefinite state

Deterministic transition from one state to another

SystemState vector

Represented by complex numbersPossible states

Superposition of statesProbabilistic transition from one state to another

The formal elements of quantum mechanics

A

B

AB

AB

Abstract state vector

Abstract state vector in dual space

Probability amplitude for going from state A to state B

Matrix element of operator

Operator

The great law of quantum mechanics

From The Feynman Lectures on Physics, Vol. 3

That is a statement, not a question

I have not understood howyou passed from A to B

Are thereany questions?

The unforgiving logic of P. A. M. Dirac

Observables in Quantum Mechanics

• Represented by real operators• Describe possible states (eigenvectors) which are associated with possible outcomes of measurements (eigenvalues)• Before the measurement: calculate probabilities of different outcomes• After the measurement: only one outcome

Example

Expectation values for different cases

EdCandidatesEd

MaryCodeZipMary

JohnIncomeJohn

_

?

Hilbert space and human life

Human life according to Classical Mechanics

Hamilton’s Equations

Human life according to Quantum Mechanics

Schroedinger Equation

Schroedinger

Heisenberg

Dirac Feynman

The different forms of quantum mechanics

333231

232221

131211

aaa

aaa

aaa

Wave FunctionMatrix Mechanics

Symbolic Method Path Integral

A

B

BA

1900 - Max Planck, studying the black body radiation, discovers the “brick”.

Planck’s constant h = 6.55 x 10-27 erg sec can be considered as the building block of quantum mechanics.

h

h

2π=

A new, downsized model of the ‘brick’ is introduced

2

1

2

1

1925 - The ‘brick’ is split in half (Uhlenbeck and Goudsmit introduce the spin).

Particles position and momentum and Heisenberg uncertainty principle

BOSONS and FERMIONS

A wrong representation of the hands of God building matter

A more realistic representation of the hands of God building matter

Identical particles are not distinguishable

Quantum Mechanics divides the Universe into two Categories

• Every particle in the universe is either a boson or a fermion, that is to say everything in the universe is made up of bosons and fermions.

• What distinguishes a boson from a fermion?

• What are the effects of this categorization?

What distinguishes a boson from a fermion

1) Bosons have spin integer, fermions have spin semi-integer

2) The possible states for a system of bosons (at least two) are symmetric3) The possible states for a system of fermions (at least two) are antisymmetric

4) Two bosons interfere with the same phase5) Two fermions interfere with the opposite phase.

1fAmplitude 2fAmplitude

2

2ff y ProbabilitCaseBoson 1

2

2ff y ProbabilitCaseFermion 1

+

+

Boson

+

-

Fermion

+

-

Pauli or ExclusionPrinciple

Shapes represent quantum states, colors represent particles

(Symmetric under exchange)

(Antisymmetric under exchange)

(Null for fermions under exchange)

Effects due to boson like features

• Bosons are very gregarious and tend to congregate together. If bosons exist in a state, there is a tendency for another boson to enter that state.

• The laser is an example of this tendency of the bosons to come together

• Superfluidity of Helium-4 (not Helium-3 which emulates a fermion) at low temperature is a macroscopic example of the result of the tendency of bosons to get into the same

state of motion.

Effects due to fermion like features

Fermions tend to avoid each other. If a fermion exists in a state, another fermion will not want to enter that state.

• Pauli’s Exclusion Principle

• What if electrons were bosons

Electrons as fermions (real)

Electrons as bosons (imagined)

Matter under different assumptions

From The Feynman Lectures on Physics, Vol. 3

Classical and Quantum Statistics

Bosons

Fermions

Fermi sphere

The different nature of bosons and fermions

My army of bosons will move

and attack as one man

Unknown Barbarian King

Everyone in my army of fermions will occupy his place and defend the empire

Unknown Roman Emperor

New States of Matter

Bose_EinsteinCondensate

Degenerate FermiGas

What they are Macroscopic Quantum SystemsPredicted 1930s 1930sRealized 1995 2001Nobel prize 2001 (Cornell, Wieman,

Ketterle)-

Atoms used Rubidium 87 Lithium 6Made possible by Optical bowls (laser containment)How is observed Velocity Distribution after expansionWhy it is important Permits extrapolations to unobservable states

of matter

THE PERIODIC TABLE

(Ability and Weirdness)

Quantum Mechanics and Weirdness - Thoughts about the periodic table

I 1 2II 3 10III 11 18IV 19 36V 37 54VI 55 56 71 86VII 87 92

57 R a r e E a r t h s 70

Energy(n)

Angular momentum()

Including m(2 +1)

Including s (spin)(×2)

TotalStates

1 0 1 2 22 0,1 1,3 2,6 83 0,1,2 1,3,5 2,6,10 184 0,1,2,3 1,3,5,7 2,6,10,14 325 0,1,2,3,4 1,3,5,7,9 2,6,10,14,18 506 0,1,2,3,4,5 1,3,5,7,9,11 2,6,10,14,18,22 727 0,1,2,3,4,5,6 1,3,5,7,9,11,13 2,6,10,14,18,22,26 98

K 2L 2 6M 2 6 10N 2 6 10 14O 2 6 10 14 18P 2 6 10 14 18 22Q 2 6 10 14 18 22 26

s p d f

K 1L 2 3M 4 5 6N 7 8 9 10O 11 12 13 14 15P 16 17 18 19 20 21Q 22 23 24 25 26 27 28

s p d f

K 1L 2 3M 4 5 7N 6 8 10 13O 9 11 14P 12 15 17Q 16

s p d f

FORMATION OF THE PERIODIC TABLE

Low Angular Momentum

High Angular Momentum

Spherical symmetry, angular momentum, and weirdness

Sociological implications of the periodic table

• Consider the order of the states as some kind of social order, or rank, or job position. In a rigid, hierarchical society, positions would be occupied according to certain parameters (e.g. diplomas, family connections, religious or ethnical factors, etc.). In a more intelligent society, people of higher ability pass in front of others and acquire a higher social status. This process has some similarity to the buildup of the periodic table. Thus nature rewards ability.

• The external shells, which are responsible for the chemical behavior of the elements, consist of s and p electrons only. The “weirder” d and f electrons are left behind, and are used to fill incomplete shells, so in a sense they hide behind less weird electrons at a higher level. Thus, nature tends to hide weirdness.,

SECOND QUANTIZATIONand

QUANTUM FIELDS

Second Quantization

36

25

24

13

12

11

36

25

24

13

12

11

123)(123 r

Occupation number representation

This operator creates or destroys particles

Fixed number of particles

One- particle space

(Hilbert space}

N- particle space

Many particle space

(Fock space)

Symmetric or

antisymmetric states

Collection of

n-particle states

Principle of symmetrization

QUANTUM MECHANICAL SPACES

VIRTUAL PARTICLES

• Virtual particles are like words, they can result in attraction or repulsion

• Virtual particles have a very short lifetime

• An exchange of momentum can be interpreted as the action of a force over a time interval

Photons Electromagnetic field

Phonons Cooper pairs, superconductivity

Mesons Nucleons

Gluons Quarks

Hideki Yukawa

Quantum Fields

A classical field is easy to visualize and understand A quantum field is an operator which is a function of position To understand a quantum field one needs to understand the local creation and annihilation operators Everything (energy, number of particles, total momentum, etc.) can be expressed in terms of the creation and annihilation operators A quantum field is expressed in terms of creation and annihilation operators A quantum field is a nice way to express the duality particle wave that pervades QM What are the eigenvalues and eigenvectors of a quantum field?

Quantum Cooking - Potatoes a la Brillouin

Leon Brillouin, 1927

THANK YOUAND MAY YOU HAVE

A HAPPY TRANSITION TO ASTATE OF HIGHER

ANGULARMOMENTUM