Quantum Mechanics: what is it and why is it interesting?

Dr. Neil ShenviDepartment of ChemistryYale University

Talk Outline1. The history of quantum mechanics2. The explanatory power of quantum mechanics3. What is quantum mechanics?

a. The postulates of quantum mechanicsb. The weirdness of the postulates

4. The usefulness of quantum mechanics5. The philosophy of quantum mechanics

Classical mechanics is the mechanics of everyday objects like tables and chairs

Sir Isaac Newton

1. An object in motion tends to stay in motion.2. Force equals mass times acceleration3. For every action there is an equal and opposite reaction.

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 UltravioletCatastrophe

The HydrogenSpectrum

The Stern-GerlachExperiment

Newtonian explanations for these phenomena were wildly insufficient

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The Stern-Gerlach experiment involved passing atomic “magnets” through a magnetic field

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-

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Ag atoms

Question 1. How many beams do we expect to emerge from the magnet?

A. 1B.2C.3D. 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

1900 – Planck’s constant

1913 – Bohr’s model of the atom

1925 – Pauli exclusion principle

1926 – Schrodinger equation

1948 – Feynmann’s path integral formulation

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1954 – Everett’s many-worldtheory

Quantum mechanics applies to all objects, no matter how big or small

Quantum mechanics (atoms and molecules)

Classical mechanics (large molecules)

Thermodynamics (collections of molecules)

Mechanical Engineering(macroscopic objects)

Penmanship (letters)

Spelling (words)

Grammar(sentences)

Creative writing (books)

However, the effects of quantum mechanics are most noticeable only for very small objects

How small is very small?

1 meter Looks classical

1 micrometer Looks classical

1 millimeter Looks classical

1 nanometer Looks quantum!

Nonetheless, quantum mechanics is still very important.

How important is very important? Without quantum mechanics:

All atoms would be unstable.Universeexplodes

Chemical bonding would be impossible.

All moleculesdisintegrate

Many biological reactions would not occur.

Life doesnot exist

Neil Shenvi’s dissertation title:Vanity of Vanities, All is Vanity

Minimal consequences

Talk Outline1. The history of quantum mechanics2. The explanatory power of quantum mechanics3. What is quantum mechanics?

a. The postulates of quantum mechanicsb. The weirdness of the postulates

4. The usefulness of quantum mechanics5. 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?

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R

R

Barrier Height = ?

Question 2. What is the approximate activation energy for nuclear decay?

A. 10 kcal / molB.100 kcal / molC.100,000 kcal / molD.10,000,000 kcal / mol

Pu U + He 2+238

94

234

92

4

2

Most chemical reactions have an activation energy of < 20 kcal/mol !

Quantum mechanical tunneling is responsible for spontaneous fission

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U Th + He 2+238

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90

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Quantum tunneling

Spontaneous fission through quantum tunneling is the basis for nuclear power, nuclear weapons (unfortunately), smoke detectors, and artificial heart generators.

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?

“Classical” H2 molecule Quantum H2 molecule

There are no stable solutions to the four-body problem in Newtonian mechanics

Overlap of the hydrogen 1s orbitals stabilizes the H2 molecule

Question 3. Hydrogen molecule (H2) is held together by:

A.Attraction between the two H nucleiB.The decreased kinetic energy of the electronsC.Repulsive forces between the electronsD.Glue

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

Chlorophyll A

E

Photonabsorption

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 Outline1. The history of quantum mechanics2. The explanatory power of quantum mechanics3. What is quantum mechanics?

a. The postulates of quantum mechanicsb. The weirdness of the postulates

4. The usefulness of quantum mechanics5. 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.

( )xx

Pr( )xx

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 isequal 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

The particle is here.

With some high probability, the particle is probably somewhere around here

Position = x Wavefunction = (x)

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 Quantum particleWavefunction = (x)

Hello, my name is:

Classical particlemy position is 11.2392…Angmy momentum is -23.1322… m/s

“I can tell you my exact position, but then I can’t tell you my momentum. I can tell you my exact momentum, but then I can’t tell you my position. I can give you a pretty good estimate of my position, but then I have to give you a bad estimate of my momentum. I can…”

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Question 4. What is the name of the law that limits our knowledge of the simultaneous position and momentum of particles?

A.Pauli’s exclusion principleB.Planck’s lawC.The Heisenberg uncertainty principleD.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:

Gray

Multicolored

Gray Multicolored

Valid states:

+

Gray AND Multicolored

Postulate 2: The motion of a nonrelativistic particle is governed by the Schrödinger equation

2 2

2ˆ ( )

2( ) ( )

dV x

m dxx E x

H E

ˆ( ) ( )i t H tt

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).

Time-dependent S.E.:

Time-dependent S.E.:

Molecular S.E.:

The Weirdness of Postulate 2: A quantum mechanical particle can tunnel through barriers rather than going over them.

Classical ball Quantum ball

Classical ball does not have enough energy to climb hill.

Quantum ball tunnels through hill despite insufficient energy.

The Weirdness of Postulate 2: Quantum particles take all paths.

This consequence is stated rigorously in Feymnann’s path integralformulation of quantum mechanics

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

Postulate 3: Measurement of a quantum mechanical system is associated with some linear, Hermitian operator Ô.

Interesting facts about the measurement postulate:1. It implies that certain properties can only achieve a discrete set of measured values2. It implies that measurement is inherently probabilistic.3. It implies that measurement necessarily alters the observedsystem.

ˆ ˆO O *ˆ ˆ( ) ( ) ( )O dx x O x x

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

Aftermeasurement

For a known state, outcomeis probabilistic.

For a known state, outcomeis deterministic.

or

The Weirdness of Postulate 3: Measurement necessarily alters the observed system

Classical Elephant: Quantum Elephant:

Before measurement

Aftermeasurement

Measurement changes the state of the system.

State of the system is unchanged by measurement.

The Weirdness of Postulate 3: Properties are actions to be performed, not labels to be read

Classical Elephant: Quantum Elephant:

The ‘position’ of an object exists independently of measurement and is simply ‘read’ by the observer

Position = hereColor = greySize = large

‘Position’ is an action performed on an object which produces some particular result

Position:

In other words, properties like position or momentum do not exist independent of measurement! (*unless you’re a neorealist…)

Talk Outline1. The history of quantum mechanics2. The explanatory power of quantum mechanics3. What is quantum mechanics?

a. The postulates of quantum mechanicsb. The weirdness of the postulates

4. The usefulness of quantum mechanics5. 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

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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

Images originally created by IBM.

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The measurement theorem enables secure quantum cryptography by guaranteeing that eavesdropping is detectable

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.

Alice BobEavesdropper

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.

A quantum computer can perform certain operations much faster than any classical computer

Smith, A 555-1032Smith, A B 555-4023Smith, A S 555-9192Smith, Amos 555-1126Smith, B A 555-7287Smith, Bob 555-1102Smith, Bob L 555-1443Smith, Cynthia 555-3739Smith, David 555-4487

Smith, A 555-1032Smith, A B 555-4023Smith, A S 555-9192Smith, Amos 555-1126Smith, B A 555-7287Smith, Bob 555-1102Smith, Bob L 555-1443Smith, Cynthia 555-3739Smith, David 555-4487

Searching an unordered database:

Factoring large numbers16238476016501762387610762691722612171239872103974621876187120736238461298739826348971218611023796918631982763192761

21= ? x ?whimper

16238476016501762387610762691722612171239872103974621876187120736238461298739826348971218611023796918631982763192761

21= ? x ? 162384760165011238798712

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Talk Outline1. The history of quantum mechanics2. The explanatory power of quantum mechanics3. What is quantum mechanics?

a. The postulates of quantum mechanicsb. The weirdness of the postulates

4. The usefulness of quantum mechanics5. 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 Quantum physics

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“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 Quantum physicscause

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effect

Physics no longer rigorously provides an answer to the question “what caused this event?”

(MacBeth) (MacBeth)

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Third, within the Copenhagen interpretation, human consciousness appears to have a distinct role

When does the wave function collapse during measurement?

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time

Wavefunction….wavefunction…wavefunction…………particle!

“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

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“…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 world2. Efficacy of human reason3. Metaphysical realism4. Regularity of universe5. Spatial uniformity of universe6. Temporal uniformity of universe7. Causality8. Contingency of universe9. Desacralization of universe10. Methodological reductionism (Occam’s razor)11. Value of scientific enterprise12. Validity of inductive reasoning13. Truthfulness of other scientists

It makes things complicated…

1. Rationality of the world2. Efficacy of human reason3. Metaphysical realism4. Regularity of universe5. Spatial uniformity of universe6. Temporal uniformity of universe7. Causality8. Contingency of universe9. Desacralization of universe10. Methodological reductionism (Occam’s razor)11. Value of scientific enterprise12. Validity of inductive reasoning13. Truthfulness of other scientists

Weirdness of Quantum mechanics??

?? Copenhagen interpretation

EPR Experiment: Pick one (only)

Neo-realism

Many worlds interpretation

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Probabilistic nature of QM

Concluding Quotes

The more success the quantum theory has the sillier it looks. - A. Einstein

[QM] has accounted in a quantitative way for atomic phenomena with numerical precision never before achieved in any field of science. N. Mermin

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

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