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Complex Correspondence Principle
Carl BenderPhysics DepartmentWashington University
in collaboration with
Daniel HookTheoretical PhysicsImperial 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:
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!)
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:
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
Example: complex PT-symmetric random walk
With a complex unfair coin!
P(heads) = -ia + ½ P(tails) = ia + ½
Difference of two solutions
The arbitrary constant is in the hyperasymptoticcontribution to the asymptotic approximation!