Resonances and antibound states of P\"oschl-Teller potential: Ladder operators and SUSY partners

2 MB·

by  D. Cevik

Resonances and antibound states of Po¨schl-Teller
potential: Ladder operators and SUSY partners
D.C¸evika,M.Gadellab,S¸. Kurua,andJ.Negrob
6
1 aDepartmentofPhysics,FacultyofScience,AnkaraUniversity,06100Ankara,Turkey
0 bDepartamentodeF´ısicaTeo´rica,Ato´micayO´pticaandIMUVA,,UniversidaddeValladolid,E-47011
2
Valladolid,Spain

n
a
J
January21,2016

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Abstract
h
p
WeanalyzetheonedimensionalscatteringproducedbyallvariationsofthePo¨schl-Teller
-
h potential,i.e.,potentialwell,lowandhighbarriers. WeshowthatthePo¨schl-Tellerwelland
t
a lowbarrierpotentialshavenoresonancepoles,butaninfinitenumberofsimplepolesalongthe
m imaginaryaxiscorrespondingtoboundandantiboundstates. Aquitedifferentsituationarises
[ onthePo¨schl-Tellerhighbarrierpotential,whichshowsaninfinitenumberofresonancepoles
and no other singularities. We have obtained the explicit form of their associated Gamow
1
states. We have also constructed ladder operators connecting wave functions for bound and
v
antiboundstatesaswellasforresonancestates. Finally,usingwavefunctionsofGamowand
4
3 antiboundstatesinthefactorizationmethod,weconstructsomeexamplesofsupersymmetric
1 partnersofthePo¨schl-TellerHamiltonian.
5
0
. 1 Introduction
1
0
6
Onedimensionalmodelsinquantummechanicsarerelevantastheymayservetotestawiderange
1
: ofquantumproperties. Theyarealsousefulinthestudyofsphericallysymmetricthreedimensional
v
models. Although many studies of quantum one dimensional models have been addressed to the
i
X
analysis of bound or scattering states, there are also a big number of works concerning unstable
r
quantum states, which occur quite often in nature. Resonances can be identified with unstable
a
quantumstates,whicharetwoequivalentmannersforthedescriptionofthesamereality[1,2,3,4].
In addition, there exists another type of not normalizable states with real negative energy called
antiboundstates[1,5,6,7].
Resonances are defined as pairs of poles of the analytic continuation of the scattering matrix
(S matrix). Inthemomentumrepresentation,thisanalyticcontinuationisgivenbyameromorphic
function S(k) on the complex plane. Then, resonance poles are symmetrically located on the
1
lower half of the complex momentum plane with respect to the imaginary axis. In the energy
representation,theanalyticcontinuationoftheSmatrixismeromorphiconatwosheetedRiemann
surface [1]. Now, each pair of resonance poles are complex conjugated of each other with real
part E and imaginary part ±Γ/2. These parameters E and Γ are the same that characterize a
R R
quantumunstablestate: theresonantenergyE (whichisthedifferencebetweentheenergyofthe
R
decaying state and the decay products) and the width Γ (which is related with the inverse of the
half life of the unstable state). These states can also be represented by wave functions, which are
eigenfunctions of the Hamiltonian with complex eigenvalue E ±iΓ/2. Since the Hamiltonian
R
is usually taken to be self adjoint on a Hilbert space, these wave functions called Gamow states
(resonancestates),cannotbenormalized[1,8,9]. Therearesomeotherdefinitionsofresonances
and quantum unstable states, not always equivalent, see references and a brief review in [10]. In
thepresentpaperforpracticalreasons,wedealwithresonancesaspairofpolesoftheS matrixin
themomentumrepresentation. Fordetailsconcerningotherformalismsweaddresstotheliterature
inthesubject[5,11,12,13,14,15].
In the study of the analytic properties of the S matrix in the momentum representation [5],
one sees the existence of three types of isolated singularities. One is the mentioned resonance
poles, which may have multiplicity one or higher [16, 17]. In addition, it may exist simple poles
on the positive part of the imaginary axis, (ik, k > 0) with energy E = −(cid:126)2|k|2/2m. Each one
determines the existence of one bound state ψ and viceversa for each bound state there exists one
ofsuchpoles. Inthiscaseψ isnormalizable,i.e.,squareintegrable. Simplepolesonthenegative
partoftheimaginaryaxis(ik,k < 0)correspondtoothertypeofstatescalledantiboundorvirtual
states. Wave functions of antibound states are not square integrable; furthermore they blow up
at the infinity. Their physical meaning is sometimes obscure (see [1, 6] and references quoted
therein). Itisalsopossiblethepresenceofasimplepoleattheoriginwithoutphysicalmeaning.
Thesethreetypesofstates: resonance,boundandantibound,correspondingtothesingularities
of the S matrix, can also be obtained by imposing purely outgoing conditions to the solutions of
theSchro¨dingerequation,asweshallseeinthenextsections.
Coming back to resonance poles, we know that they come into pairs. In the energy represen-
tation, these pairs are complex conjugate of each other, so that if z = E −iΓ/2 is one such
R R
pole, z∗ = E + iΓ/2 is another one. Then, for each resonance, there are two Gamow states,
R R
thesocalleddecayingGamowstate,ψD,satisfyingHψD = z ψD andthegrowingGamowstate,
R
ψG with HψG = z∗ψG. The decaying Gamow state ψD decays exponentially to the future, i.e.,
R
e−itHψD = e−iERte−tΓ/2ψD, whilethegrowingGamowvectordecaysexponentiallytothepast
(and grows exponentially to the future, hence its name), i.e., e−itHψG = e−iERtetΓ/2ψG. These
formulas make sense in an appropriate rigged Hilbert space [8, 9]. In the momentum represen-
tation, poles on the forth quadrant correspond to decaying Gamow states and poles on the third
quadrantcorrespondtogrowingGamowstates.
In the present paper we shall deal with the hyperbolic Po¨schl-Teller potential characterized
by a parameter λ. Depending on the values of such parameter, this potential admits bound and
antiboundorevenresonancestates. TheircorrespondingpolesoftheS matrixwillbedetermined
analytically, contrarily to most of known resonance models where the poles have to be computed
2
bynumericalmethods[7,18,19,20].
The factorization method has been used since the early times of quantum mechanics in order
to obtain the spectrum of some Hamiltonians by algebraic means [21, 22]. For the hyperbolic
Po¨schl-Teller potential, we may distinguish three different situations: potential well, low barrier
andhighbarrier. Inthefirstcase,thereexistsboundandantiboundstateswhichareobtainedfrom
eachotherthroughladderoperators. Inthesecond, thereexistantiboundstatesonly, althoughthe
situationissimilartotheformer. Finally,thehighbarrierhasaninfinitenumberofresonancesand
thecorrespondinggrowinganddecayingGamowstatesarerelatedbytwodifferenttypesofladder
operators. The ladder operators form an algebra supported by eigenfunctions of the Hamiltonian,
whichingenerallieoutsidetheHilbertspace. Uptonow,thistypeofladderoperatorswereapplied
toboundstatesbuttheyhaveneverbeenappliedtoantiboundstatesandresonances. Thisisavery
important result: resonance and antibound states share the same algebraic properties as bound
states.
AnotherapplicationofthefactorizationmethodistofindHamiltonianhierarchiesstartingfrom
a given Hamiltonian, see [21] and references therein. In order to obtain a Hamiltonian of a hier-
archy, one uses in general an eigenfunction without zeros of the initial Hamiltonian, often corre-
sponding to the ground state. However, one rarely uses an antibound state or a Gamow state to
constructsupersymmetricpartners[23,24,25]. Withtheseideasinmind,wegivesomeexamples
in which we build supersymmetric partners of Po¨schl-Teller potentials using wave functions of
antiboundandGamowstates.
This paper is organized as follows: In the next section, we review some basic and impor-
tant facts, concerning the hyperbolic Po¨schl-Teller potential and introduce the basic notation. In
Section 3, we discuss some of its scattering properties. We show that all poles of the S matrix
corresponding to the purely outgoing boundary conditions, giving the energies and momenta of
bound,antiboundandresonancestates,canbeobtainedanalyticallyandexactly. Thisisaveryex-
ceptionaloutcome,sinceforthevastmajorityofworkedpotentialsthesepolescanonlybeobtained
bynumericalmethods. Theconstructionofladderoperatorsrelatingeigenfunctionsforboundand
antibound states, growing and decaying Gamow states is done in Section 4. In section 5, we get
SUSYpartnersofthePo¨schl-TellerHamiltonian,usingantiboundandGamowstates. Wecloseour
presentationwithconcludingremarks.
2 The hyperbolic Po¨schl-Teller potential
LetusconsiderthefollowingonedimensionalHamiltonian:
(cid:126)2 d2 (cid:126)2 α2λ(λ−1)
H = − − , (1)
2m dx2 2m cosh2αx
where the second term in the right hand side of (1) is known as the real hyperbolic Po¨schl-Teller
potential. Here, α is a fixed constant while λ is a parameter. Along the present paper, we shall
consider three possibilities for λ each one giving a different shape for the potential. They will be
studiedseparately:
3
V
4
2
x
-4 -2 2 4
-2
-4
-6
-8
Figure 1: Plot of the Po¨schl-Teller potential for different values of λ. The continuos line corresponds to
λ=3.5(well),thedashinglinetoλ=0.6(lowbarrier),thedottedlinetoλ=1/2+i2(highbarrier).
• λ > 1,potentialwell,
• 1 ≤ λ < 1,lowbarrier,
2
• λ = 1 +i(cid:96); (cid:96) > 0,highbarrier.
2
The justification for the assigned names comes from their shapes shown in Fig. 1. Obviously,
for the value λ = 1 the potential vanishes. For integer values of λ, greater than one, it is well
knownthattheresultingpotentialisreflectionless.
ThetimeindependentSchro¨dingerequationproducedbytheHamiltonian(1),hasbeenwidely
studied[30,31]. Nevertheless,theforthcomingpresentationisquiterelevantinordertofollowour
arguments. If we denote by U(x) the wave function, the time independent Schro¨dinger equation
obtainedaftertheHamiltonian(1)isgivenby:
(cid:20) α2λ(λ−1)(cid:21)
U(cid:48)(cid:48)(x)+ k2+ U(x) = 0, (2)
cosh2αx
2mE d2U
wherek2 = andU(cid:48)(cid:48)(x) = . Then,letusintroducethefollowingnewvariable
(cid:126)2 dx2
y(x) := tanhαx (3)
andthenewfunctionν(y),
U(y) = (1+y)r(1−y)sν(y), (4)
wherer andsare
ik ik
r = , s = − . (5)
2α 2α
4
Withthesechoices,equation(2)becomestheJacobiequation,
(cid:20) (cid:21)
2ik
(1−y2)ν(cid:48)(cid:48)(y)+ −2y ν(cid:48)(y)+λ(λ−1)ν(y) = 0. (6)
α
In order to write (6) in the standard form of the hypergeometric equation, we need to use the
followingchangeofvariable
y+1
z := , (7)
2
sothat(6)takestheform:
(cid:20) (cid:21)
ik
z(1−z)ν(cid:48)(cid:48)(z)+ −2z+1 ν(cid:48)(z)+[λ(λ−1)]ν(z) = 0. (8)
α
Note that ν(cid:48)(z) denotes dν/dz, etc. We have finally reached the hypergeometric equation z(1−
z)ν(cid:48)(cid:48)(z)+[c−(a+b+1)z]ν(cid:48)(z)−abν(z) = 0, with a = λ, b = 1−λ and c = ik/α+1.
Two independent solutions are given in terms of hypergeometric functions: F (a,b;c;z) and
2 1
z1−c F (a − c + 1,b − c + 1;2 − c;z) (provided that c is not an integer [26].) Therefore, the
2 1
generalsolutionofequation(2)canbereachedaftersomeevidentmanipulationsandisgivenby:
(cid:18) (cid:19)
ik 1+tanhαx
U(x) = A(1+tanhαx)ik/2α(1−tanhαx)−ik/2α F λ,1−λ; +1;
2 1
α 2
+B2ik/α(1+tanhαx)−ik/2α(1−tanhαx)−ik/2α
(cid:18) (cid:19)
ik ik ik 1+tanhαx
× F λ− ,1−λ− ;1− ; , (9)
2 1
α α α 2
whereAandB arearbitraryconstants.
TheSmatrixconnectstheasymptoticformsoftheincomingwavefunctionwithoutgoingwave
function. Fortunatelytheasymptoticbehaviorofthehypergeometricfunctionsiswellknown[26]
andtheasymptoticformof(9)is:
• Forx (cid:55)−→ +∞
(cid:34) Γ(cid:0)ik +1(cid:1) Γ(cid:0)ik(cid:1) Γ(cid:0)1− ik(cid:1) Γ(cid:0)ik(cid:1)(cid:35)
U+(x) = A α α +B α α eikx
Γ(cid:0)ik +1−λ(cid:1) Γ(cid:0)ik +λ(cid:1) Γ(1−λ)Γ(λ)
α α
(cid:34) Γ(cid:0)ik +1(cid:1) Γ(cid:0)−ik(cid:1) Γ(cid:0)1− ik(cid:1) Γ(cid:0)−ik(cid:1) (cid:35) (10)
+ A α α +B α α e−ikx
Γ(1−λ)Γ(λ) Γ(cid:0)λ− ik(cid:1)Γ(cid:0)1−λ− ik(cid:1)
α α
= A(cid:48)eikx+B(cid:48)e−ikx
• Forx (cid:55)−→ −∞
U−(x) = Aeikx+Be−ikx. (11)
5
We recall that α is a given positive constant. In the sequel, we shall fix α = 1 for simplicity.
Then, wecandefinetheS matrixthatrelatestheasymptoticallyincomingwavefunctionwiththe
asymptoticallyoutgoingwavefunction[27]:
    
B S S A
11 12
  =    . (12)
A(cid:48) S S B(cid:48)
21 22
The matrix elements S of the S matrix are usually written in terms of the elements T of the
ij ij
transfer matrix T which relates the asymptotic wave functions in the negative infinity and in the
positiveinfinityisdefinedas
    
A(cid:48) T T A
11 12
  =    , (13)
B(cid:48) T T B
21 22
inthefollowingform:
 
−T 1
1 21
S =   . (14)
T
22 T T −T T T
11 22 21 12 12
The explicit form of the transfer matrix T [28, 29], obtained from (10), (11) and (13), is the
following:
 
Γ(ik+1)Γ(ik) Γ(1−ik)Γ(ik)
 Γ(ik+1−λ)Γ(ik+λ) Γ(1−λ)Γ(λ) 
T =   . (15)
 
Γ(ik+1)Γ(−ik) Γ(1−ik)Γ(−ik)
 
Γ(λ)Γ(1−λ) Γ(λ−ik)Γ(1−λ−ik)
It is easy to check that in this case detT = 1 and SS† = S†S = 1. Thus, we have obtained the
explicit form of the S matrix in the momentum representation, which will be henceforth denoted
byS(k).
Now, we define the purely outgoing states of the Schro¨- dinger equation in this case, as the
solutions characterized by (10) and (11) such that: A = B(cid:48) = 0. In other words, the asymptotic
behavior consist in outgoing waves to the right and to the left of the potential range. From the T
matrixequation(13),B(cid:48) = T A+T B. Therefore,thevaluesofksatisfyingthepurelyoutgoing
21 22
boundary conditions reduce to the solutions of T (k) = 0. As it is seen from (14), this equation
22
characterizes the poles of S(k) which are related with purely outgoing states. For such values of
k,accordingto(9),thewavefunctionscorrespondingtooutgoingstatesaregiven(uptoaconstant
factor)by
U(x) = 2ik/α(1+tanhαx)−ik/2α(1−tanhαx)−ik/2α
(cid:18) (cid:19)
ik ik ik 1+tanhαx
× F λ− ,1−λ− ;1− ; . (16)
2 1
α α α 2
6
3 Three types of Po¨schl-Teller potentials
Alongthispresentsection, weintendtoanalyzeallkindoffeaturesthatemergefromascattering
analysis of the three types of hyperbolic Po¨schl-Teller potentials under our study. This includes
scatteringstates,resonances,boundandantiboundstates. Weshallfollowtheorderbeginningwith
thepotentialwell,thenlowbarriertoconcludewiththehighbarrier.
3.1 Potentialwell(λ > 1)
Oneofthemostinterestingobjectsinthestudyofscatteringistheexplicitformsofthereflection
and transmission coefficients. The point of departure is now an asymptotic incoming plane wave
fromtheleftthatafterinteractionwiththepotentialcomesintoareflectedandatransmittedplane
waves characterized by k ∈ R . This means that in (10) and (11), we take A = 1 and B(cid:48) = 0.
Then,weobtainthefollowingreflectionr andtransmissiontamplitudes:
T Γ(ik)Γ(λ−ik)Γ(1−λ−ik)
21
r = B = S = − = ,
11
T Γ(−ik)Γ(1−λ)Γ(λ)
22
(17)
1 Γ(λ−ik)Γ(1−λ−ik)
t = A(cid:48) = S = = .
21
T Γ(1−ik)Γ(−ik)
22
Then,thereflectionandthetransmissioncoefficientsaregivenby
(cid:12)(cid:12)Γ(ik)Γ(λ−ik)Γ(1−λ−ik)(cid:12)(cid:12)2
R = |r|2 = (cid:12) (cid:12) ,
(cid:12) Γ(−ik)Γ(1−λ)Γ(λ) (cid:12)
(18)
(cid:12)(cid:12)Γ(λ−ik)Γ(1−λ−ik)(cid:12)(cid:12)2
T = |t|2 = (cid:12) (cid:12) .
(cid:12) Γ(1−ik)Γ(−ik) (cid:12)
WecancheckthatT+R = 1fork ∈ R. InFig.2,weplotT(k)versusR(k)forλ = 3.5. Werecall
that whenλis an integer, then the transmission coefficient is equal to one: T = 1. Consequently,
R = 0andwehaveareflectionlesspotential.
Now, consider that k ∈ C. As was settled earlier, singularities of S(k) corresponding to
outgoingstates,aredeterminedviatheequationT (k) = 0,whichinourcasetakestheform:
22
1 Γ(1−ik)Γ(−ik)
= = 0. (19)
t(k) Γ(λ−ik)Γ(1−λ−ik)
Therefore, such singularities of S(k) coincide with the singularities of t(k) or T(k). Since the
Gamma function has no zeros, the solutions of (19) are restricted to the poles of the two Gamma
functionsinthedenominator. Therefore,solutionsof(19)satisfyeitherλ−ik = −nor1−λ−ik =
−n,withn = 0,1,2,.... Ifwecallk (n)andk (n)tothesolutionsofthefirstandsecondtype,
1 2
respectively,wehavethat
k (n) = −i(n+λ), k (n) = −i(n−λ+1). (20)
1 2
7
1.0
T
0.8
0.6
0.4
0.2
R
k
0.0 0.5 1.0 1.5 2.0
Figure2: Wellpotential: PlotofT(k)andR(k)forλ=3.5.
Figure3: Wellpotential: PlotofT(k)forλ=3.5,andcomplexvaluesk =k +ik . Thesingularitiesare
r i
shownatk (n):i2.5,i1.5,i0.5,−i0.5,−i1.5,−i2.5,−i3.5(left). Attheright,itisshowntheprofileof
2
T(k)whenk =0.ThiscoincideswiththetransmissioncoefficientofFig.2(extendedto−∞<k <+∞).
i
U
1.0
0.5
x
-5 5
-0.5
Figure 4: Well potential: The plot of bound state wave functions with λ = 3.5 and n = 0 (continuous),
n=1(dotted)andn=2(dashed). Itscorrespondingvaluesofk (n)are: i2.5,i1.5andi0.5.
2
8
U U
60
100
40
80
20
60
x
40 -1.0 -0.5 0.5 1.0
-20
20
-40
x
-1.0 -0.5 0.5 1.0
-60
Figure 5: Wellpotential: Plotoftheantiboundwavefunctionsforthefirstthreeevenvaluesofn, n = 4
(continuousline),n=6(dashedline)andn=8(dottedline)withλ=3.5(left).Attheright,itisdisplayed
thewavefunctionsforthethreefirstoddvaluesofn,n=3(continuousline),n=5(dashedline)andn=7
(dottedline).
When λ > 1, where λ is not an integer, solutions k (n) are all located in the negative part
1
of the imaginary axis. Poles on the negative imaginary axis are called antibound poles. Their
correspondingrealenergiesareeigenvaluesoftheHamiltonianandtheirrespectiveeigenstatesare
calledantiboundstates. Wavefunctionsforantiboundstatesarenotsquareintegrableanddivergeat
theinfinity. Allthismeansthatourpotentialshowsaninfinitenumberofequallyspacedantibound
poles. Now, let us focus our attention in the second identity in (20), k (n). The inequality n −
2
λ+1 < 0hasatleastonesolutionandthenumberofitssolutionsisalwaysfinite. Consequently,
thesolutionsk (n)giveafinitenumberofpolesinthepositiveimaginarysemiaxis, whichdefine
2
boundstates[5]andaninfinitenumberofantiboundpoles.
No resonances appear for these specific values of λ (λ > 1). In Fig. 3, bound and antibound
poles are shown in the plot of T(k), k ∈ C (as mentioned above, such singularities coincide with
the poles of T). In Fig. 4, we plot the first three bound state wave functions and in Fig. 5 first six
antiboundstatewavefunctionsforthevalueλ = 3.5.
3.2 Lowbarrier(1 ≤ λ < 1)
2
Thetransmissionandthereflectioncoefficientsarerespectivelygivenby:
sinh2(πk) sin2(πλ)
T = , R = . (21)
sin2(πλ)+sinh2(πk) sin2(πλ)+sinh2(πk)
Obviously,T +R = 1fork ∈ R.
Inthiscase,thesingularitiesofS(k),k ∈ C,arealsogivenbyequations(20). However,since
1 ≤ λ < 1,wealwayshavethatn+λ > 0andn−λ+1 > 0,sothatnoboundstatesexisthere.
2
Instead, we have two different series of antibound states, where the antibound poles are given by
k (n)andk (n)asin(20). ThisisillustratedinFig.6,whereT(k),forλ = 0.75,isrepresented.
1 2
The plot of T(k), R(k) and the shape of wave functions for antibound states are quite similar to
thepreviouscase.
9
Figure 6: Low barrier potential: Plot of T(k) for λ = 0.75 and complex values k = k + ik . The
r i
singularitiesareshownatk (n):−i0.25,−i1.25,−i2.25,−i3.25,−i4.25(left). Attheright,itisshown
2
theprofileofT(k)whenk =0. Thiscoincideswiththeshapeofthetransmissioncoefficient.
i
3.3 Highbarrier(λ = 1 +i(cid:96))
2
Tostartwith,letusgivetheexpressionsforthetransmissionandreflectioncoefficients:
sinh2(πk) cosh2(π(cid:96))
T = , R = . (22)
cosh2(πk)+sinh2(π(cid:96)) cosh2(πk)+sinh2(π(cid:96))
Again, T +R = 1 for k ∈ R. This is possibly the most interesting case, as it shows resonance
phenomena. Here, weareassumingthat(cid:96) > 0. Then, bothseriesofpolesolutionscanbewritten
as:
(cid:18) (cid:19) (cid:18) (cid:19)
1 1
k (n) = (cid:96)−i n+ , k (n) = −(cid:96)−i n+ , (23)
1 2
2 2
where n = 0,1,2,... . For each value of n, solutions k (n) and k (n) give a pair of resonance
1 2
poles. Notethat, asexpected, theyarelocatedinthelowerhalfofthek planesymmetricallywith
respect to the imaginary axis. Let us write each pair of resonance poles as k (n) = (cid:96)−iγ and
1 n
k (n) = −(cid:96)−iγ withγ = n+1/2. Then,thecorrespondingenergylevelsare:
2 n n
(cid:126)2 Γ (cid:126)2 Γ
z = k (n)2 = E −i , z∗ = k (n)2 = E +i (24)
R 2m 1 R 2 R 2m 2 R 2
with
(cid:126)2 (cid:126)2
E = (cid:0)(cid:96)2−γ2(cid:1) , Γ = 4(cid:96)γ . (25)
R 2m n 2m n
Asseenontheseformulas,bothrealandimaginarypartsofresonancesdependonn. Theredo
notexistanyothersingularitiesofS(k)likeboundorantiboundpoles,seeFig.7. InFig.8weplot
the modulus and the real part of wave functions of the poles k (n) for the first three even values
1
of n. For the odd values of n the wave functions are odd, so they include zeros in the origin. We
shouldrecallthat,asaconsequenceofthegeneraltheory,themodulusofthesewavefunctionsare
exponentiallygrowingattheinfinity[32].
10