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Unified Theory of AnnihilationCreation Operators for Solvable
(‘Discrete’) Quantum Mechanics
§1. Introduction
§2. General theory
§3. Examples
3.1. quantum mechanics
3.2 discrete QM
§4. Summary
Shinshu University
Satoru Odake
Collaboration with Ryu Sasaki (YITP)
quant-ph/0605215  J.Math.Phys.47(2006)102102
quant-ph/0605221  Phys.Lett.B641(2006)112-117
1 November 2006
Joint Meeting of Pacific Region Particle Physics Communities
parallel session “Field Theory Motivated Theoretical Developments”
at Sheraton Waikiki Hotel
title
§1. Introduction
The annihilation and creation operators are the most
basic and important tools in quantum mechanics.
Modern quantum physics is almost unthinkable without them.
Ex : Quantum Field Theory
field operator
(in the interaction
picture)
collection of
the annihilation-creation operators
for the harmonic oscillator
In this talk
we consider the bound states of (‘discrete’) quantum
mechanical systems of one degree of freedom.
intro1
Let us recall the harmonic oscillator.
annihilation-creation operator
eigenfunctions
Schrödinger picture
intro2-1
classical theory

 time evolution
harmonic coordinate
quantum theory

 Heisenberg operator
negative freq. part 
intro2-2
creation
Heisenberg picture
annihilation
 positive freq. part
solvable system
the entire energy spectrum
and the corresponding
eigenvectors
are known explicitly.
Schrödinger picture
annihilation and creation operators
creation op.
annihilation op.
existence of ann./cre. op.  solvable system

There are many solvable systems in (‘discrete’)
quantum mechanics (of single degree of freedom).
Examples: QM
dQM
Almost all examples have the shape invariant property.
intro3
next
Examples of solvable systems
quantum mechanics
: (second order) differential operator
number of states
infinite
finite
orthogonal
polynomial
infinite
finite
orthogonal polynomial : Hermite (
All of these systems are shape invariant .
Intro3-1
), Laguerre (
), Jacobi (
)
back
Examples of solvable systems
‘discrete’ quantum mechanics
: (second order analytic) difference operator
infinite number of states
orthogonal polynomial
Meixner-Pollaczek polynomial
continuous Hahn polynomial
 Hermite
continuous dual Hahn polynomial
 Laguerre
Wilson polynomial
Askey-Wilson polynomial
 Jacobi
Askey-scheme of hypergeometric
orthogonal polynomials
These systems are single-particle cases of
the Ruijsenaars-Schneider-van Diejen (RSvD) systems.
All of these systems are shape invariant .
Intro3-2
back
Question
prev
Do these solvable quantum mechanical systems have the
annihilation-creation operators?
We want to find the annihilation-creation operators such that
· as differential (or difference) operators
· as n-independent expression (n : n of n-th eigenstate )
Answer
For most of the solvable (‘discrete’) quantum mechanical
systems, we can construct them.
Our method
The annihilation-creation operators are defined as
the positive/negative frequency parts of
the exact Heisenberg operator solution
for the ‘sinusoidal coordinate’.
intro4
§2. General Theory
Question
Do these solvable quantum mechanical systems have the
annihilation-creation operators?
· as differential (or difference) operators
· as n-independent expression (n : n of n-th eigenstate
)
Answer
polynomial of degree n in a real variable
closure relation
 we can construct the desired annihilation-creation
operators.
Remark
gen1
For the usual quantum mechanics, the closure relation implies the first condition.
The closure relation holds for the examples in §1 satisfying the first condition.
classical level
Poisson bracket
closure relation


is easily calculated.
is obtained. sinusoidal time-evolution
: ‘sinusoidal coordinate’
gen2
quantum level
closure relation
commutation relation

is easily calculated.
 Heisenberg op.
is obtained.
two “frequencies”
gen3
Heisenberg picture
This Heisenberg op. contains all the dynamical information.
Therefore we have
From our assumption
, we obtain
 from ‘hermitian conjugate’
These overdetermined conditions and
determine the
entire energy spectrum
completely for each Hamiltonian.
gen4
three term recursion relation
measure
orthogonal
orthogonal polynomial
Orthogonal polynomials of single variable satisfy
the three term recursion relation:
gen5
Dynamical and unified definition of
the annihilation-creation operators
Annihilation/creation operator (
) is
the positive/negative frequency part
of the Heisenberg operator of the ‘sinusoidal coordinate’.
action on the eigenvector
consistency
For a system with finite number of states (
gen6
states)
minimal requirement for annihilation-creation operators
It should be stressed that there is no a priori principle for fixing the
normalization of the operators. Sometimes it is convenient to introduce the
annihilation-creation operators with a different normalization.
We can show that our annihilation-creation operators are hermitian
conjugate to each other:
Remark :
gen7
, in general
§3. Examples
§3.1. quantum mechanics
Ex.0: harmonic oscillator
Ex.1: x2+1/x2 potential
Ex.2: 1/sin2x potential
see
see
see
Ex.3: Pöschl-Teller potential
Ex.4: soliton potential
see
Ex.5: Morse potential
see
Hermite polynomial
Laguerre polynomial
Jacobi polynomial
see Jacobi polynomial
Jacobi polynomial
Laguerre polynomial
Solution for the closure relation
go
next
QM
Ex.0: harmonic oscillator
classical level closure relation

quantum level closure relation

QM0-1
energy spectrum

annihilation-creation operators
algebra
coherent state
AOCS for
:
back
QM0-2
Ex.1: x2+1/x2 potential
(A1 Calogero system,
harmonic oscillator with the centrifugal barrier)
classical level closure relation

quantum level closure relation

QM1-1
energy spectrum

annihilation-creation operators
algebra
coherent state
AOCS for
back
QM1-2
1/sin2x
Ex.2:
potential
(A1 Sutherland potential,
symmetric Pöschl-Teller potential)
classical level closure relation

quantum level closure relation

QM2-1
energy spectrum

annihilation-creation operators
similarity transformation in term of the ground state wavefunction

QM2-2
algebra
coherent state
AOCS for
AOCS for
back
QM2-3
Ex.3: Pöschl-Teller potential
classical level closure relation

quantum level closure relation

QM3-1
energy spectrum

annihilation-creation operators
coherent state
AOCS for
:
AOCS for
:
back
QM3-2
Ex.4: soliton potential
(symmetric Rosen-Morse potential)
classical level closure relation

quantum level closure relation

QM4-1
energy spectrum

annihilation-creation operators
algebra
back
QM4-2
Ex.5: Morse potential
classical level closure relation

quantum level closure relation

QM5-1
energy spectrum

annihilation-creation operators
back
QM5-2
Solution for the closure relation
Hamiltonian
quantum level closure relation


‘sinusoidal coordinate’ :
 trigonometirc, hyperbolic,
exponential, quadratic, linear in x

the obtained Hamiltonian is a member of
the known solvable (shape invariant) systems. back
QMsol
§3.2. discrete QM
prev
(deformed harmonic oscillator)
Ex.1: Meixner-Pollaczek polynomial
Ex.2: continuous Hahn polynomial
see
see
Ex.3: continuous dual Hahn polynomial
Ex.4: Wilson polynomial
see
see
Ex.5: Askey-Wilson polynomial
see
next
dQM
Ex.1: Meixner-Pollaczek polynomial
classical level closure relation

quantum level closure relation

dQM1-1
(deformed harmonic oscillator)
energy spectrum

annihilation-creation operators
Degasperis-Ruijsenaars,
Ann.of Phys. 293(2001)92-109
algebra
coherent state
AOCS for
back
dQM1-2
Ex.2: continuous Hahn polynomial
classical level closure relation

quantum level closure relation

dQM2-1
energy spectrum

annihilation-creation operators
coherent state
AOCS for
:
AOCS for
:
back
dQM2-2
Ex.3: continuous dual Hahn polynomial
classical level closure relation

dQM3-1
quantum level closure relation

energy spectrum

annihilation-creation operators
coherent state
AOCS for
dQM3-2
:
back
Ex.4: Wilson polynomial
classical level closure relation

dQM4-1
quantum level closure relation

energy spectrum

dQM4-2
annihilation-creation operators
coherent state
AOCS for
:
AOCS for
:
back
dQM4-3
Ex.5: Askey-Wilson polynomial
classical level closure relation
dQM5-1
quantum level closure relation

energy spectrum

dQM5-2
annihilation-creation operators
coherent state
AOCS for
:
back
dQM5-3
§4. Summary
prev
We have shown that most solvable quantum mechanics of one
degree of freedom have exact Heisenberg operator solution.
The annihilation-creation operators (a(–)/a(+) ) are defined as
the positive/negative frequency parts of the exact Heisenberg
operator solution for the ‘sinusoidal coordinate’.
These (a(±)) are hermitian conjugate to each other.
comments
·
: raising/lowering op. of the polynomial
c.f. Koornwinder, math.CA/0601303
· coherent state go
· shape invariant system go
future problem
· classification of the ‘sinusoidal coordinate’ for discrete QM
· generalization to multi-particle systems
· application of our annihilation-creation operators to physical
systems
summary
END
comment 1
coherent state
There exist several definitions for coherent states.
One definition of the coherent state is the eigenvector of the annihilation
operator (AOCS, Annihilation Operator Coherent State):
Let


temporal stability
It should be remarked that the concrete form of the AOCS depends
on the specific normalization of the annihilation operator.
Which coherent state is useful depends on the physics of the system. back
Comment1
comment 2
shape invariant system
factorization
shape invariance
construction of isospectral Hamiltonians
formal definition of the annihilation-creation
operators used within the framework of
shape-invariant quantum mechanics:
annihilation-creation operators
The operator
is rather formal and it cannot be
expressed as a differential or a difference operator.
For the systems with equi-spaced spectrum,
we can construct parameter shift operators X
as a differential or a difference operator.
Comment2
back