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

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? • The postulates of quantum mechanics • 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 Stern-Gerlach Experiment The Hydrogen Spectrum The Ultraviolet Catastrophe ? Newtonian explanations for these phenomena were wildly insufficient

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? • A. 1 • 2 • 3 • D. A diffuse cloud + ? ? ? ? ? - Exactly two well-defined beams emerge from the magnet! Ag atoms

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 Increasing weirdness 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) Spelling (words) Quantum mechanics (atoms and molecules) 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: Universe explodes All atoms would be unstable. Chemical bonding would be impossible. All molecules disintegrate Many biological reactions would not occur. Life does not exist Minimal consequences Neil Shenvi’s dissertation title: Vanity of Vanities, All is Vanity

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? • Question 2. What is the approximate activation energy for nuclear decay? • A. 10 kcal / mol • 100 kcal / mol • 100,000 kcal / mol • 10,000,000 kcal / mol 238 234 4 Pu  U + He 2+ 94 92 2 E R Barrier Height = ? Most chemical reactions have an activation energy of < 20 kcal/mol ! R

Quantum mechanical tunneling is responsible for spontaneous fission 238 234 4 U  Th + He 2+ 92 90 2 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 * … * Quantum mechanics yields the general structure of the periodic table from a very simple model of atoms Classical mechanics offers no explanation for the general structure of the periodic table

Classical puzzle #4: Why do atoms form chemical bonds? • Question 3. Hydrogen molecule (H2) is held together by: • Attraction between the two H nuclei • The decreased kinetic energy of the electrons • Repulsive forces between the electrons • Glue “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

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.

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 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 • Question 4. What is the name of the law that limits our knowledge of the simultaneous position and momentum of particles? • Pauli’s exclusion principle • Planck’s law • The Heisenberg uncertainty principle • The Dirac equation Classical particle Quantum particle Wavefunction = (x) “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…” Hello, my name is: Classical particle my position is 11.2392…Ang my momentum is -23.1322… m/s ? ? ? ?

The Weirdness of Postulate 1: a particle can be put into a superposition of multiple states at once Classical elephant: Quantum elephant: Valid states: Valid states: Gray Gray Multicolored Multicolored + Gray AND Multicolored

Postulate 2: The motion of a nonrelativistic particle is governed by the Schrödinger equation Time-dependent S.E.: Time-dependent S.E.: Molecular S.E.: 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 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. Classical mouse Quantum mouse The Schrodinger equation indicates that there is a nonzero probability for a particle to take any path Classical particles take a single path specified by Newton’s equations. This consequence is stated rigorously in Feymnann’s path integralformulation of quantum mechanics

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 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…)

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 H H O 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 555-1032 Smith, A B 555-4023 Smith, A S 555-9192 Smith, Amos 555-1126 Smith, B A 555-7287 Smith, Bob 555-1102 Smith, Bob L 555-1443 Smith, Cynthia 555-3739 Smith, David 555-4487 Smith, A 555-1032 Smith, A B 555-4023 Smith, A S 555-9192 Smith, Amos 555-1126 Smith, B A 555-7287 Smith, Bob 555-1102 Smith, Bob L 555-1443 Smith, Cynthia 555-3739 Smith, David 555-4487 Factoring large numbers 1623847601650176238761076269172261217123987210397462187618712073623846129873982634897121861102379691863198276319276121= ? x ? 1623847601650176238761076269172261217123987210397462187618712073623846129873982634897121861102379691863198276319276121= ? x ? whimper 162384760165011238798712 X 87230987183740987123761

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 100% 1000000 10-10 99.99..% “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 physics cause ? 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

? Weirdness of Quantum mechanics ? ? Copenhagen interpretation ? It makes things complicated… 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 EPR Experiment: Pick one (only) ? Many worlds interpretation Neo-realism ? ? 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

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