Download Quantum Mechanics: what is it and why is it interesting? Dr. Neil Shenvi

Quantum Mechanics:
what is it and why is it
interesting?
Dr. Neil Shenvi
Department of Chemistry
Yale University
Talk Outline
1. The history of quantum mechanics
2. The explanatory power of quantum mechanics
3. What is quantum mechanics?
a. The postulates of quantum mechanics
b. The weirdness of the postulates
4. The usefulness of quantum mechanics
5. The philosophy of quantum mechanics
Classical mechanics is the mechanics of everyday
objects like tables and chairs
1. An object in motion tends to stay in motion.
2. Force equals mass times acceleration
3. For every action there is an equal and
opposite reaction.
Sir Isaac Newton
Classical mechanics reigned as the dominant theory of
mechanics for centuries
1687 – Newton’s Philosophiae
Mathematica
1788 – Lagrange’s Mecanique
Analytique
1834 – Hamiltonian mechanics
1864 – Maxwell’s equations
1900 – Boltzmann’s
entropy equation
However, several experiments at the beginning of
the 20th-century defied explanation
The Ultraviolet
Catastrophe
The Stern-Gerlach
Experiment
Newtonian explanations for
these phenomena were wildly
insufficient
The Hydrogen
Spectrum
?
The Stern-Gerlach experiment involved passing
atomic “magnets” through a magnetic field
Question 1. How many
beams do we expect to
emerge from the
magnet?
+
-
?
?
?
?
?
Ag atoms
A. 1
B. 2
C. 3
D. A diffuse cloud
Exactly two well-defined
beams emerge from the
magnet!
Quantum mechanics was developed to explain these
results and developed into the most successful
physical theory in history
Increasing weirdness
1900 – Planck’s constant
1913 – Bohr’s model
of the atom
1925 – Pauli exclusion principle
1926 – Schrodinger equation
1948 – Feynmann’s path
integral formulation
1954 – Everett’s many-world
theory
Quantum mechanics applies to all objects, no matter
how big or small
Mechanical Engineering
(macroscopic objects)
Creative writing
(books)
Thermodynamics
(collections of molecules)
Grammar
(sentences)
Classical mechanics
(large molecules)
Quantum mechanics
(atoms and molecules)
Spelling
(words)
Penmanship
(letters)
However, the effects of quantum mechanics are most
noticeable only for very small objects
How small is very small?
1 meter
Looks classical
1 millimeter
Looks classical
1 micrometer
Looks classical
1 nanometer
Looks quantum!
Nonetheless, quantum mechanics is still very important.
How important is very important?
Without quantum mechanics:
Many biological reactions
would not occur.
Life does
not exist
Chemical bonding would be
impossible.
All molecules
disintegrate
All atoms would be unstable.
Universe
explodes
Neil Shenvi’s dissertation title:
Vanity of Vanities, All is Vanity
Minimal
consequences
Talk Outline
1. The history of quantum mechanics
2. The explanatory power of quantum mechanics
3. What is quantum mechanics?
a. The postulates of quantum mechanics
b. The weirdness of the postulates
4. The usefulness of quantum mechanics
5. The philosophy of quantum mechanics
Quantum mechanics is essential for understanding
fundamental concepts in physics, chemistry, and
biology
•
•
•
•
•
Decay of nuclear isotopes
Stability of the atom
The periodic table
Chemical bonding
Photoabsorption spectra
Classical puzzle #1: How can nuclear decay ever
occur at room temperature?
238
94 Pu
234
 92 U
+
Question 2. What is the
approximate activation
energy for nuclear
decay?
4
2+
2 He
E
A. 10 kcal / mol
B. 100 kcal / mol
C. 100,000 kcal / mol
D. 10,000,000 kcal / mol
R
Barrier
Height = ?
R
Most chemical reactions
have an activation
energy of < 20 kcal/mol !
Quantum mechanical tunneling is responsible for
spontaneous fission
238
92 U
234
 90 Th
+
4
2+
2 He
Spontaneous fission
through quantum tunneling
is the basis for nuclear
power, nuclear weapons
(unfortunately), smoke
detectors, and artificial
heart generators.
E
R
Quantum tunneling
R
Classical puzzle #2: why are atoms stable?
Bohr (i.e. “planetary”)
model of the atom
Problem 1: why
don’t electrons fall
into the nucleus?
Problem 2: why
don’t atoms
disintegrate on
collision?
Quantum mechanics shows that electrons can only
populate discrete orbitals around the nucleus
Quantum atom
Atom collapse is
prohibited
Atoms are stable to
collision
Classical puzzle #3: Where does the structure of the
periodic table come from?
Quantum solutions to electrons
confined to a sphere
Periodic table of elements
…
*
*
Classical mechanics offers no
explanation for the general structure
of the periodic table
Quantum mechanics yields the
general structure of the periodic
table from a very simple model of
atoms
Classical puzzle #4: Why do atoms form chemical
bonds?
Question
“Classical”3.H2 Hydrogen
molecule
molecule (H2) is held
together by:
A. Attraction between the
two H nuclei
B. The decreased kinetic
energy of the electrons
C. Repulsive forces between
There are no stable solutions to the
the electrons
four-body problem in Newtonian
D.
Glue
mechanics
Quantum H2 molecule
Overlap of the hydrogen 1s orbitals
stabilizes the H2 molecule
Classical puzzle #5: Why do molecules absorb light
only at particular frequencies?
Chlorophyll A
Quantum mechanics predicts that molecules have
discrete energy levels, leading to discrete absorption
frequencies
E
Photon
absorption
Chlorophyll A
In theory, quantum mechanics allows us to predict
the properties of atoms and molecules from scratch,
without ever appealing to experiment
Quantum mechanics allows the prediction of:
• Atomic properties: ionization energy, UV absorption spectra
• Molecular structure: bond lengths, bond angles, dissociation
energies
• Spectral features: infrared absorption, microwave absorption
• Chemical features: rate constants, enthalpy of reaction
• Biochemical features (often only in theory): crystal structure
binding affinity
The caveat: the larger the system, the more difficult the
calculations become.
Talk Outline
1. The history of quantum mechanics
2. The explanatory power of quantum mechanics
3. What is quantum mechanics?
a. The postulates of quantum mechanics
b. The weirdness of the postulates
4. The usefulness of quantum mechanics
5. The philosophy of quantum mechanics
The laws of quantum mechanics are founded upon
several fundamental postulates
The Fundamental Postulates of Quantum Mechanics:
Postulate 1: All information about a system is provided
by the system’s wavefunction.
Postulate 2: The motion of a nonrelativistic particle is
governed by the Schrodinger equation
Postulate 3: Measurement of a system is associated with
a linear, Hermitian operator
Postulate 1: All information about a system is
provided by the system’s wavefunction.
 ( x)
Pr( x )
x
x
Interesting facts about the wavefunction:
1. The wavefunction can be positive, negative, or complex-valued.
2. The squared amplitude of the wavefunction at position x is
equal to the probability of observing the particle at position x.
3. The wave function can change with time.
4. The existence of a wavefunction implies particle-wave duality.
The Weirdness of Postulate 1: Quantum particles
are usually delocalized, meaning they do not have a
well-specified position
Classical particle
Quantum particle
Position = x
Wavefunction = (x)
The particle
is here.
With some high probability,
the particle is probably
somewhere around here
The Weirdness of Postulate 1: At a given instant in
time, the position and momentum of a particle
cannot both be known with absolute certainty
Classical particle
Hello, my name is:
Classical particle
my position is
11.2392…Ang
my momentum is -23.1322… m/s
Question
What is the
Quantum 4.
particle
name
of the =law
Wavefunction
(x)that limits
our knowledge of the
“I can tell you my exact position, but then I
simultaneous
positionI can
andtell
can’t tell you my momentum.
you my exact momentum,
but then I can’t
momentum
of particles?
tell you my position. I can give you a
pretty good estimate of my position, but
then Pauli’s
I have to give
you a bad estimate of
A.
exclusion
my momentum. I can…”
principle
B. Planck’s law
C. The Heisenberg
uncertainty
?
principle
D. The? ?Dirac
? equation
The Weirdness of Postulate 1: a particle can be put
into a superposition of multiple states at once
Classical elephant:
Valid states:
Quantum elephant:
Valid states:
Gray
Gray
Multicolored
Multicolored
+
Gray AND Multicolored
Postulate 2: The motion of a nonrelativistic particle is
governed by the Schrödinger equation

i
 (t )  Hˆ  (t )
t
Time-dependent S.E.:
Time-dependent S.E.:


2
 2m dx
2
Molecular S.E.:
d
2
Ĥ   E 

ˆ
 V ( x )   ( x )  E  ( x)

Interesting facts about the Schrödinger Equation:
1. It is a wave equation whose solutions display interference effects.
2. It implies that time evolution is unitary and therefore reversible.
3. It is very, very difficult to solve for large systems (i.e. more
than three particles).
The Weirdness of Postulate 2: A quantum mechanical
particle can tunnel through barriers rather than going
over them.
Classical ball
Classical ball does not have
enough energy to climb hill.
Quantum ball
Quantum ball tunnels through
hill despite insufficient energy.
The Weirdness of Postulate 2: Quantum particles
take all paths.
Classical mouse
Quantum mouse
Classical particles take a
single path specified by
Newton’s equations.
The Schrodinger equation
indicates that there is a
nonzero probability for a
particle to take any path
This consequence is stated rigorously in Feymnann’s path integral
formulation of quantum mechanics
Postulate 3: Measurement of a quantum mechanical
system is associated with some linear, Hermitian
operator Ô.
Oˆ   Oˆ 
Oˆ   dx * ( x) Oˆ ( x)( x)
Interesting facts about the measurement postulate:
1. It implies that certain properties can only achieve a discrete set
of measured values
2. It implies that measurement is inherently probabilistic.
3. It implies that measurement necessarily alters the observed
system.
The Weirdness of Postulate 3: Even if the exact
wavefunction is known, the outcome of measurement
is inherently probabilistic
Classical Elephant:
Quantum Elephant:
Before
measurement
or
After
measurement
For a known state, outcome
is deterministic.
For a known state, outcome
is probabilistic.
The Weirdness of Postulate 3: Measurement
necessarily alters the observed system
Classical Elephant:
Quantum Elephant:
Before
measurement
After
measurement
State of the system is
unchanged by measurement.
Measurement changes
the state of the system.
The Weirdness of Postulate 3: Properties are actions
to be performed, not labels to be read
Classical Elephant:
Quantum Elephant:
Position = here
Color = grey
Size
= large
Position:
The ‘position’ of an object exists
independently of measurement and is
simply ‘read’ by the observer
‘Position’ is an action performed on an
object which produces some particular
result
In other words, properties like position or momentum do not exist
independent of measurement! (*unless you’re a neorealist…)
Talk Outline
1. The history of quantum mechanics
2. The explanatory power of quantum mechanics
3. What is quantum mechanics?
a. The postulates of quantum mechanics
b. The weirdness of the postulates
4. The usefulness of quantum mechanics
5. The philosophy of quantum mechanics
Many technologies depend crucially on quantum
mechanical effects
•
•
•
•
NMR spectroscopy
Scanning tunneling microscope
Quantum cryptography
Quantum computation
The quantized character of nuclear spin is the basis
of NMR and MRI technology
B
O
H
H H
H
H
H
H
H
9.3
2.0
ppm
The energy difference between the spin up and spin down states
of protons is what enables NMR spectrometers to differentiate
between different types of hydrogen
Electron tunneling between tip and sample is the
basis for the scanning tunneling electron microscope
e-
E
tunneling
tip
tip
sample
z
Images originally created by IBM.
The measurement theorem enables secure quantum
cryptography by guaranteeing that eavesdropping is
detectable
Alice
Eavesdropper

Bob

To steal the data, Eve must measure the quantum
particles. But since measurement alters the state of the
particle, her presence can always be detected.
C.H. Bennett and G. Brassard "Quantum Cryptography: Public Key Distribution and Coin Tossing",
Proceedings of IEEE International Conference on Computers Systems and Signal Processing, Bangalore
India, December 1984, pp 175-179.
A quantum computer can perform certain operations
much faster than any classical computer
Searching an unordered database:
Smith, A
Smith, A B
Smith, A S
Smith, Amos
Smith, B A
Smith, Bob
Smith, Bob L
Smith, Cynthia
Smith, David
555-1032
555-4023
555-9192
555-1126
555-7287
555-1102
555-1443
555-3739
555-4487
Smith, A
Smith, A B
Smith, A S
Smith, Amos
Smith, B A
Smith, Bob
Smith, Bob L
Smith, Cynthia
Smith, David
555-1032
555-4023
555-9192
555-1126
555-7287
555-1102
555-1443
555-3739
555-4487
Factoring large numbers
16238476016501762387610762691722612171239872103974621876187
12073623846129873982634897121861102379691863198276319276121
=
? x?
16238476016501762387610762691722612171239872103974621876187
12073623846129873982634897121861102379691863198276319276121
=
whimper
? x?
162384760165011238798712
X
87230987183740987123761
Talk Outline
1. The history of quantum mechanics
2. The explanatory power of quantum mechanics
3. What is quantum mechanics?
a. The postulates of quantum mechanics
b. The weirdness of the postulates
4. The usefulness of quantum mechanics
5. The philosophy of quantum mechanics
Quantum mechanics has many important
implications for epistemology and metaphyics
•
•
•
•
•
The possibility of almost anything
The absence of causality/determinism
The role of human consciousness
The limits of human knowledge
The cognitive dissonance of reality
First, quantum mechanics implies that almost no event
is strictly impossible
Classical physics
100%
Quantum physics
99.99..%
1000000
-10
10
“the random nature of quantum physics means that there is always a minuscule, but
nonzero, chance of anything occurring, including that the new collider could spit out
man-eating dragons [emph. added]” - physicist Alvaro de Rujula of CERN regarding the
Large Hadron Collider, quoted by Dennis Overbye, NYTimes 4/15/08
Second, quantum mechanics abrogates notions of
causality and (human?) determinism
Classical physics
cause
Quantum physics
effect
effect
H + T
H
?
(MacBeth)
(MacBeth)
Physics no longer rigorously provides an answer to the
question “what caused this event?”
?
Third, within the Copenhagen interpretation, human
consciousness appears to have a distinct role
When does the wave function collapse during measurement?
|
Wavefunction….wavefunction…wavefunction…………particle!
time
“The very study of the physical world leads to the conclusion that the concept of
consciousness is an ultimate reality” “it follows that the being with a consciousness
must have a different role in quantum mechanics than the inanimate object” – physicist
Eugene Wigner, Nobel laureate and founder of quantum mechanics
Fourth, the fact that the wavefunction is the ultimate
reality implies that there is a severe limit to human
knowledge
|
KEEP OUT
“…classical mechanics took too superficial a view of the world: it dealt with appearances.
However, quantum mechanics accepts that appearances are the manifestation of a
deeper structure (the wavefunction, the amplitude of the state, not the state itself)” –
Peter Atkins
Finally, quantum mechanics challenges our
assumption that ultimate reality will accord with our
natural intuition about what is reasonable and normal
Classical physics
Quantum physics
I think it is safe to say that no one understands quantum mechanics. Do not keep
saying to yourself, if you can possibly avoid it, 'But how can it possibly be like
that?' … Nobody knows how it can be like that. – Richard Feynman
What effect does QM have on the fundamental
assumptions of the science?
1. Rationality of the world
2. Efficacy of human reason
3. Metaphysical realism
4. Regularity of universe
5. Spatial uniformity of universe
6. Temporal uniformity of universe
7. Causality
8. Contingency of universe
9. Desacralization of universe
10. Methodological reductionism (Occam’s razor)
11. Value of scientific enterprise
12. Validity of inductive reasoning
13. Truthfulness of other scientists
It makes things complicated…
?
?
?
?
?
?
?
1. Rationality of the world
Weirdness of Quantum mechanics
2. Efficacy of human reason
3. Metaphysical realism
Copenhagen interpretation
4. Regularity of universe
5. Spatial uniformity of universe
EPR Experiment: Pick one (only)
6. Temporal uniformity of universe
7. Causality
8. Contingency of universe
Many worlds interpretation
9. Desacralization of universe
10. Methodological reductionism (Occam’s razor)
Neo-realism
11. Value of scientific enterprise
12. Validity of inductive reasoning
13. Truthfulness of other scientists
Probabilistic nature
of QM
Concluding Quotes
[QM] has accounted in a quantitative way for atomic
phenomena with numerical precision never before achieved
in any field of science. N. Mermin
The more success the quantum theory has the sillier it looks.
- A. Einstein
I do not like it, and I am sorry I ever had anything to do with
it. -E. Schrödinger
Acknowledgements
•
•
•
•
Dr. Christina Shenvi
Prof. John Tully
Prof. K. Birgitta Whaley
Prof. Bob Harris
Cartoons provided by: prescolaire.grandmonde.com and www.clker.com