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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
Region of broken
PT symmetry
PT phase
transition
Region of unbroken
PT symmetry
The PT phase transition has
now been seen experimentally!
Laboratory verification using
table-top optics experiments!
Observing
PT symmetry using optical wave guides:
• 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)
The observed PT phase transition
People at this meeting who have
worked on PT quantum mechanics
Thrust Cigar Moth
Recalled Iran
Hah! Minum Nipple
Accuse Zinc Mule
Bill to Milkman
Mat Off John
Mafia Had Zealts
Nag Jck
Jars Nth Loon
Jag Verse
Shh! Ask Veg
Gnaw Knish
(with apologies!)
People at this meeting who have
worked on PT quantum mechanics
Thomas Curtright
Andre LeClair
Philip Mannheim
Luca Mezincescu
Kimball Milton
John Moffat
Ali Mostafazadeh
Jack Ng
John Ralston
S G Rajeev
K V Shajesh
Kwang Shin
(with apologies!)
PT. There is
a network
that ties us
together.
Extending classical mechanics
into the complex domain...
Find all solutions, real
or complex, to
Hamilton’s equations:
Motion on the real axis
Motion of particles is governed by Newton’s Law:
F=ma
In freshman physics this motion is restricted to the
REAL AXIS.
Harmonic oscillator:
Particle on a spring
Back and forth motion
on the real axis:
Turning point
Turning point
Harmonic oscillator:
Motion in the
complex plane:
Turning point
Turning point
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
2
3
H  p  ix
(
= 1)
2
H = p - (ix)
(11 sheets)
Conventional correspondence principle
Classical probability
(1/speed)
Quantum probability
th
16
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 PTsymmetric 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 is
impossible 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 hyperasymptotic
contribution to the asymptotic approximation!
Quantized
bundle
Separatrix
Paths in the complex plane
Bad Stokes’ wedge
Good 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 version
of the pup tent!!
(with ripples on the canopy)
These people are
amazed that
classical mechanics
and quantum
mechanics can be
extended into the
complex plane,
and that the
correspondence
principle continues
to hold!