Exponential Type Complex and non-Hermitian Potentials within Quantum Hamilton-Jacobi Formalism

0.1 MB·English

by  Ozlem Yesiltas

Exponential Type Complex and non-Hermitian Potentials
within Quantum Hamilton-Jacobi Formalism
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0 O¨zlem Yes¸iltas¸ 1 and Ramazan Sever 2 ,
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2

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2 1TurkishAtomicEnergyAuthority,NuclearFusionandPlasmaPhysicsLaboratory,06983,Ankara,Turkey
2

2DepartmentofPhysics,MiddleEastTechnicalUniversity,06531,Ankara,Turkey
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February 1, 2008
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h Abstract
p
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t PT-/non-PT-symmetricandnon-HermitiandeformedMorseandPo¨schl-Tellerpotentialsarestud-
n
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u ied firsttimeby quantum Hamilton-Jacobi approach. Energy eigenvalues and eigenfunctions are
q
: obtained bysolving quantum Hamilton-Jacobi equation.
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r PACSRef:03.65.Db,03.65.Ge
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Keywords:PT-symmetry;QuantumHamiltonJacobi,Morse,Po¨schl-Teller
∗Correspondingauthor:[email protected]
1
1 Introduction
Recently, theso called PT-symmetricquantummechanics has attracted wideattention[1]. In Bender
and Boettcher’s work in 1998 [1], it was shown that the class of non-Hermitian, Hamilton operators
such as H = p2 + x2(ix)ǫ(ǫ > 0) has a real spectrum due to its PT-symmetry where P and T are
the parity and time reversal operators respectively [2]. Exact solution of the Schro¨dinger equation
forvariouspotentialswhich arecomplexare generallyofinterest. ItisalsoknownthatPT-symmetry
does not necessarily lead to completely real spectrum, and an extensive kind of potentials of real
or complex form are being faced with in various fields of physics. In particular, the spectrum of
the Hamiltonian is real if PT-symmetry is not spontaneously broken. Recently, Mostafazadeh has
generalized PT symmetry by pseudo-Hermiticity [3]. In fact, a Hamiltonian of this type is said to
be η- pseudo Hermitian if H+ = ηHη 1, where + denotes the operator of adjoint. In [4] new class
−
of non-Hermitian Hamiltonians with real spectra was proposed which are obtained using pseudo-
symmetry. Moreover, completeness and orthonormality conditions for eigenstates of such potentials
are proposed [5]. In the study of PT-invariant potentials various techniques from a great variety of
quantum mechanical fields have been applied such as variational methods, numerical approaches,
Fourieranalysis,semi-classicalestimates,quantumfieldtheoryandLiegrouptheoreticalapproaches
[5-14]. In addition, PT-symmetric and non-PT symmetric and also non-Hermitian potential cases
such as oscillator type potentials [15], a variety of potentials within the framework of SUSYQM
[16-19], exponential type screened potentials [20], quasi/conditionally exactly solvable ones [21],
PT-symmetric and non-PT symmetric and also non-Hermitian potential cases within the framework
ofSUSYQM viaHamiltonianHierarchyMethod[22]and someothersare studied[23-25].
The QHJ formalism, which is a formulation of quantum mechanics was investigated as a theory re-
lated to the classical transformation theory [26-27]. It was formulated by Leacock and Padgett in
1983[28-29]. WithintheQuantumHamiltonianJacobiapproach (QHJ), whichfollowsclassicalme-
chanics, not only the the energy spectrum of exactly solvable(ES) and quasi-exactly solvable(QES)
models in quantum mechanics but eigenfunctions can also be determined [30-36]. The advantage
of this method is that it is possible to determine the energy eigenvalues without having to solve
for the eigenfunctions. In this formalism, singularity structure of the quantum momentum function
2
p(x) which is a quantum analog of classical momentum function p determines the eigenvalues of
c
the Hamiltonian. An exact quantization condition is formulated as a contour integral, representing
the quantum action variable, in the complex plane. The quantization condition leads to the number
nodesofthewavefunction. Thewavefunctionisrelatedtothequantummomentumfunction(QMF).
TheequationsatisfiedbytheQMFis anon-lineardifferentialequation,called as quantumHamilton-
Jacobiequation. ThereisaboundaryconditioninthelimitQMFwhichisusedtodeterminephysically
acceptable solutionsfor the QMF [29-37]. In the applications, Ranjani and her collaborators applied
the QHJ formalism, to Hamiltonianswith Khare-Mandal potential and Scarf potential, characterized
by discreteparity andtimereversal (PT)symmetries[31].
The purpose of the present work has been to apply the QHJ formalism to the Hamiltonian in one
dimensionwithnon-hermitianexponentialtypepotentialsinordertoseepossiblesingularitiesforthe
QMFwhichdetermineeigenvaluesand convenienteigenfunctions.
The organization of the paper is as follows. In Sec. II, we briefly introduce the Quantum Hamilton-
Jacobi formalism. In Sec. III and IV,solutionsofPT-/non-PT-symmetricandnon-Hermitianforms of
thewell-knownpotentialsare presented byusingQHJmethod. WediscusstheresultsinSec. V.
2 Quantum Hamilton-Jacobi Formalism
In quantumtheory,oneassumesthatfunctionW(x,E) satisfies(2m = 1)[33],
∂2W(x,E) ∂W(x,E) 2
ih¯ + = (E V(x)) (1)
− ∂x2 " ∂x # −
Eq. (1) willbecalled as theQHJequation. Themomentumfunction
∂W(x,E)
p(x,E) = (2)
∂x
will be called as the QMF. In the limit h¯ 0, the QHJ equation turns into the classical Hamilton-
→
Jacobiequation. Then, QMFturnsintotheclassical momentumfunctionintheh¯ 0limit:
→
p(x,E) p (x,E) = E V(x) (3)
c
→ −
q
3
In termsofp(x,E) theQHJequation,Eq.(1)can bewrittenas
p2(x,E) ih¯p′(x,E) [E V(x)] = 0. (4)
− − −
LeacockandPadgett[28,29]proposedusingthefollowingquantizationconditionfortheboundstates
in order to obtain eigenvalues. C is a contour that encloses the moving poles between the classical
turningpointsand theintegral
1
J(E) = p(x) dx (5)
2π
IC
iscalled thequantumaction variable. Moredetailscan befoundin thepaperofBhallaet all[30-37].
Then
J = nh¯ = J(E) (6)
givestheexactenergyeigenvalues(n = 0,1,2,...)[28-33]. Leacock[28,29]definesthewavefunction
inorder toconnect QHJequationto theScho¨dingerequation,
i
ψ(x,E) exp W(x,E) (7)
≡ h¯
(cid:20) (cid:21)
henceψ(x,E)satisfiestheScho¨dingerequationandthephysicalboundaryconditions. Thequantiza-
tionconditionbecomes [33]
p(x,E)dx = 2πi (Res) = nh (8)
k
IC k
X
where (Res) is sum of the residues. In the QHJ equation, if V(x) has a singular point, p(x,E)
k k
will alsPo have singularpoint in that zone [33]. These singularitiesare known as fixed singularpoints
which are energy independent. Other types of singular points are the moving singular points. They
can only be poles with residue ih¯. Suppose b = 0 then, moving singularities are in the form of
− 6
[30-37]
4
b
p(x,E) +... (9)
∼ (x x )r
0
−
intheQHJequation. Ifthepotentialisnotsingularatx = x thenrmustbeequaltooneandb = ih¯
0
−
[33].
3 Generalized Morse Potential
Thegeneralized Morsepotentialisgivenby[19]
V(x) = V e 2αx V e αx (10)
1 − 2 −
−
In ordertoapply QHJmethod,wewritethepotentialrelationin Eq.(4)(h¯ = 2m = 1)
p2 ip′ [E V e 2αx +V e αx] = 0 (11)
1 − 2 −
− − −
Substitutionofthetransformationofy = √V e αx in Eq.(11)gives:
1 −
V
p2(y,E)+iαyp′(y,E) E y2 + 2 y = 0 (12)
−" − √V1 #
Definep = iαyφandχ = φ+ 1 inordertotransformEq.(11)intoaRiccatitypedifferentialequation
2y
as,
1 1 V
χ′ +χ2 + + E y2+ 2 y = 0 (13)
4y2 α2y2 " − √V1 #
As itcan beseen from Eq. (13), χhas apoleonlyat y = 0 and fory = 0 defineχas,
b
χ = 1 +a +a y (14)
0 1
y
SubstituteEq.(14)in(13)and equatecoefficients of 1 yields
y2
5
1
b = (α 2√ E) (15)
1
2α ± −
When itcomes tothediscussionofthebehaviourofχat infinity,oneexpandsχas:
λ λ
χ = a + + 1 (16)
0 y y2
and find λ as
V
λ = 2 (17)
±2α√V
1
Onecan seethat thebehaviourofχis b1+n forlargey. Hence,
y
b +n = λ (18)
1
In order to find the wavefunction, χ(y) can be written as the sum of the Laurent expansions around
differentsingularpoints,plusaconstantC . Hence
1
′
b P (y)
χ(y) = 1 + n +C (19)
1
y P (y)
n
whereP (y)isa nthdegreepolynomial. SubstituteEq. (19)in(13)and get
n
P′′ 2P′ 2b b2 b +E/α2 +1/4 2b C 1 V
n + n 1 +2C + 1 − 1 + 1 1 +C2 + 2 = 0. (20)
Pn Pn y 1! y2 y 1 − α2 α2y√V1!
For large y onecan find C = 1. The wavefunction in terms ofχ can be written by usingeq. (19)
1 ±α
and (7)as
′
b P 1 1
ψ(y) = exp 1 + n dy . (21)
Z y Pn − α − 2y! !
6
InEq. (21)thecorrect valueofC isusedasC = 1 becauseoftheconditionforthewavefunction
1 1 −α
whichisknownasy ,ψ(y) 0. ItisseenfromEq. (15)thatb hastwovaluesandnoparticular
1
→ ∞ →
valuehas been chosen. UsingEq. (18), theenergy eigenvaluesforanyn-th statebecome,
α2 V 2
E = (2n+1)+ 2 (22)
n
− 4 "− α√V1#
IfweuseEq.(22),(15)and C = 1/αin Eq.(20),itbecomes
1
−
V
yP′′(y)+( 2 +1 y)P′(y)+nP (y) = 0 (23)
n 2α√V − n n
1
whichis aLaguerredifferentialequation. Therefore, thewavefunctionis obtainedas
ψn(y) = Cn e−αy y(−n±2αV√2V1) Ln2αV√2V1(y). (24)
V2
whereC isanormalizationconstantandL2α√V1(y)areLaguerrefunctions. Thewavefunctionsatis-
n n
fies theboundaryconditionthat isy , ψ(y) 0.
→ ∞ →
3.1 Non-PT symmetric and non-Hermitian Morse Potential
In equation(10), ifthepotentialparameters are defined as V = (A+iB)2, V = (2C +1)(A+iB)
1 2
and α = 1,then thepotentialbecomes[19],
V(x) = (A+iB)2e 2x (2C +1)(A+iB)e x (25)
− −
−
whereA, B and C arearbitrary real parameters andi = √ 1. TheQHJequationis
−
p2 ip′ [E (A+iB)2e 2x +(2C +1)(A+iB)e x] = 0 (26)
− −
− − −
Usingthetransformationintheform ofy = (A+iB)e x in theEq. (26),then usingp(y) = iyφand
−
χ = φ+ 1 , Eq. (26)becomes
2y
7
1 1
χ′ +χ2 + + E y2 +(2C +1)y = 0 (27)
4y2 y2 −
h i
As itisseen from Eq. (27), χhas apoleonlyat y = 0and fory = 0 define χfortheEq.(27)as,
b
χ = 1 +a +a y (28)
0 1
y
UsingtheEq. (28)in(27), b isfound as
1
1 4E
b = i | | (29)
1
2 ± s 2
Followingthesameprocedurethatis giveninsection 3,energy is foundas
E = (n C)2. (30)
n
− −
Now choose the potential parameters in Eq.(10) as V is real and V = A+iB, the Morse potential
1 2
can bewrittenin thefollowingform
V(x) = V e 2iαx (A+iB)e iαx (31)
1 − −
−
and followingthesameprocedure, theenergy isobtainedas
2
A+iB
E = α4 (n+1/2) (32)
n
 − 2α V 
1
|− |
 q 
Accordingto Eq. (32), thespectrumisreal in thecaseofIm(V ) = 0.
2
3.2 PT symmetric and non-Hermitian Morse Potential
When α = iα andV ,V arereal, theMorsepotentialbecomes
1 2
8
V(x) = V e 2iαx V e iαx (33)
1 − 2 −
−
Thus,theenergy eigenvaluesareobtainedas
2
1 V
E = α4 (n+ )+ 2 (34)
n
 2 2α V 
1
|− |
 q 
IfwetaketheparametersofEq.(39)asV = ω2,V = Dandα = 2then,correspondingeigenvalues
1 2
−
foranyn-th stateare obtainedas
D
E = (2n+1+ )2 (35)
n
2ω
are consistentwiththeresults[10-11,19].
4 Po¨schl-Teller Potential
Thegeneral form ofthePo¨schl-Tellerpotentialis
e 2αx
−
V(x) = 4V (36)
− 0(1+qe 2αx)2
−
TheQHJequationis givenas
e 2αx
p2 ip′ E +4V − = 0 (37)
− − 0(1+qe−2αx)2!
If we take y = i√qe αx and use the transformations as p = iαyφ and χ = φ + 1 , the QHJ
± − − 2y
equationturnsinto
1 1 4V y2
χ′ +χ2 + + E 0 = 0. (38)
4y2 α2y2 " − q (1 y2)2#
−
9
As itisseen from Eq.(38),χhas polesat y = 0 and 1. χisexpandedat y = 0and b isfound
1
±
1
b = (α 2√ E) (39)
1
2α ± −
At y = 1,onecan expand χas
′
b
χ = 1 +a′ +a′(1 y) (40)
1 y 0 1 −
−
′
and b isfoundas
1
1
′
b = (qα α2q2 +8qV ) (41)
1 2qα ± 0
q
At y = 1, onecan expandχas
−
′′
b
χ = 1 +a′′ +a′′(1+y) (42)
1+y 0 1
′ ′′
Substitute Eq. (42) in (38), to obtain b = b . One can look at the behavior of χ at infinity with
1 1
expandingχas
λ λ
χ = A + + (43)
0 y y2
From Eq. (43)and (38), λ isfoundas
1
λ = (α 2√ E) (44)
2α ± −
′ ′′
and behaviorofχis b1+b1+b1+2n forlarge y. Hence,
y
′ ′′
λ = b +b +b +2n (45)
1 1 1
In ordertofind thewavefunctions,χis writtenas
10