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

Extending quantum mechanics into the complex domain

This Hamiltonian is PT symmetric

Region of brokenPT symmetry

Region of unbrokenPT symmetry

PT phasetransition

The PT phase transition has now been seen experimentally!

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:

The observed PT phase transition

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!)

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!)

PT. There isa networkthat ties us together.

Find all solutions, real or complex, to

Hamilton’s equations:

Extending classical mechanicsinto the complex domain...

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.

Harmonic oscillator: Particle on a spring

Turning point Turning point

Back and forth motionon the real axis:

Harmonic oscillator:

Turning point Turning point

Motion in thecomplex plane:

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

32 ixpH ( = 1)

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

Conventional correspondence principle

Classical probability(1/speed)

Quantum probability

16th Eigenstate

Complex classical harmonic oscillator

Classical probability in the complex plane

Pup Tent

Complex quantum probability

Potential is PT symmetric means

Local conservation law:

Probability contour

Example: complex PT-symmetric random walk

With a complex unfair coin!

P(heads) = -ia + ½ P(tails) = ia + ½

Condition I

Ground state of harmonic oscillator

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

Toy model

Leading asymptotic behavior:

Full asymptotic behavior:

Where is the arbitrary constant?!?

Difference of two solutions

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

Separatrix

Quantizedbundle

Paths in the complex plane

Good Stokes’ wedge

Bad Stokes’ wedge

Conditions II and III:Real part of the probability

CONVERGENT!!!

Going into and out of the bad Stokes’ wedge

Probability contours in the complex plane

More interesting contours...

First excited state – one node

Second excited state – two nodes

This is the quantum versionof the pup tent!!

(with ripples on the canopy)

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