Quantum bound states with zero binding energy

1 August 1994 PHYSICS LETTERS A

Physics Letters A 190 (1994) 357-362

ELSEVIER

Quantum bound states with zero binding energy Jamil Daboul a,1, Michael Martin Nieto b,2 a Phystcs Department, Ben Gurton Umverstty of the Negev, Beer Sheva, Israel b Theorettcal Dlwston, Los Alamos Nattonal Laboratory, Umverstty of Cahforma, Los Alamos, NM 87545, USA Recewed 30 May 1994, accepted for pubhcatlon 10 June 1994 Communicated by J.P. Vlgler

Abstract After reviewing the general properties of zero-energy quantum states, we gave the explicit solutions of the Schrodmger equation w~th E = 0 for the class ofpotenuals V = -17l/r ", where -c~ < u < oo. For v > 2, these solutions are normallzable and correspond to bound states, if the angular momentum quantum number 1 > 0 (These states are normahzable, even for l = 0, ff we increase the space d~menslon, D, beyond 4, i e for D > 4.) For v < - 2 the above solutions, although unbound, are normahzable This is true even though the corresponding potentmls are repulswe for all r We discuss the physics of these unusual effects

1. Introduction In studying q u a n t u m , nonconfinmg, potential systems, care is gwen to describing both the discrete, normahzable, b o u n d states, which exxst for energy E < 0, and also the non-normallzable, free (including reson a n t ) states, with energy E > 0. However, usually httie is saxd about the zero energy states. From systems such as the Coulomb problem, where the E = 0 state is m the continuum, it can easily be assumed that all E = 0 states are in the c o n t i n u u m and not normahzable. However, there are at least two known examples where, for discrete values of the coupling constant, the E = 0 state is bound. It is the purpose of this note to explore this phenomenon. We will demonstrate an exactly solvable system of power-law potentials where the E = 0 states are b o u n d for continuous values of the coupling con-

stant. We will lllucldate the physics of this situaUon and also demonstrate that there also exist normalxzable states which cannot be interpreted as b o u n d states. Finally, we will show that by increasing the dlmenslonahty of the problem, an effective centrifugal barrier is created which causes states to be bound, even if the expectation value of the angular m o m e n t u m operator, L 2, vamshes.

2. Background Consxder the radial Schrbdmger equation with angular-momentum q u a n t u m n u m b e r l,

[ h2 ( d2 2 d ERr = -)'-'m ~ + r dr

l(1+ 1)) ] r2 q- V(r) RI.

(1) For the Coulomb problem, the effective potentml

I E-mad" daboul@bguvms bgu ac d 2 E-mad: [email protected] El~xaer Soenee B.V.

SSDI0375-9601 ( 9 4 ) 0 0 4 6 4 - Z

h 2 l ( l + 1)

U(r)-

2m

r2

e2

r

(2)

358

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J Daboul, M M Nteto / Phystcs Letters A 190 (1994) 357-362

' ' '

0 o

'

'

' /

'

-i

/

-4

-4

~6

Fig 1 A dlmenmonless representation of the Coulomb effective potential We show the effective potential

8 o = h 2 / 2 m a 2, We t a k e l = 2, D = 11, b = 25, and

U = liP 2 - 4/p

p0 = 1

has the form shown in Fig. 1. The effective potential asymptotes to zero from below as r --* o~, so that a particle with zero binding energy, E = 0, has a positive kinetic energy and is free to travel out to lnfimty. (Note that m this case the zero-energy solution, m addition to being continuously connected to the continuum, is also a limit point o f the bound states, whose energies, E n , go as - l / n 2 ) The physical situation is very different, however, if the potential approaches zero from the top as r ~ o~ This is the case, for example, m the "standard" discussion o f alpha decay. Consider a phenomenologxcal descnption o f alpha-decay with a Morse potentml. Then, the effective potential, with the angular-momentum barrier included, is U = 80[1(/+ 1)/p 2 + D{-2exp [-2b(p+ exp [ - b ( p -

Po)]}],

P0)] (3)

where here, and later, we use the notation r

p--= - , a

Fig 2 The Morse effective potential of Eq (3) m umts of

0

2

2ma 2 '

(4)

a being a distance scale and 80 being an energy scale. In Fig. 2 we plot an example o f the Morse potential Early q u an t u m -m e c h a m c a l text books [1] discussed the energy ranges E > 0 and E < 0 for this type o f potential, but often did not include the E = 0 case in the discussion. Even in the famous lecture notes o f F er m i [2 ] the W K B tunnel time to the outside was discussed only for E > 0, even though one can see that it goes to mfimty as E ~ 0 (This last

2

5

3

-2

-4

6

-8

Fig 3 The effective potential U ( p ) of Eq (5), in units of 80, for a spherical-well, with l = 1 and V0 = n28o or g = n With these parameters, there is exactly one bound state, at E=0 argument provides an intuitive understanding o f the bound-state result we are discussing.) N o w we make this point analytic by consldenng a solvable system whose effective potential has the features of the alpha-decay potential. This potential is

~2

80----

m

U(r)

-

h2 l(1+ 2m

1)

r2

Vo = 8 o [ l ( l + 1 ) / p 2 - g2],

ra

- -

r2

= £ o l ( l + 1 ) / p 2,

(5)

where g 1s a dimensionless "coupling constant" In this case, the effective potential, shown in Fig. 3, is infinite at r = 0, falls below zero, rises above zero at some r = a, and then goes to zero from above as

359

J Daboul, M M Nteto / Ph.vszcsLetters A 190 (1994) 357-362 r ~ o~. This Is the spherical box, discussed in many places. The E = 0 solution ~s even simpler than the general case. To our knowledge, this was the first example o f a b o u n d state with zero binding energy being explicitly demonstrated for a wide audience [3,4] 3 First consider the interior, r < a. The solution is a spherical Bessel function, which insures that the wave function is finite at the origin,

Indeed, in Fig. 3 we plot the effective potential U (r) in (5) in units o f Eo for l = 1 and V0 = rr2£0 or g = n. Then, an E = 0 b o u n d state exists and corresponds to the first zero, x a = ~, o f do(xa) = s m ( x a ) / x a . Another example o f this type is the focusing potential o f D e m k o v and Ostrovskn [6,7], here written m the form

R l ( r < a) ,,~ J t ( x r ) = J t ( g P ) ,

where w is a dimensionless coupling constant This system has normahzable, E = 0 solutions only for the following discrete values o f w [6,7 ],

x=

~

=

g/a,

(6)

which is ,,~ r-l/2Jtt + 1/2). F r o m the Schrodinger equation, the exterior solutions (r > a, where V = 0) go as a power law. The choice is the negative power law since we are interested in normahzable solutions,

V =

WN

wC0 p2(pr + p - r ) 2 ,

=

4r 2

N=n+ R l ( r > a) ,.~ 1/r l+l.

(7)

(

x > 0,

N + ~-~ - I

(1-1)l,

N +

(11)

,

(12)

n=nr+l+l,

nr = 0, 1,2,.

(13)

The matching condition at r = a is that dln(Rt) 1 dRt dr - Rt dr

(8)

3. Zero-energy bound states and singular discrete states for power-law potentials

be continuous at r = a. This means that 0 = (l +

1)Jl(lCa)

+

(xa)j~(xa) = (xa)Jt-t (xa).

(9) The first equahty is the physical condition. The second equahty is a standard mathematical result o f spherical Bessel functions. Therefore, the spherical well is a different situation than the Coulomb case, where E = 0 ts a limit point o f the bound-state spectrum The spherical well has a finite bound-state spectrum. In general the E = 0 solutions o f the spherical well are not normalizable. However, for a given l, an E = 0 solution is normalizable if Vo is such that x a Is equal to a zero o f the spherical Bessel function JI-i. That is, J t - ~ ( x a ) = O.

(10)

These zeros can easily be calculated and are tabulated

[51. 3 In Ref [ 3 ] Downs acknowledges a letter from Scinff. Tins commumcatlon clearly helped inspire the discussion of th~s problem which Sch~ff inserted into the third edmon of Ins textbook (see the discussions around pages 127 and 279 in Ref. [4] )

N o w we present an infinite class o f potentials which is exactly solvable for E = 0, and has the property that many o f the E = 0 states are b o u n d Elsewhere we will go into more detail on this system for both the classical case [8 ] and the q u a n t u m case [9 ]. F o r convenience we parametrlze these potentials as V(r) -

Y r~

--

g2£0

--

p~ ,

c~ < v < oo,

(14)

where g is a dimensionless coupling constant. It will be useful to interchange the variables v and/z, which are related by !2( v - 2 ) = # ,

v=2(#+l).

(15)

In Fig. 4 we show an example o f such a potential where the E = 0 solution will be a b o u n d state (v > 2). We now demonstrate that the Schrodinger equation is exactly solvable for all E = 0 and all - ~ < v < ~ . To do this, set E = 0 in Eq. ( 1 ), change variables to p, and then multiply by _p2. One finds 0=

(p2 d2 ~-~+2p

_~p

g2) - l ( 1 + 1) + ~-~ R t ( r ) . (16)

J Daboul,M M Nteto/PhystcsLettersA 190(1994)357-362

360

Integrals o f products o f Bessel functions are well studmd and are comphcated when the orders and arguments approach each other [ 11 ]. However, this integral is convergent and is given in R e f [ 11], p 407, Eq (1).

1 r(½ + l/a) _r'((l + ½)/lal- i/a) I1 = 2~,/2 F(1 + I/a) F(1 + (l + ½)/lal + l/a)' (21) if the following two conditions are satisfied, 2l+____11 + 1 > - 2 + 1 > 0 Fig 4 The effective potential obtained from Eq (14) for v = 4 in umts of ½£0, as a function of p = r/a The form

It3 U(ff) = 4/p 2 - 1/p 4 The above is a well-known differential equation o f mathematical physics [10]. F o r v # 2 or a # 0, the solution can be directly gdven as 2g

1

Rl(r)=-~-~J((2l+l)/lv-2l)(lv_2,-~O,-2)/2) ) = pl/2J,(t+,/2)/lal)(l~[p~ ),

a # 0

(17,

The other possible solution o f Eq (16), revolving the functions Y, is ruled out on physical grounds (see the Appendix o f Ref. [ 9 ] ) Also, note that the power in the argument o f the solution is not an absolute value o f a (The singular, free, v = 2 or a = 0 case will be discussed in Ref. [9 ].) We now find out under what circumstances these states are normahzable. The normalization constants for the wave functions would have to be o f the form oo

I"VF 2 =

jfr2dr'2__ ---~'-J((l+l/2)/llal)( g )

(18)

0

Changing variables first from r to p, and then from p to z = g/lalP ~, and being careful about the hmits o f integration for all #, one obtains

a3(g) Nt -2 = ~ -~

2/1.

It,

(19)

d z 1"2 z(l'-~'-f/~;a((l+l/2)/ll~l)(Z'.

(20)

where

It

=

0

lal

a

(22)

(In obtaining the final result in Eq. (21) the doubling formula for F ( 2 z ) was used.) Eqs (21) and (22) lead to two sets of normalizable states. The first is when a>0orv>2,

1>½.

(23)

These are ordinary b o u n d states and result because the effective potential asymptotes to zero from above, as m Fig. 4. In this case, for E = 0, the wave function can reach infinity only by tunneling through an infinite forbidden region. That takes forever, and so the state is bound. Note that the condition on l in Eq. (23) is the m i n i m u m nonzero angular m o m e n t u m allowed in q u a n t u m mechanics, lm,n = 1 This agrees with the classical orbit solution which is b o u n d for any nonzero angular m o m e n t u m [8 ] Notice that the above E = 0 solutions exist for all g2 > 0, and not just for discrete values o f the coupling constant. The reason for this surprising result is the scaling property o f power-law potentials. A change o f the coupling constant, g2, by a positive factor, to a2g 2, can be accounted for by changing the argument o f the wave function from g/lalP ~ to ag/lalp ~ This is essentially a change o f the length scale. F o r - 2 ~< v ~< 2 o r - 2 ~< a ~< 0 (as well as the solutions with l = 0 and a > 0 or v > 2) the solutions are free, continuum solutions. However, there is one remaining class of normahzable solutions which is quite surprising F o r any l and all v < - 2 or a < - 2 , the reader can verify that the conditions o f Eq. (22) are also satisfied. Thus, even though one here has a repulsive potential that falls off faster than the inverse-harmonic oscillator and the states are not bound, the solutions are normallzable!

J Daboul, MM Nteto/PhyswsLettersA 190 (1994) 357-362 The corresponding classical soluUons yield mfimte orbits, for which the particle needs only a finite time to reach infinity [8 ]. But it is known that a classical potential which yields trajectones wRh a finite travel time to infinity also y~elds a discrete spectrum m the quantum case [ 12 ]. This conclusion is m agreement with the situation here. Although normahzable quantum solutions exmt not just for E = 0 but also for a continuous range of E, by imposing special boundary condlt~ons a discrete subset can be chosen which defines a self-adjomt extension of the Hamlltonlan [ 1315] Th~s system has many other interesting features, both classically and quantum mechamcally. We refer the reader elsewhere to d~scussions of these properties

[8,9].

II,D =

j

dz 2 Z(1+2/$)Ji(l+D/2-1)/ll~D (Z)

(27)

0

We see that the above integral is equal exactly to that m (20), except that 1 is replaced by the effective quantum number l,fr = l + ½(D - 3).

(28)

Therefore,

1 r(½ + l/it) It,o = 2/tl/2 F(1 + 1~It) F ( ( l + 0 / 2 - 1)/litlr ( 1 + (1 + 0 / 2 - 1)/litl

×

1/it) + 1/it)'

(29)

which Is defined and convergent for 21+D-2

One can easily generahze the problem of the last section to arbitrary D space dimensions. Doing so y~elds another surprising physical result. To obtain the D-dimensional analogue of Eq. (16), one simply has to replace 2p by ( D - 1 )p and 1(I + 1 ) by 1(l + D - 2) [16],

0 = fp2 d 2 ~,

-d--~ + ( D -

1)p~p

+ ) -r~)

-l(l+D-2)

Rt,n.

(24)

The solutions also follow similarly as

RI,D = ~

1

2g

J((2l+O-2)/iv-2l) (IV _ 21~'Uv-2)/2) )

1 pD/2-1 Jt(I+DI2-1)II#I)(],' ~)

•

(25)

To find out which states are normahzable one first has to change the integration measure from r 2 dr to r o - 1 dr and again proceed as before. The end result is that if the wave functions are normal~zable, the normal~zatlon constant ~s given by

-~

-~]

It.o,

+ 1>-2 + 1 >0.

IItl

4. Bound states in arbitrary dimensions

where

361

This yields the surprising result that there are bound states for all v > 2 or It > 0 when l > 2 - ½D. Exphcitly th~s means that the minimum allowed 1 for there to be zero-energy bound states are D=2, D=

/ram = 2 , 3,

/man = 1,

D=4,

lmm= 1,

D>4,

lmm=0

(31)

This effect of d~menslons Is purely quantum mechanical and can be understood as follows: Classically, the number of dlmens~ons revolved in a central potential problem has no mtnnslc effect on the dynamics. The orbit remains in two dimensions, and the problem ~s deoded by the form of the effective potential, U, which contains only the angular momentum barher and the dynamical potential. In quantum mechanics there are actually two places where an effect of d~menslon appears. The first is in the factor l(l + D - 2) of the angular-momentum bamer. The second is more fundamental. It ~s due to the operator Uqm =

(26)

(30)

It

D-ld p dp"

(32)

The contrlbuUon of Uqm to the "effective potential" can be calculated by using the ansatz

J Daboul, M M Nteto/Physws LettersA 190 (1994) 357-362

362

1 Rt,D(p) -- p~D_l)/2"Xl,D(p)

(33)

This transforms the D-dimensional radial Schrodlnger equation into a one-dimensional Schrodinger equation,

0 =

--~

+ Ut,D(p) XI,D

(34)

In Eq (34), the effective potential Ut,D(p) is given by

Ut,o(p) = --

(D-1)(D-3) 4p 2

l¢ff(l~ff + 1 ) p2

.q.-

V(p),

+

l(l + D - 2) p2 + V(p) (35)

with left given in Eq. (28) Since the Schrodinger equation (34) depends only on the combination lcff, the solution Xt,o(P) does not depend on l and D separately This explains, in particular, the values of lmm given In Eq. (31 ) Although the above ansatz is well known, the dimensional effect has apparently not been adequately appreciated One reason may be attributed to the fact that in going from D = 3 to D = l, leff remains equal to ! However, in our problem this effect leads to such a counter-intuitive result, that it cannot be overlooked The dimensional effect essentially produces an additional centrifugal barrier which can b i n d the wave function at the threshold, even though the expectation value of the angular m o m e n t u m vanishes. Note that this is in distinction to the classical problem, where there would be no "effective" centrifugal barrier to prevent the particle from approaching r ~ e¢

5. Summary After obtaining exact, E = 0 solutions of the SchrodInger equation for power-law potentials, we demonstrated three interesting effects ( 1 ) There exist b o u n d states at the threshold, for all l > 0 and all 7 > 0 These states persist if one changes the coupling constant 7 by a positive factor (In contrast, E -- 0 b o u n d states exist for the spherical well and the focusing potentials only for very special values of the coupling constants, and never for all l > 0 simultaneously )

(2) There exist normahzable solutions for u < - 2 , i.e., for highly repulsive potentials, singular at p ~ oc (3) For higher-space dimensions, each additional dimension adds a half unit to the effective angularm o m e n t u m q u a n t u m number, ldr, of Eq. (28) An effective centripetal barner, solely due to this dimensional effect, i.e, for L 2 = 0, is capable of producing a b o u n d state. This result is a remarkable manifestation of q u a n t u m mechanics and has no classical counterpart

References [ 1] A Messiah, Quantum mechamcs, Vol 1 (Wdey, New York) Ch IX, Art 7, p 355 [2] J Orear, J H Rosenfeld and R A Schluter, note compilers, Nuclear physics A course given by Ennco Fermi, revised Ed (University of Chicago Press, Chicago, 1950) Ch III, Sec. C, p 58 [3] BW Downs, Am J Phys 30 (1962) 248 [4] L I Schlff, Quantum mechamcs, 3rd Ed (McGrawHill, New York, 1968) [5] Nanonal Bureau of Standards, Tables of sphencal Bessel funcnons, Vols. I, II (Columbia Umv Press, New York, 1947) [6] Y N Demkov and V N Ostrovskn, Soy Phys JETP 35 (1972) 66 [Zh Eksp Teor Fiz 62 (1972) 125] [7] Y Katagawa and AO Barut, J Phys B 16 (1983) 3305, 17 (1984) 4251 [8] J Daboul and M M Nleto, Am J Phys, on the classical solunons of the E = 0, power-law system, m preparation [9] J Daboul and M M Nleto, Am J Phys, on the quantum solutions of the E = 0, power-law system, m preparation [ 10 ] W Magnus, F Oberhettlnger and R P Sore, Formulas and theorems for the specml functions of mathematical physics, 3rd Ed (Springer, Berlin, 1966) p 77 [11] G N Watson, A treatise on the theory of Bessel functions, 2nd Ed (Cambridge Unlv Press, Cambridge, 1966) Secs 13 41-13 43 [ 12] A S Wlghtman, m 1964 Carg~se lectures m theoretical physics, Vol 2, ed M l_~vy (Gordon and Breach, New York, 1967) p 171, see Sec VIII, p 262 [13] K M Case, Phys Rev 80 (1950) 797 [14] F A Berezm and MA Shubln, The Schrodmger equaUon (Kluwer, Dordrecht, 1991 ) [15] C Z h u a n d J R Klauder, Found Phys 23 (1993)617, Am J Phys 61 (1993)605 [16] M M Nleto, Am J Phys 47 (1979) 1067