rescuing quantum mechanics from atomic physics · quantum mechanics from atomic physics ... (e.g....

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RESCUINGQUANTUM MECHANICS

FROMATOMIC PHYSICS

Edwin F. TaylorMassachusetts Institute of Technology

June 2002

[email protected]

www.eftaylor.com

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Thirty-one years ago [1949!],Dick Feynman told me about his"sum over histories" version ofquantum mechanics. "Theelectron does anything it likes,"he said. "It just goes in anydirection at any speed, forwardor backward in time, however itlikes, and then you add up theamplitudes and it gives you thewave-function." I said to him, "You're crazy."

But he wasn't. -- Freeman Dyson, 1980

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Using Feynman's path integral approach,based on the Principle of Least Action,there is no longer any differencebetween classical mechanics andquantum mechanics, except for a trivialadjustment to the mathematics. . .But . . .students are still taught classicalmechanics the old-fashioned way, andthen forced to train themselves into anew way of thinking in order to studyquantum mechanics using the . . .Schrödinger equation. By the time mostpeople learn about Feynman's approach(if they ever do), . . . it is hard toappreciate its simplicity, and galling torealize that they could have saved timeand effort by learning quantum theory(and classical theory!) Feynman's way inthe first place.

-- John and Mary Gribbin

Richard Feynman, A Life in Science

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OUTLINE

1. Quantum mechanics, child of atomicphysics, has outgrown its parent.

2. Sum over histories is a simple,fundamental, and powerfulintroduction to quantum mechanics.

3. Excerpts from incompletely realizedstory line on sum over histories.

4. Modern applications in context

5. Longer story line connectinggeneral relativity, Newtonianmechanics, & quantum mechanics

6. Abbie Hoffman : "Steal this Book"

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1. Quantum mechanics, child of atomicphysics, has outgrown its parent.

Quantum mechanics grew out of theeffort to decode the atom.

The "story line" of quantum mechanicshas remained (pseudo!) historical –atomic physics.

Summary of first three chapters ofFrench and Taylor QM text:"Waves are particles and particles arewaves. Here is a wave equation.Let's go!"

Solving the Schrödinger equation is aprofessional specialty for those whomanipulate the atom and its constructsBut many current fields make different(more basic?) uses of quantummechanics

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NON-SCHRÖDINGER-APPLICATIONS

Photon quantum mechanics(Holbrow et al et al et al)

Quantum seeing in the dark

Delayed choice experiments

Entangled states

EPR experimentssssssss

Quantum information (qubit vs. bit)

Quantum computing

Quantum teleportation

Quantum cryptography

Feynman diagrams (Schroeder)

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2. "Explore all paths!" is a simple, basic,and powerful introduction to quantummechanics. Excerpts from anincompletely realized story line

SCHRÖDINGEREQUATION

ATOMIC PHYSICSAND POINTS WEST

PROPAGATORSTATIONARY STATES

APPLICATIONS

PHOTONReflections 2-D Motions

ELECTRONFree electron& in Potential

ELECTRONWavefunction

PROPAGATOR Free Electron SHO Potential

QUANTUM MECHANICS TREE TRUNK

EXPLORESAMPLE Paths

EXPLOREALL Paths

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SCHRÖDINGEREQUATION

ATOMIC PHYSICSAND POINTS WEST


APPLICATIONS


ELECTRONFree electron& in Potential



VERBAL: Adding Arrows

COMPUTER sums Action along classical worldline, hunts for worldline of minimum action

COMPUTER-aided heuristic derivation of propagator for free particle.

COMPUTER sums stopwatch rotations along each path, adds alternative path arrows.

COMPUTER sums arrows forworldlines from all initial events at t 1 to each event at t 2.

COMPUTER DOES THE HEAVY LIFTING

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Introduce notationwith modern applications

QUANTUM SEEING IN THE DARK

ENTANGLED STATES = SUPERPOSITION STATES FOR TWO (OR N) PARTICLES

EPR EXPERIMENTS

DELAYED CHOICE EXPERIMENTS

SUM OVER PATHS IMPLIES SUPERPOSITION STATES

|ψ> = a |x> + b |y>

= e |R> + g |L>

amplitude = <x|ψ>

Alternative representations

probability = |<x|ψ>|

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state: |ψ>

NOTATION: Photon Polarization States(with real equipment)

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"xy ANALYZER LOOP" for photons

CALCITE|ψ>|ψ>

a |x>

b |y>

|ψ> = a |x> + b |y>

ANY state | > can be dismantled andreconstituted. This is the meaning ofstatement that |x> and |y> form acomplete set.

Arbitrary orientation of xy axes impliesthere are an infinite number ofalternative two-state complete sets.

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Use quarter-wave plates to convert to"RL analyzer," yielding yet anothercomplete set (& photon with spin one)

|ψ> = e |R> + g |L>

CAL-CITE

|ψ>e |R>

g |L>

|ψ>

λ/4 PLATES

The states in ANY two-state completeset can be assigned the arbitraryindices 0 and 1.

The result is the QUBIT, the centralactor in quantum information,computing, teleportation, cryptography

|Q> = a |0> + b |1>

Replaces BIT (0 or 1) in classicalcomputing

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3. Additional modern applications

FEYNMAN DIAGRAMS(ANOTHER KIND OF “PATH”)

QUANTUM INFORMATION(QUBIT VS. BIT)

QUANTUM COMPUTING

QUANTUMCRYPTOGRAPHY

QUANTUM TELEPORTATION

APPLICATION OF SUM OVER PATHS TO OTHER FIELDS

DIRECT APPLICATIONS

PRINCIPLE OFLEAST ACTION

INTRODUCTORYMECHANICS(e.g. Nobody)

ADVANCED MECHANICS(e.g. Sussman & Wisdom)

CLASSICAL THEORY OF FIELDS(e.g. Landau & Lifshitz)

QUANTUM FIELD THEORY(e.g. Kaku)

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SOME DETAILS:OF STORY LINE

START WITH PHOTONS

SCHROEDINGEREQUATION

PROFESSIONAL FIELDS



B R A N C H E S


ELECTRONFreeIn Potential

Rotation frequency = quantum stopwatch

KE—PE h




hE

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CONTINUE STORY LINEMOVE ON TO ELECTRONS


PROFESSIONAL FIELDS



B R A N C H E S




KE—PE h




hE

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SEAMLESS TRANSITIONTO CLASSICAL MECHANICS

In principle, the electron takes aninfinite range of alternative paths,but there is a "pencil" of paths thatmake the major contributions to theresulting vector

.

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The “pencil” of paths makingmajor contribution to final arrowshrinks as the mass increases.

For mass = mass of electron:

For mass = 4 times mass of electron

For mass = 1000 times mass of electron(half the mass of a proton)

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INTRODUCE THE WAVEFUNCTION


PROFESSIONAL FIELDS



B R A N C H E S




KE—PE h




hE

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PROPAGATOR CONVERTSSAMPLING OF PATHS

TO ALL PATHS


PROFESSIONAL FIELDS



B R A N C H E S




KE—PE h




hE

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PROPAGATOR summarizes result ofALL paths from one initial event a toone final event b. (Feynman calls itthe KERNEL, hence symbol K)

arrow at

event b

= K b, a( )

arrow at

event a

K(b,a) simply rotates arrow a, changesits length to contribute to arrow b.

Final arrow at b is sum of propagatedarrows from initial events.

Sum over initial arrows goes tointegral over initial wavefunction:

xb ,tb( ) = K b, a( ) xa , ta( )−∞

+∞∫ dxa

(Until now just words, no equations.)

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Heuristic derivation of propagator forfree particle, using computer and trialand error.

GIVEN propagator for simple harmonicoscillator in computer program, candiscover stationary states as specialcase of "sloshing" probabilities forarbitrary initial wavefunction.

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SUMMARY OF A STORY LINE

SPECIAL RELATIVITYGENERAL RELATIVITY

“Go straight!”(Follow worldline of maximal aging.)

“Explore all paths!”

Principle ofMaximal Aging

Lowspeed

Increasedmass

Pencil of contributingworldlines narrows as mass increases.

(“Maximal” and “least” generalize to “extremal.”)

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FEYNMAN DIAGRAMS(ANOTHER KIND OF “PATH”)

QUANTUM INFORMATION(QUBIT VS. BIT)

QUANTUM COMPUTING

QUANTUMCRYPTOGRAPHY

QUANTUM TELEPORTATION

APPLICATION OF SUM OVER PATHS TO OTHER FIELDS

DIRECT APPLICATIONS

PRINCIPLE OFLEAST ACTION

INTRODUCTORYMECHANICS(e.g. Nobody)

ADVANCED MECHANICS(e.g. Sussman & Wisdom)

CLASSICAL THEORY OF FIELDS(e.g. Landau & Lifshitz)

QUANTUM FIELD THEORY(e.g. Kaku)

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Not only does the least-actionprinciple offer a means offormulating classical mechanicsthat is more flexible and powerfulthan Newtonian mechanics, [butalso] variations on the least-action principle have proveduseful in general relativity theory,quantum field theory, andparticle physics.As a result, this principle liesat the core of much ofcontemporary physics. Thomas A. Moore Macmillan Encyclopedia of Physics

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HISTORY OF SUM OVER HISTORIES:

Online classes on National TeachersEnhancement Network, Montana State

One semester class at MIT

NEXT STEPS

Abbie Hoffman: Steal this Book

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HANDOUT

Rescuing Quantum Mechanics from Atomic Physics

Edwin F. Taylor

Massachusetts Institute of Technology

[email protected]

June 2002

Initial steps in story line based onRichard P. Feynman, QED: The Strange Theory of Light and Matter1985, Princeton University Press, ISBN 0 691 02417 0

Advanced treatment of path integral approach to quantum mechanics:R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path IntegralsMcGraw-Hill, 1965, Library of Congress 64-25171 (Before ISBNs!)FULL of typos and errors. For list of corrections, see:http://www.oberlin.edu/physics/dstyer/TeachQM/Hibbs.pdf

WEBSITE: www.eftaylor.com/

Available from quantum mechanics page of this website:

1. Transparencies for this talk

2. All computer programs (Mac and Windows) illustrated in this talk, and more.

3. Student workbook to accompany computer programs.

4. Link to following article:"Teaching Feynman's sum-over-paths quantum theory"Edwin F. Taylor, Stamatis Vokos, and John M. O'Meara, and Nora S. ThornberComputers in Physics, Vol. 12, No 2, Mar/Apr 1998, pages 190 –198

rescuing quantum mechanics from atomic physics · quantum mechanics from atomic physics ... (e.g....

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