Non-Hermitian Hamiltonian versus E=0 localized states

0.15 MB·

by  S. Habib Mazharimousavi

Non-Hermitian Hamiltonian versus E = 0
localized states
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S. Habib Mazharimousavi
2
Department of Physics, Eastern Mediterranean University,
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a G Magusa, north Cyprus, Mersin 10,Turkey
J
E-mail: [email protected]

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February 2, 2008

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h
p
Abstract
-
t
n We analyze the zero energy solutions, of a two dimensional system
a which undergoes a non-radial symmetric, complex potential V (r,φ). By
u
virtueofthecoherentstatesconcept,thelocalizedstatesareconstructed,
q
and the consequences of the imaginary part of the potential are found
[
both analytically and schematically.

1
v
1 Introduction
9
4
5 Sincetheearlyyearsofquantummechanicstheexactsolvabilityofquantumme-
1
chanical models have attracted much attention. Some exactly solvable models
.
1 have already become typical standard examples in quantum mechanical text-
0
books. However, it was believed that the reality of the spectra of the Hamil-
8
tonians, describing quantum mechanical models, is necessarily attributed to
0
: their Hermiticity. It was the non-Hermitian PT-symmetric Hamiltonians pro-
v
posed by Bender and Boettcher [1] that relaxed the Hermiticity condition as
i
X a necessity for the reality of the spectrum [1–7]. Herein, P denotes the par-
r ity (PxP = x) and the anti-linear operator T mimics the time reflection
a (TiT = i). R−ecently, Mostafazadeh [8] has introduced a broader class of non-
−
Hermitian pseudo-Hermitian Hamiltonians (a generalization of PT-symmetric,
therefore). In these settings [8–19], a Hamiltonian H is pseudo-Hermitian if
it obeys the similarity transformation: ηHη 1 = H where η is a Hermitian
− †
invertible linear operator. On the other hand, the study of the E = 0 bound
stateshavefound manyapplicationsin variousfields [20-24]. Longago,Daboul
andNieto[25,26]havediscoveredthat,anattractiveradialpowerlowpotential,
V(r) r ν for ν < 2 and ν > 2, passes through the E = 0 normalizable
−
∼ −
solutions. More recently Makowski and G´orska, established the classical cor-
respondence localized states of a system with zero energy and a general form
1
of power low potentials [27]. In their work, it was shown that the classical
trajectoriesoftheparticlepreciselymatchedwiththelocalizedquantumstates.
Inthis workwe attemptto understandthe consequencesofadding animag-
inarytermtoapotentialwhosezeroenergylevelpassesthroughthe E =0nor-
malizable bound state solution (namely V(r) = Γr 4). We believe that this
−
−
kind of study is necessary, to find relations, if any between the non-Hermitian
quantum mechanics and the classical mechanics.
Theorganizationofourpaperisasfollows: InSec. (2)wegiveananalytical
solution to the Schr¨odinger equation of a zero energy particle under our cho-
sen complex potential. We continue in Sec. (3) by adapting a closed form of
the localized states from literature and then we set up and plot the classically
equivalent coherent states of the system. We conclude our paper with Sec. (4).
2 Analytic solution of the Schr¨odinger equation
Two dimensional Schr¨odinger equation for a zero energy particle under a com-
plex, non-radial symmetric potential is given by
ℏ2 1 ∂ ∂ 1 ∂2
r + +V (r,φ) ψ(r,φ)=0 (1)
−2m r∂r ∂r r2∂φ2
(cid:20) (cid:18) (cid:18) (cid:19) (cid:19) (cid:21)
where
Γ Λ
V (r,φ)= eiφ (2)
−r4 − r2
and for our purpose, Γ and Λ are some non-negative real constants. Before we
gofurther,findingoutthesymmetricpropertiesofV (r,φ)andconsequentlythe
Hamiltonian of the system may give some connections between this potential
and the well known -symmetric type potentials which are studied in the
PT
literature.
Let us introduce an operator Θ which is defined as
Θ:i i,φ 2π φ (3)
→− → −
inwhichi=√ 1andφisthe usualazimuthalangle. Onecaneasilyshowthat
−
Θ is a non-Hermitian, invertible operator whose inverse is given by
Θ 1 =Θ (4)
−
and it can be decomposed into the two other operators and Π such that
T
Θ=Π (5)
T
in which the definition of these operators are given by
Π : φ 2π φ (6)
→ −
: i i.
T →−
2
Π is Hermitian and invertible such that Π 1 =Π=Π , and is the usual
− †
T
time reversaloperator. It is remarkable to observe that the Hamiltonian of the
particle under the potential (2) is Π -symmetric which means
T
H =HΠ . (7)
T
Also it is said that the Π -symmetry of a Hamiltonian H is unbroken if all
T
of the eigenfunctions of H are simultaneously eigenfunctions of Π . It is easy
T
to show that if the Π -symmetry of a Hamiltonian H is unbroken, then the
T
spectrum of H is real (to see the proof, one may see [30]).
We come back to the equation (1) and as usual, we take
ψ(r,φ)=R(r)Φ(φ) (8)
and after making substitution, in Eq. (1) we get a set of two equations as
1 ∂ ∂ l2 2mΓ
r + R(r)=0 (9)
r∂r ∂r − r2 ℏ2r4
(cid:20) (cid:18) (cid:19) (cid:21)
and
∂2Φ(φ) 2mΛ
+ eiφΦ(φ)= l2Φ(φ) (10)
∂φ2 ℏ2 −
where l is a constant to be identified.
By introducing a new dimensionless variable ρ as
r
ρ= (11)
a
◦
where a is a positive constant, the Eqs. (9) and (10) become
◦
1 d d l2 γ2
ρ + R(r)=0 (12)
ρdρ dρ − ρ2 ρ4
(cid:20) (cid:18) (cid:19) (cid:21)
and
d2Φ(φ)
+λ2eiφΦ(φ)= l2Φ(φ). (13)
dφ2 −
in which
2mΓ
γ2 = (14)
ℏ2a2
◦
and
2mΛ
λ2 = . (15)
ℏ2
In the angularpartofthe Schr¨odingerequationwechange the variable,and
introduce
χ=eiφ (16)
3
hence the Eq. (13) reads
d2Φ(χ) dΦ(χ)
χ2 +χ l2+λ2χ Φ(χ)=0. (17)
dχ2 dχ −
(cid:0) (cid:1)
This is the modified Bessel ODE, such that its complete solution is well known
as
Φ(χ) = C I (2λ√χ)+C K (2λ√χ)
1 2l 2 2l
or (18)
Φ(φ) = C I 2λeiφ/2 +C K 2λeiφ/2
1 2l 2 2l
(cid:16) (cid:17) (cid:16) (cid:17)
in which I (z) and K (z) are the modified Bessel functions.
ν ν
This solution should satisfy the following boundary condition
Φ(φ)=Φ(φ+2π) (19)
or equivalently
C I 2λeiφ/2 +C K 2λeiφ/2 = (20)
1 2l 2 2l
(cid:16) (cid:17) (cid:16) (cid:17)
C I 2λei(φ+2π)/2 +C K 2λei(φ+2π)/2 .
1 2l 2 2l
(cid:16) (cid:17) (cid:16) (cid:17)
Since I (z)andK (z)aretwoindependentsolutionsofthe modifiedBessel
ν ν
ODE , a class of solution is possible when we put C =0. Therefore we get
2
I 2λeiφ/2 =I 2λei(φ+2π)/2 . (21)
2l 2l
(cid:16) (cid:17) (cid:16) (cid:17)
In accordance with the following property of the modified Bessel functions
[28]
I zemπi =emπνiI (z) (22)
ν ν
where z is a complex variable,(cid:0)m is a(cid:1)n integer and ν is a real number, one can
choose
z =2λeiφ/2 (23)
and m=1 in Eq. (22) to get
I zeiπ =ei2lπI (z). (24)
2l 2l
This equality is valid for all z (cid:0)and l(cid:1)in their domains, but if one considers l to
be an integer, we will get
I zeiπ =I (z) (25)
2l 2l
which is equivalent to Eq. (22). T(cid:0)heref(cid:1)ore l is found to be an integer i.e.
l=0, 1, 2,... (26)
± ±
4
Of course as a different possibility, one can chooseC =0 to find a different
1
class of solution but the following property [28]
K zeimπ =e mπνiK (z) πisinmνπcscνπI (z) (27)
ν − ν ν
−
it comes to our s(cid:0)etting(cid:1)as
K2l zeiπ =e−π2liK2l(z) πisin2lπcscνπI2l(z) (28)
−
which obviously do(cid:0)es no(cid:1)t admit any solution.
As a result, the solutions of the angular part of the Schr¨odinger equation
can be written explicitly as
Φ (φ) = C I 2λeiφ/2 (29)
l lλ 2l
l = 0,1,2,(cid:16)... (cid:17)
where,sinceI (z)=I (z),wejustconsiderl tobe nonnegative,andC are
n n lλ
−
the normalization constants given by
1 1
C = = . (30)
lλ s 02π|Φl(φ)|2dφ s 02π I2l 2λeiφ/2 2dφ
R R (cid:12) (cid:0) (cid:1)(cid:12)
(cid:12) (cid:12)
Onemaynotethatlstillcanbeinterpretedastheangularquantumnumber,
since
∂
< Lˆ >=<Φ iℏ Φ >= (31)
l l
Φl |− ∂φ|
∂
< C I 2λeiφ/2 iℏ C I 2λeiφ/2 >=
lλ 2l lλ 2l
|− ∂φ|
(cid:16) (cid:17) (cid:16) (cid:17)
iℏ C 2 < I 2λeiφ/2 i λeiφ/2I 2λeiφ/2 +lI 2λeiφ/2 >=lℏ.
lλ 2l 2l+1 2l
− | | |
(cid:16) (cid:17) h (cid:16) (cid:17) (cid:16) (cid:17)i
The radialpartofthe Schr¨odingerequation,canbe consideredasthe Bessel
ODE if one defines
1
ξ = (32)
ρ
and therefore the Eq. (12) becomes
d2 d
ξ2 R(ξ)+ξ R(ξ)+ γ2ξ2 l2 R(ξ)=0 (33)
dξ2 dξ −
(cid:0) (cid:1)
which admits a complete solution
R (ξ)=C J (γξ)+C Y (γξ). (34)
lγ 1 l 2 l
A physical, normalizable solution which for l >1 corresponds to the bound
state is given by [25,26]
R (ξ) = N J (γξ) (35)
lγ lγ l
l = 2,3,...
5
in which N are normalization constants given by
lγ
2 (l+1)!
N = . (36)
lγ
a γs(l 2)!
◦ −
NowwearereadytowritethecompletesolutionoftheSchr¨odingerequation
(i.e. wave function), by using the Eqs. (29) and (35) as
2 (l+1)! γa
ψl,γ,λ(r,φ)=Clλa γs(l 2)!I2l 2λeiφ/2 Jl r◦ . (37)
◦ − (cid:16) (cid:17) (cid:16) (cid:17)
We notice that, with l > 1, the only complex part of ψ (r,φ) is the
l,γ,λ
modifiedBesselfunction. As amatteroffact,the effectofΘ(introducedinEq.
(3)) on ψ (r,φ) is equivalent to the effect of Θ on I 2λeiφ/2 . Therefore,
l,γ,λ 2l
by using the expansion form of the modified Bessel function, this equation can
(cid:0) (cid:1)
be written as
2 (l+1)! γa ∞ λ2(s+l)
ψl,γ,λ(r,φ)=Clλa γs(l 2)!Jl r◦ Γ(s+2l+1)s!eiφ(s+l) (38)
◦ − (cid:16) (cid:17)Xs=0
in which,clearly this is invariant under Θ (i.e., Θ ψ (r,φ)=ψ (r,φ)).
l,γ,λ l,γ,λ
2.1 A realized approach to the problem
In this section we will consider the following radial symmetric real potentials
Γ Λ
V (r)= (39)
± −r4 ± r2
where Γ and Λ are some positive constants as before. One should notice that
thenegativebranchofthe abovepotentialissameasthepotentialinEq. (2)in
anattractiveform,andthepositivebranchofitissamebutinarepulsiveform.
The Schr¨odingerequation (1), after the usual separationmethod and changeof
variable, with the potential (38) comes to a set of two separated equations as
1 d d ˜l2 γ2
ρ + R(ρ)=0 (40)
"ρdρ(cid:18) dρ(cid:19)− ρ2 ρ4#
and
d2Φ(φ)
= l2Φ(φ) (41)
dφ2 −
where
˜l2 =l2 λ2
±
in which the positive (negative) sign is related to the +Λ ( Λ),and the other
r2 −r2
factorsaredefined asbefore. One caneasilyshow that, the finalsolutionof the
6
Schr¨odinger equation with the potentials (38) can be written as
˜l+1 !
1 2 1
ψ (r,φ)= eilφJ (42)
l,γ,λ √2πa◦γvuu(cid:16)˜l−2(cid:17)! ˜l(cid:18)r(cid:19)
u
t(cid:16) (cid:17)
in which ˜l must be greater then 1 to have bound states. In the sequel we will
use the closed forms of the infinite number of degenerate wave functions (i.e.,
these states have same energy equal to zero), presented in the Eq.s (37) and
(41) to construct the classical correspondence localized states.
3 Zero energy localized states
Inreferences[25-27]itwasshownthatthetrajectoryofaclassicalparticlewhich
experiencesarealpotentialintheformoftheΓ partofthepotentialconsidered
−
in this work (2) (i.e. Γ) is given by
−r4
x= a(1+cos(φ φ ))
y = a2sin(φ φ )− ◦ (43)
(cid:26) 2 − ◦
which represents a circle with radius a/2 (i.e., if one set φ = 0, this becomes
(x a/2)2+y2 =(a/2)2) such that ◦
−
a= 2mΓ/L2 (44)
p
and L is the conserved angular momentum of the particle. We notice that, the
classical correspondence localized state (see [27] and the references therein) of
the potential (2) while Λ 0 must have probability peak in accordance with
→
the classicaltrajectory(i.e., a circle with radius a/2asimplied by Eq. (42)[27]
). OuraiminthefollowingistoseetheeffectoftheΛ partofthepotential(2)
−
on the shape of the classical correspondence localized states. To this end, first
we find the localized states of the original potential (2) in a closed analytical
form,thenwefollowsimilarlybutforthecasewhenthepotentialisintheforms
of Eq. (38).
Anavailablemethodtoderivethecorrespondinglocalizedstatesbyusingthe
solutionsofthe Schr¨odingerequationgiveninthe previoussections,isbasedon
theconceptofdeformedoscillatoralgebras[27,29]. Thereforeasuitableexplicit
form of the localized state over the infinite number of degenerate states ( with
E =0) ψ (r,φ) reads (see [27,29] and the references therein)
lγλ
1 N N 1/2
Ψ = τkψ (r,φ) (45)
N N/2 k k,γ,λ
√2π 1+ τ 2 k=0(cid:18) (cid:19)
| | X
(cid:16) (cid:17)
in which k =l 2 and τ =Aeiθ◦, where A and θ are some real constants.
− ◦
7
3.1 Results
The classicalcorrespondence localized states of a particle undergoes the poten-
tials (2) and (38), by choosing N =7 may be written as
8√1π 7 C(2+k)λ (k7)(kk!+3)!I(4+2k) 2λeiφ2 J(2+k) r1 V =Vc
Ψ7 = 161πkk=P7=00r(k7()˜l(−˜l+2)1q!)!eilφJ˜l r1 (cid:16) (cid:17) (cid:0) (cid:1) V =V± .
 P (cid:0) (cid:1) (46)
inwhichγa andAaresettobeoneandforconvenienceV andV refertothe
c
◦ ±
potentials (2) and (38) respectively. Within the Fig.s (1) to (7) some density
plot of Ψ 2 are given in terms of different values of λ.In the Fig. (1) we plot
7
| |
Ψ 2 with λ = 0 (i.e., V = V = Γ) and the classical trajectory of the
| 7| c ± −r4
particle ( this figure was reported in the Ref.[27]). In the Fig.s (2) to (4), (a),
(b) and (c) refer to the potentials V , V and V , respectively. In the Fig.s (5)
+ c
−
and (6), (a) refers to V and (b) refers to V . Finally Fig. (7) refers to V .
+ c c
3.2 Behavior of the complexified modified Bessel func-
tions
In this section we give a short description of the behavior of the complexified
modified Bessel functions I (z) to explain why with V = V the plot of the
2l c
probability density with large values of λ are very localized around φ=0. It is
notdifficulttoseethat,inthecaseofV =V theΛ partoftheoriginalpotential
c
−
(2) goes into the φ-part of the Schr¨odinger equation and therefore the entire
effect of this term appears in the φ-part of the wave function. Therefore the
contribution of C I 2λeiφ/2 in the final form of the wave function, instead
lλ 2l
of the usual φ part in the wave functions (i.e., 1 eilφ) of the case of V = V
− (cid:0) (cid:1) √2π ±
is the reason of the great localization about small φ. Let us write
φ φ
I 2λeiφ/2 =I 2λcos +i2λsin (47)
2l 2l
2 2
(cid:16) (cid:17) (cid:18) (cid:18) (cid:19) (cid:18) (cid:19)(cid:19)
which shows that the square root of the real (imaginary) part of the potential
directlygoesthroughthereal(imaginary)partoftheargumentofI 2λeiφ/2 .
2l
To see the behavior of I 2λeiφ/2 in terms of the real (imaginary) part of its
2l (cid:0) (cid:1)
argument we rewrite the last equation as
(cid:0) (cid:1)
φ φ
I I 2λ cos +i2λ sin (48)
2l 2l 1 2
→ 2 2
(cid:18) (cid:18) (cid:19) (cid:18) (cid:19)(cid:19)
which in the limit of λ =λ =λ turns out to be I 2λeiφ/2 . Figures (8),(9)
1 2 2l
and (10) show that once λ vanishes I 2 does not change much, but once λ
2 | 2l| (cid:0) (cid:1) 1
becomes zero, I 2 decreases strongly. Also once λ takes a larger value (it
2l 1
| |
does notmatter whatis the value ofλ ), I 2 takesmuchhigher value closeto
2 2l
| |
8
φ=0or2π.We concludethereforethat, the imaginarypartofthe argumentof
I (and imaginarypart of the potential therefore) does not contribute much in
2l
the localization of the Ψ 2 around small φ. In contrast, the real part, and its
7
| |
φ dependent part (i.e., cosφ) causes such a great localization.
−
4 Conclusion
Inconclusion,weconcentrateourselvesandwishtocommentonthefigures. Fig.
(1)is ourreferencefigure,i.e.,the classicalcorrespondencelocalizedstatewhen
the Λ part of the potential vanishes. Fig.s (2-4) clearly show that the effects
−
of the Λ part in the V and V cause a higher localization while in the V we
c +
− −
see a lower localization. It is remarkable to observe that, the magnitude of the
effects(withinthesethreecases)V isgreaterthanothers. Fig.s(5,6)showthat
c
as λ takes large value the radius of the localized state corresponding to the V
+
decreases while the φ distribution of the Ψ 2 does not change. For the case of
7
| |
potentialV the radiiofthe localizedstatesarefixedwhile theφdistributionof
c
theprobabilitydensity Ψ 2ischangedsothattheparticleseemstobelocalized
7
| |
more around φ=0. In the Fig. (7) we see the effect of the Λ partin the V as
c
−
agreatlocalizationaboutφ=0.Obviouslythe figuresimply thatinthecaseof
V =V , the particle is localized around φ=0, which is a direct consequence of
c
the Λ partof the potential. We notice that, our approachto the problem is in
−
aclosedanalyticalform,whereallnumericalresults arebasedonthe analytical
solutions. There is no need to comment that any other approachwill definitely
lack the advantages of an exact analytical solution.
Acknowledgement 1 TheauthorwouldliketothankProfessorMustafaHalil-
soy and Professor Omar Mustafa for fruitful discussions and useful comments.
Acknowledgement 2 AlsoIwouldliketothanktheanonymousrefereeforthe
valuable comments and suggestions.
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