Complex Correspondence Principle Carl Bender Physics Department Washington University in...

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Complex Correspondence Principle Carl Bender Physics Department Washington University in collaboration with Daniel Hook Theoretical Physics Imperial College

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Page 1: Complex Correspondence Principle Carl Bender Physics Department Washington University in collaboration with Daniel Hook Theoretical Physics Imperial College.

Complex Correspondence Principle

Carl BenderPhysics DepartmentWashington University

in collaboration with

Daniel HookTheoretical PhysicsImperial College

Page 2: Complex Correspondence Principle Carl Bender Physics Department Washington University in collaboration with Daniel Hook Theoretical Physics Imperial College.

Extending quantum mechanics into the complex domain

This Hamiltonian is PT symmetric

Page 3: Complex Correspondence Principle Carl Bender Physics Department Washington University in collaboration with Daniel Hook Theoretical Physics Imperial College.

Region of brokenPT symmetry

Region of unbrokenPT symmetry

PT phasetransition

Page 4: Complex Correspondence Principle Carl Bender Physics Department Washington University in collaboration with Daniel Hook Theoretical Physics Imperial College.

The PT phase transition has now been seen experimentally!

Page 5: Complex Correspondence Principle Carl Bender Physics Department Washington University in collaboration with Daniel Hook Theoretical Physics Imperial College.

Laboratory verification using table-top optics experiments!

• Z. Musslimani, K. Makris, R. El-Ganainy, and D. Christodoulides, PRL 100, 030402 (2008)

• K. Makris, R. El-Ganainy, D. Christodoulides, and Z. Musslimani, PRL 100, 103904 (2008)

• A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, Phys. Rev. Lett. 103, 093902 (2009)

Observing PT symmetry using optical wave guides:

Page 6: Complex Correspondence Principle Carl Bender Physics Department Washington University in collaboration with Daniel Hook Theoretical Physics Imperial College.

The observed PT phase transition

Page 7: Complex Correspondence Principle Carl Bender Physics Department Washington University in collaboration with Daniel Hook Theoretical Physics Imperial College.

People at this meeting who have worked on PT quantum mechanics

Thrust Cigar MothRecalled IranHah! Minum NippleAccuse Zinc MuleBill to MilkmanMat Off JohnMafia Had ZealtsNag JckJars Nth LoonJag VerseShh! Ask VegGnaw Knish

(with apologies!)

Page 8: Complex Correspondence Principle Carl Bender Physics Department Washington University in collaboration with Daniel Hook Theoretical Physics Imperial College.

People at this meeting who have worked on PT quantum mechanics

Thomas CurtrightAndre LeClairPhilip MannheimLuca MezincescuKimball MiltonJohn MoffatAli MostafazadehJack NgJohn RalstonS G RajeevK V ShajeshKwang Shin

(with apologies!)

Page 9: Complex Correspondence Principle Carl Bender Physics Department Washington University in collaboration with Daniel Hook Theoretical Physics Imperial College.

PT. There isa networkthat ties us together.

Page 10: Complex Correspondence Principle Carl Bender Physics Department Washington University in collaboration with Daniel Hook Theoretical Physics Imperial College.

Find all solutions, real or complex, to

Hamilton’s equations:

Extending classical mechanicsinto the complex domain...

Page 11: Complex Correspondence Principle Carl Bender Physics Department Washington University in collaboration with Daniel Hook Theoretical Physics Imperial College.

Motion on the real axis

Motion of particles is governed by Newton’s Law:

F=maIn freshman physics this motion is restricted to theREAL AXIS.

Page 12: Complex Correspondence Principle Carl Bender Physics Department Washington University in collaboration with Daniel Hook Theoretical Physics Imperial College.

Harmonic oscillator: Particle on a spring

Turning point Turning point

Back and forth motionon the real axis:

Page 13: Complex Correspondence Principle Carl Bender Physics Department Washington University in collaboration with Daniel Hook Theoretical Physics Imperial College.

Harmonic oscillator:

Turning point Turning point

Motion in thecomplex plane:

Page 14: Complex Correspondence Principle Carl Bender Physics Department Washington University in collaboration with Daniel Hook Theoretical Physics Imperial College.

The classical particle can enter the classically forbidden region!

But its motion is orthogonal to the real axis!

This is like total internal reflection:

Glass Vacuum

Page 15: Complex Correspondence Principle Carl Bender Physics Department Washington University in collaboration with Daniel Hook Theoretical Physics Imperial College.

32 ixpH ( = 1)

Page 16: Complex Correspondence Principle Carl Bender Physics Department Washington University in collaboration with Daniel Hook Theoretical Physics Imperial College.

(11 sheets)H = p - (ix)2

Page 17: Complex Correspondence Principle Carl Bender Physics Department Washington University in collaboration with Daniel Hook Theoretical Physics Imperial College.
Page 18: Complex Correspondence Principle Carl Bender Physics Department Washington University in collaboration with Daniel Hook Theoretical Physics Imperial College.

Conventional correspondence principle

Classical probability(1/speed)

Quantum probability

Page 19: Complex Correspondence Principle Carl Bender Physics Department Washington University in collaboration with Daniel Hook Theoretical Physics Imperial College.

16th Eigenstate

Page 20: Complex Correspondence Principle Carl Bender Physics Department Washington University in collaboration with Daniel Hook Theoretical Physics Imperial College.

Complex classical harmonic oscillator

Page 21: Complex Correspondence Principle Carl Bender Physics Department Washington University in collaboration with Daniel Hook Theoretical Physics Imperial College.

Classical probability in the complex plane

Page 22: Complex Correspondence Principle Carl Bender Physics Department Washington University in collaboration with Daniel Hook Theoretical Physics Imperial College.

Pup Tent

Page 23: Complex Correspondence Principle Carl Bender Physics Department Washington University in collaboration with Daniel Hook Theoretical Physics Imperial College.

Complex quantum probability

Potential is PT symmetric means

Local conservation law:

Page 24: Complex Correspondence Principle Carl Bender Physics Department Washington University in collaboration with Daniel Hook Theoretical Physics Imperial College.

Probability contour

Page 25: Complex Correspondence Principle Carl Bender Physics Department Washington University in collaboration with Daniel Hook Theoretical Physics Imperial College.

Example: complex PT-symmetric random walk

With a complex unfair coin!

P(heads) = -ia + ½ P(tails) = ia + ½

Page 26: Complex Correspondence Principle Carl Bender Physics Department Washington University in collaboration with Daniel Hook Theoretical Physics Imperial College.

Condition I

Page 27: Complex Correspondence Principle Carl Bender Physics Department Washington University in collaboration with Daniel Hook Theoretical Physics Imperial College.

Ground state of harmonic oscillator

This equation looks easy, but it isimpossible to solve exactly!

Page 28: Complex Correspondence Principle Carl Bender Physics Department Washington University in collaboration with Daniel Hook Theoretical Physics Imperial College.

Toy model

Leading asymptotic behavior:

Full asymptotic behavior:

Where is the arbitrary constant?!?

Page 29: Complex Correspondence Principle Carl Bender Physics Department Washington University in collaboration with Daniel Hook Theoretical Physics Imperial College.

Difference of two solutions

The arbitrary constant is in the hyperasymptoticcontribution to the asymptotic approximation!

Page 30: Complex Correspondence Principle Carl Bender Physics Department Washington University in collaboration with Daniel Hook Theoretical Physics Imperial College.

Separatrix

Quantizedbundle

Page 31: Complex Correspondence Principle Carl Bender Physics Department Washington University in collaboration with Daniel Hook Theoretical Physics Imperial College.

Paths in the complex plane

Good Stokes’ wedge

Bad Stokes’ wedge

Page 32: Complex Correspondence Principle Carl Bender Physics Department Washington University in collaboration with Daniel Hook Theoretical Physics Imperial College.

Conditions II and III:Real part of the probability

CONVERGENT!!!

Page 33: Complex Correspondence Principle Carl Bender Physics Department Washington University in collaboration with Daniel Hook Theoretical Physics Imperial College.

Going into and out of the bad Stokes’ wedge

Page 34: Complex Correspondence Principle Carl Bender Physics Department Washington University in collaboration with Daniel Hook Theoretical Physics Imperial College.

Probability contours in the complex plane

Page 35: Complex Correspondence Principle Carl Bender Physics Department Washington University in collaboration with Daniel Hook Theoretical Physics Imperial College.

More interesting contours...

Page 36: Complex Correspondence Principle Carl Bender Physics Department Washington University in collaboration with Daniel Hook Theoretical Physics Imperial College.

First excited state – one node

Page 37: Complex Correspondence Principle Carl Bender Physics Department Washington University in collaboration with Daniel Hook Theoretical Physics Imperial College.

Second excited state – two nodes

Page 38: Complex Correspondence Principle Carl Bender Physics Department Washington University in collaboration with Daniel Hook Theoretical Physics Imperial College.

This is the quantum versionof the pup tent!!

(with ripples on the canopy)

Page 39: Complex Correspondence Principle Carl Bender Physics Department Washington University in collaboration with Daniel Hook Theoretical Physics Imperial College.

These people areamazed thatclassical mechanicsand quantummechanics can beextended into thecomplex plane, and that the correspondenceprinciple continuesto hold!


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