The Analytical Transfer Matrix Method Combined with Supersymmetry: Coulomb Potential

The Analytical Transfer Matrix Method Combined with Supersymmetry: Coulomb Potential

HosungSun

Department of Chemistry, Sungkyunkwan University, Suwon 440-746, Korea. *E-mail:[email protected] Received November20, 2006

Combiningtheanalytical transfermatrixmethod with supersymmetryalgebra,anew quantizationconditionis suggested.Todemonstrate theefficiencyofthenewquantization condition, the eigenenergies of theCoulomb potential areanalyticallyderived.Thescattering-ledphaseshiftsarealsodetermined andthey are thesamefor all Coulomb potential states. Itisfound that the newquantizationcondition is mathematically simple and exact.

Key Words : Coulombpotential,ATMM,Supersymmetry

Introduction

For a one-dimensionalsystem, whena particle is moving between two classical turning points, its kinetic energy cannot be larger than the potential energy. Therefore the particle cannot escape out from the potential well but oscillatesbetween thetwoturningpoints. However quantum mechanicallythe particle canenterthe classicallyforbidden region. The exponentially decaying wave function in the classically forbiddenregion shouldbe connected smoothly to the oscillating wave function in the classically allowed region atthe classicalturningpoints. The wave reflected at theturning point has a different phase, relative to the incom­

ingwavetravelingtowards theturning point. Thereforethe phase of wave function shouldchange through the turning points and the change is called a phase shift. Furthermore, there exists a scattering phenomenon between the two turning pointsunlessthe potential is constant. The scattering also causes an additional phase shift that is called the scattering-led phase shift.1

Theexistence of theadditional phaseshiftis ignoredin the well known WKB (Wentzel-Kramers-Brillouin) approxi­

mation(orquantizationcondition.).2,3 This is thereason why the WKB quantization is not exact though it is found to be exact for the harmonic oscillator potential and the Morse potential. The recently developedanalytical transfer matrix method (ATMM)suggests a newquantizationcondition that explicitly includes the additionalphase shift. The so-called complete quantummomentumfunction is suggested and the integration of thequantummomentumfunctionover position variable yields the quantizationcondition that can describe the motionof a particleoscillating within a classicallybound region. The ATMM quantization condition hasbeenapplied tovariouspotentials and is considered to beexact.4-8

We utilizethe ATMM quantization conditionto generate theeigenenergy and theadditional phaseshift oftheCoulomb potential.Thisapproachenables one to directly evaluatethe phase shift. However,thereisagreatdifficulty inevaluating the phaseshift because the phaseshift is expressedinterms of eigenfunctions. That is, one has to predetermine the eigenfunctions of all eigenstates to use the ATMM quanti­

zation. For theone-dimensionalCoulomb potential,theeigen­ functionsare, of course,known. Evenso,the mathematical evaluation of thephaseshift isstill very difficult becausethe eigenfunctionsofexcitedstates are verycomplicated.9

Inthis work, adoptingthe supersymmetry algebra, a new and simple quantization condition for evaluatingthe phase shift and eigenenergy is suggested.10-12 Since the supersym­

metricpartnerHamiltonians share thecommoneigenenergies of the original Hamiltonian, only the ground state eigen­

functions (or superpotentials) of partner Hamiltonians, instead of allthe eigenfunctions of theoriginalHamiltonian, are required to evaluate the phase shift when the new quantization condition is used. It reduces a mathematical difficulty a lot.

Inthe followingsectionstheCoulombpotential system is explicitly defined. The supersymmetry and the ATMM quantization condition are introduced. Particularly the supersymmetry adoptedversion of the ATMM quantization condition is presented. The eigenenergy and thephase shift of theCoulomb potential arepresented and analyzed. Finally the information obtained fromthis work is providedin the conclusion section.

CoulombPotenti^지

The Coulomb potential energyfunction is widely used in chemistry.13 And the most well known one is the proton­

electron attraction potential in a hydrogen atom. The one­

dimensional Coulomb potential problem canbeanalytically solved. The eigenfunctions of the Coulomb potential are expressed in terms of the Laguerre polynomials and the eigenenergies are obtained in a closedform. Let us define theone-dimensionalCoulombpotential explicitly bytaking a simpleexample.

Theradial Schrodinger equation for ahydrogenlike atom containing oneelectron of mass Me anda nucleus ofmass M, for a certainangularmomentumquantumnumber l,is

h d , h l(l +1)ze2' ---+--- ---

2^dr2 M 『 _r 甲ni(r) =En咻 r) (0 <r <8) (1)

where 卩_=[Me/(1 + Me/M)] is the reduced mass, Ze is the nuclear charge and, of course, e is the unit charge. The attraction potential between the electron and the nucleus -(Ze /r) iszero at r— 8 . The Schrodinger equation Eq.

(1) can be exactly solved to generate the N-dependent eigenenergyENwhere

Z 4

En= 쑼法 _{N = 1,} 2, 3, ... ₍₂₎

The Nis called the principal quantum number. The wave function (or eigenfunction) 甲ni(r) depends not only on l but alsoon N.甲ⁿⁱ(r⁾ is equivalentto r^Nl(r) where Rni(F)is the customarily defined radial function of a hydrogenlike atom.13 The boundary condition isthatRnQ)goestozeroat infinity. Now the angular momentum quantum number l is limited as l = 0, 1, 2, ..., N- 1 in order to satisfy the boundarycondition.

TheSchrodingerequationEq. (1)can berewritten as

[壬₊-

dr r

2~|

q-

r ^甲Nl(r) = £ⁿ甲Nl(r⁾ (0 <r <8) (3)

2

where q2=卩_： h

For convenience anew quantum number n=N 一 l 一 1(n = 0, 1, 2, .) is introduced. n is, innature, equivalent to the numberof nodes in theradialwavefunction 甲n(r).

In summary,theCoulomb potential systemofacertain l is as follows:13-15

and £ⁿ = 一 꼬.

「d2 1

H 甲(r K -d/+ V( r)甲(r) = £甲 r)

(0

wheretheCoulombpotential V(r) =~으- + l('+1)

r r ⁽⁵⁾

q4

and theeigenenergy £n =--- --- for n=0, 1, 2,....

Eqs. (4) and (5) are theCoulombpotentialsystem whose eigenenergy and phase shift will be evaluated using the ATMM quantization condition1 with the aid of super- symmetry.10-12 In the next section the supersymmetry is briefly reviewed and a newATMM quantization condition combined with supersymmetryisintroduced.

ATMMCombinedwithSupersymmetry

First some characteristics of supersymmetry are briefly presented. According tothe supersymmetric quantummech­ anics,12 the raisingand lowering operators A*? for the s-th supersymmetricpartnerHamiltonian H(s) aredefined as

A^s⁾ = -으 + W^(s)( r) s =0 1 2

z丄土 + dr^{、” \、}r) s v/,丄,z-, ♦… (7) The superscript(s) is the hierarchyindex so that itindicates

the s-th partner Hamiltonian. In this work, the s=0 case is theoriginalHamiltonian. Forexample, H°)is equivalent to the original H in Eq. (4). Likewise, "0)(r) = V(r), 甲 Kr)=甲r), and 當=£ⁿ. Here the superpotential W(s)(r) is, in essence, the minus log derivative of the ground state(n=0)wave function 甲0" r) ofHamiltonian H(^s, i.e.,

d甲0s)(r)/dr W( s)(r) =--- 0스-)^—

甲"(r) Anditsatisfies afollowing Riccatiequation

dW：--- = [ W(s)(r)]2 ^{- Us)}(r)+ 鶯.

dr 0

(8)

(9) Then the original Hamiltonian in Schrodinger equation Eq. (4)can be rewritten as

H(o)(= H)=-乌 _{+ V}o)(r) = A⁺0 A^-0⁾ + f⁰0⁾ dr

= -으; + [ W⁽0⁾(r)]2 ^{- 씨}*")₊ £0). (10)

dr2 dr 0

Thefirstsupersymmetric partner Hamiltonianis H⑴=a-0)^a

+

= -土 + [ W⁽¹⁾(r^)]2 ^{- 씻}W으1 ₊ 話 .

dr2 dr 0 ⁽¹¹⁾

In generalthe partner Hamiltoniancanbe generated for any index s, i.e.,

H，)=a

+

= -丄 + [W⁽s)(^ r)]2 ^- 의W으) +理

dr2 dr

From the supersymmetry algebra we easily following relations,12

勺=£0

甲(r) = A^+s)A+s -1⁾-A+1^{)^0s)(}r)

(12)

obtain the

(13) (14) 甲"(r) = C")exp[-

j

where Cs) is anormalization constant. Note that 甲$(r) is the s-thstate (n= s) wavefunctionoftheCoulomb potential Hamiltonian (H) and 甲"⁽r) is the ground state (n=0) wave function of Hs). Consequently we find that the hierarchy index sis equivalenttothequantum number n of theoriginal Hamiltonian. Therefore on we can safely use the notation of n in replacement ofs. In Figure 1,the hierarchy of supersymmetric partner Hamiltonians, Eq. (13) is illu­ strated.A similarillustration for arigidsymmetrictoprotor can befound in thisjournal.16

Now let us introduce the quantization condition of the analytical transfermatrix method.1 TheATMMquantization condition, for the Schrodinger equation Eq. (4), can be (15)

Figure 1. Hierarchy of supersymmetric partner Hamiltonians.

summarizedas

f

rL

wheren is the quantumnumber. k is themomentum,i.e., Kn(r)=，旗_-以r) (17) where V(r)isthepotential and £ⁿistheeigenenergyof then­ thstate. r^L andr^Rare inner (short distance) and outer(long distance) turningpoint, respectively, i.e., V⁽七)= V(Fr)=

£n and Fl < Fr . Kn(r) is alwayspositive between^fl and ^fr. The so-called scattering-led phase shift毎 is

rR P^負(r)

5n= "

J

(18)

rL Pn (r)

the n-th state (or ground state) of original Hamiltonian.

Needless to say these two indicescan beusedinterchange­ ably. The new quantization condition (Eqs. (19), (20), and (21))will be applied totheCoulombpotentialsystem.

The difference between the original ATMM (Eq. 16) and the new ATMM with supersymmetry (Eq. 19) lies in what kind ofeigenfunctions is used. In the original ATMM allthe eigenfunctions (including excited ones) of the original Hamiltonianare required while in the new ATMM only the eigenfucntions ofthe ground state ofpartner Hamiltonians are required. It is mucheasierto evaluate the ground state eigenfunctions thantheexcited stateeigenfunctions. Further­ morethe integrationinvolving onlythe groundstate eigen­

functions is alsoeasier.

Eigenenergy andPhaseShift of CoulombPotenti지

where "(

F and

)

is &

minus log derivativeof the n-th statewavefunction iPn(r), 膜.，Pn(r)=-dK(r )/dr

---

甲n( r) ^.Note that P°(r) = W⁽r).

The Coulomb potential is introduced earlier. Using the supersymmetryalgebra wecandeterminethesuperpotential, etc. The directway of determining V(n)(r⁾ and W⁽ⁿ⁾(r) is as follows. Solving the Riccati equation, Eq. (9) with the originalpotential V(0⁾(r), one obtains W⁽0⁾(r) and %0 . Then using the supersymmetry relationship12 of

V's +1⁾(r) = [W(s)(r^)]2 + 지" (r- + %), one can easily dr

determine V 1)(r- from W°)(r-.17With thepartner poten­

tial V 1)(r - , one again solvesthe Riccati equation to obtain W(i)(r- and so on. Thisprocesscanberepeatedly perform­

eduptoanyhierarchyn(or s).Theresults for the Coulomb potentialHamiltoniangivenin Eq. (4) are

2

V% r -=^-七 + r

(n+ l )(n r In order to evaluatethephase shift 8ⁿinEq. (18)onemust

predetermine the wave function Kn(r) of H(=H(0)), which

2

requires a lot ofalgebra. Even the Tn (r⁾ is known, the integrationisvery difficultbecause ofthe complexity ofthe

Wok r - =---^으--- (-2 (n +1 +1-

4

wavefunction. Here weadoptthe supersymmetry algebra to avoid this difficulty. The quantization condition, Eq. (16) shouldbe valid not only for H(or Kn(r⁾) but also for any partner Hamiltonian Hn)(or 甲：)(r)). Kⁿ⁽r⁾ and 甲*)(r) are, of course, differentto each otherbut the eigenenergies areidentical, i.e., £n = &? (seeEq.(13).)Therefore onecan rewrite the quantization conditiononly for the ground state (n= 0) of any n-thsupersymmetricpartnerHamiltonian H(n),

4(n +1+ 1)2

--2--- (22)

n+ 1 + 1 --- (23)

r

=%. (24)

i.e.,

r^n^

J

rL

K0ⁿ)(r)=損")-V^")( r) = J %- ”^")( r) (20)

暦= "J； 亨 W꼬drdr. (21)

Of course, the eigenenergies in Eq. (24) are identicalwith thosein Eq. (6).

Now using the new ATMM with supersymmetry, i.e., Eqs.

(19), (20), and (21), we determine the eigenenergy ofthe Coulomb potential Hamiltonian.Firstthemomentumpartis

f 忒(r-dr = JR*瓦*。dr

rL rL N r r

2

= 쯜- (

-

1

Theaboveintegral can be found in a literature.15

The phase shiftofthe n-th partner Hamiltonian H(n),狀")

inEq.(21)can berewritten as, usingEqs. (22) and (23),

The superscript (n) denotes the hierarchy of partner Hamiltonians and the subscriptnumber n (or 0) represents

+^으2 (n+ 1)(n+1+1))

r F )

^^^—{^■^■^■^BH^l^H^H^H^—^B^H^H^H^H^B^H^l^H^H^H^H^H^H^H^H^H^H^H^H^H^H^S^H^HSI^H^H^—^H^H^H^H^H^H^H^H^H^H^BH^H^H^B^H^H^H^H^H^H^H^—(26) (n+1+1)2 / r

Lety="(n+1)(n+1 + 1)_/r, y(L^>n=J(n +1)(n +1 + 1)时', yR1^ =」_(n +1)(n +1 + 1)/rR", then Eq.(26) is reduced to

시 y』⁽、y- 시)(b-y) 2(ab)3/2 Theintegration result is

船 = 쯩시(n+ l +1)-2(-勇)-3/2 8

-，똘2(n +1 + 1 )-1 (2n +21 + 1)(_-書"))-^'

+ n(n +1 ⁾1/2 (n +1 + 1 )1/2. (28) So the quantization condition Eq. (19) from Eqs. (25) and

2

(28) is, by setting z = 옼・(_-話) ,

nz - n(n+l)"(n+1 + 1)" + n(n +1 + 1^{) 2}z3

—n(n+1 + 1^{) 1(}2 n +2l + 1 )z

+ n(n +1 ⁾1/2 (n +1 + 1)1/2 = n. (29) Rearranging it,weobtain the cubic equation tosolve,

z3-(n+l)(n +1 + 1)z-(n+1 + 1 )2 = 0. (30) It has three solutions, i.e.,

z = -(n +1 + 1)/2±J⁽n+ l + 1)(n+1 - 3) /2 and z =n + l + 1, but onlythepositive solution(z = n +1 + 1)ismeaning­ ful. Therefore,

2

n +1 +1=* (시"))- . (31)

4 i,(n) dy

輦)=q(n+l)1/2 (n +1 + 1)-3/2 仁 ,______________

0

4

— q~ (n +1 + 1^{) 1} (2n+ 2l + 1)「 -- ， 시*

板，y*그成侦

+ (n+ l )1/2 (n+1 +1 )1/2 广 h 내dy丿 --- (27)

板> JGR%W3

Since 舟 and 舟 are the turningpoints, i.e., Vⁿ)(£：、= V허)(澧) = * and 0< r?) < rR, one immediately finds that y?)> yR">0 and the relations; y(Rn + y?，=

q^I』_(n+l)(n+1 + 1)andy'R^'y'^{LL^} =-話.

Though the above integrals can be found in integration tables,18 we list the expressions for the convenience of interested readers. For real numbers b > 시 > 0,

cb dy # dy ^{n 、}

I

I

시 7(y - 시)(b-y) 시 yKy - 시)(b -y) 』ab

/dy_ 시 +b

4

We finallyobtain g? =--- 으--- - that is identical with 4⁽n+1 + 1 )2

the eigenenergygiven in Eq. (24). Inconclusionthe ATMM quantization condition produces the eigenenergy of the coulomb potential easily when it is combined with superH symmetryalgebra.

Now let us determinethe scattering-ledphase shift8n. As seen in Eq. (28), 蘑 of the n-th partner Hamiltonian depends ontheeigenenergy g (=覺).Insertingthe eigen- energy of Eq. (24) intoEq.(28),ithas a simple form, i.e.,

普=n」(n +1)(n+ l + 1) ^- n(n +l). (32) ComparingEq. (5) with Eq. (22), onefindsthat V'n⁾⁽r) can be obtained byreplacing l with n +1 in V(r) (= "0)(r)). One canevaluate P^°( r), i.e.,

d%( r )/dr d { rRu(r)}/dr -HHHHHHHHHHHHHHHHHHHHHHHHH = -HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH

%(r) rRu (r) 으2 l + 1

HHHHHHHHHHHHHHHHHH - HHHHHHHHHH.

2(l + 1) r (33)

Theradialfunctionofthe Coulombpotential Ru(r) canbe found in quantum chemistry textbooks.13 Comparing the above Eq. (33) with Eq. (23), it is found W^{( n})( r) can be obtained by replacing l with n +1 in P^°(r). Recall that P0(r) ₌ W⁾(r). This is a characteristic of the soHcalled translational shape invariantpotentials and, of course, the Coulomb potential is one of them.12 Therefore one can concludethat the scatteringHledphaseshift1 oftheCoulomb potential S? can be obtained by simplyreplacingn +1 with l inS")inEq. (32),i.e.,

S"= [십/l(l +1) - l]n. (34)

Generally it is believed that S depends onthe quantum numbern. Ournumericalwork on a distorted Morse potenH tial confirmed that Sis indeed "-dependent.19 However, for the Coulomb potential, we find in Eq. (34) that S is independent of n, i.e., the scatteringHled phase shift is constant for all n states. So farithasbeen known thatSn for the harmonic oscillator potential and the Morse potential havea constant S of1/2n.5,9 Thereasonbehindthis pheno­

menon is not understood yet. Furthermore our preliminary work shows that all the shape invariant potentials have a constant Z^".210

ConclusionandDiscussion

Combining theanalyticaltransfer matrix method, Eq.(16) with supersymmetry algebra, a new and exact quantization condition, Eq. (19) is suggested. In this new quantization only the ground state eigenfunctions of supersymmetric partnerHamiltonians are necessarywhile the originalquantiH zationcondition requiresthe eigenfunctions of all states. It reduces mathematical or computational efforts a lot when theeigenenergiesof a system are evaluated.

Theusage ofthenewquantizationis successfully demonH

Figure 2. Supersymmetric nature of hydrogenlike atom energy levels.

N=4 ________4s 4d _

4,

N=3 ^3$ 3p . 3"

N=2 ^{고」 2?}

N니 Is

/=0 1=2 M3

strated for determining the eigenenergies of the Coulomb potential. During thecourseof work theso-calledscattering- ledphase shift is evaluated and analyzed. Interestinglythe phaseshift for theCoulomb potential isfound tobe constant, i.e., allthe eigenstateshavethesamephaseshift for a given l. Particularly when l =0 (s waves), & = 0, i.e., there isno phase shift. Thenthe quantization condition is reducedto a simple formof

/r

f

The above equation i.e, integration of momentum over position variable, is a well-known form of quantization.

Forexample,the WKB quantizationis

「k( r)dr=(n+；) ⁿ n = 0, 1, 2, ... (36)

As mentionedabove, the harmonicoscillator and theMorse potentials have & = 1/2n. Inserting it into Eq. (16), one immediatelyfinds that the ATMM quantization is reduced to the above WKB quantization Eq. (36). This is the reason why the WKB quantization is exact for the two poten- tials.5,15,19,21,22 Therefore &n is critical in judging whethera certain quantization condition is exact or not.

Finally we would like to mention more aboutthe super­ symmetry of hydrogenlike atoms. From Eq. (24), one immediately finds that the principal quantum number N(= n +1 + 1) and theangular momentumquantum number l(=0, 1,2,...,N - 1)are relatedto each other. Forexample,

for N= 1,thereisonly one choiceof n and l, i.e., (n=0and l =0; 1s wave or orbital). And both have the same eigen- energy because ofthesameN. But for N =2,therearettwo possibilities (doubly degenerate), i.e., (n= 1 and l = 0; 2s orbital) and (n =0 and l=1^； 2p orbital). ForN =3, there exist threedegenerate states, i.e., (n= 2 andl = 0; 3s orbital), (n= 1 and l =1；3p orbital), and (n =0 and l =2; 3d orbital).

Thisrelation is illustratedinFigure 2. Comparing Figure 2 with Figure 1, one canunderstand thathydrogenlike atoms indeed have a supersymmetric structure.23,24

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