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IRREVERENT QUANTUM MECHANICS
Giancarlo Borgonovi
May 2004
MOTIVATION
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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 20 th
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)
History
Applications
People
Results
QM
Concepts
Criticism
Implications
Interpretation
What is irreverent quantum mechanics?
A discipline for OFs to keep involved with QM:
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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
Classical
System
State vector
Represented by real numbers
Possible states
Definite state
Deterministic transition from one state to another
Quantum
System
State vector
Represented by complex numbers
Possible states
Superposition of states
Probabilistic transition from one state to another
The formal elements of quantum mechanics
A
B
BA

B A
Abstract state vector
Abstract state vector in dual space
Probability amplitude for going from state A to state B
Operator
Matrix element of operator

The great law of quantum mechanics
From The Feynman Lectures on Physics, Vol. 3
The unforgiving logic of P. A. M. Dirac
Are there
any questions?
I have not understood how
you passed from A to B
That is a statement,
not a question
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
John Income John
Mary Zip_Code Mary
Ed Candidates Ed
Hilbert space and human life
?
Human life according to Classical Mechanics
Hamilton’s Equations
Human life according to Quantum Mechanics
Schroedinger Equation
The different forms of quantum mechanics
a11 a12
a21 a22
a31 a32

a13
a23
a33
Matrix Mechanics
Wave Function
Schroedinger
Heisenberg
B
AB
A
Symbolic Method
Dirac
Path Integral
Feynman
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
A new, downsized model of the ‘brick’ is introduced
h

=
2π
1925 - The ‘brick’ is split in half (Uhlenbeck and Goudsmit introduce the spin).
1

2
1

2
Particles position and momentum and Heisenberg uncertainty principle
BOSONS and FERMIONS
Identical particles are not distinguishable
A wrong representation of the
hands of God building matter
A more realistic representation of the
hands of God building matter
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 symmetric
3) The possible states for a system of fermions (at least two) are antisymmetric
4) Two bosons interfere with the same phase
5) Two fermions interfere with the opposite phase.
Amplitude  f1
Boson Case Probabilit y  f 1  f 2
Amplitude  f 2
2
Fermion Case Probabilit y  f 1  f 2
2
Shapes represent quantum states, colors represent particles
+
Boson
+
(Symmetric
under exchange)
+
Fermion
-
+
-
(Antisymmetric
under exchange)
Pauli or
ExclusionPrinciple
(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
Matter under different assumptions
Electrons as fermions (real)
Electrons as bosons (imagined)
From The Feynman Lectures on Physics, Vol. 3
Classical and Quantum
Statistics
The different nature of bosons and fermions
Everyone in my army of fermions will occupy his
place and defend the empire
My army of bosons will move
Unknown Roman Emperor
and attack as one man
Unknown Barbarian King
Bosons
Fermi
sphere
Fermions
New States of Matter
What they are
Predicted
Realized
Nobel prize
Atoms used
Made possible by
How is observed
Why it is important
Bose_Einstein
Degenerate Fermi
Condensate
Gas
Macroscopic Quantum Systems
1930s
1930s
1995
2001
2001 (Cornell, Wieman,
Ketterle)
Rubidium 87
Lithium 6
Optical bowls (laser containment)
Velocity Distribution after expansion
Permits extrapolations to unobservable states
of matter
THE PERIODIC TABLE
(Ability and Weirdness)
Quantum Mechanics and Weirdness - Thoughts about the periodic table
I
II
III
IV
V
VI
VII
1
3
11
19
37
55
87
2
10
18
56
36
54
86
71
92
57
Energy
(n)
1
2
3
4
5
6
7
R
Angular momentum
()
0
0,1
0,1,2
0,1,2,3
0,1,2,3,4
0,1,2,3,4,5
0,1,2,3,4,5,6
a
r
e
Including m
(2 +1)
1
1,3
1,3,5
1,3,5,7
1,3,5,7,9
1,3,5,7,9,11
1,3,5,7,9,11,13
E
a
r
t
h
Including s (spin)
(×2)
2
2,6
2,6,10
2,6,10,14
2,6,10,14,18
2,6,10,14,18,22
2,6,10,14,18,22,26
s
70
Total
States
2
8
18
32
50
72
98
K
L
M
N
O
P
Q
2
2
2
6
6
10
2
2
2
2
s
6
6
6
6
p
10
10
10
10
d
14
14
14
14
f
18
18
18
22
22
26
FORMATION OF THE PERIODIC TABLE
K
L
M
N
O
P
Q
K
L
M
N
O
P
Q
1
2
4
3
5
6
7
11
16
22
s
8
12
17
23
p
9
13
18
24
d
1
2
4
3
5
7
8
11
15
10
14
17
13
p
d
f
6
9
12
16
s
10
14
19
25
f
15
20
26
21
27
28
Spherical symmetry, angular momentum, and weirdness
Low Angular Momentum
High Angular Momentum
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 QUANTIZATION
and
QUANTUM FIELDS
Second Quantization
 11 21 31 42 52 63   11 21 31 42 52 63
Fixed number of particles
3 21  ( r ) 3 21
Occupation number representation
This operator creates or destroys particles
QUANTUM MECHANICAL SPACES
Many particle space
(Fock space)
Collection of
N- particle space
n-particle states
Symmetric or
antisymmetric states
One- particle space
(Hilbert space}
Principle of symmetrization
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 YOU
AND MAY YOU HAVE
A HAPPY TRANSITION TO A
STATE OF HIGHER
ANGULAR
MOMENTUM