SUSY-WKB is neither exact nor never worse than WKB for all solvable potentials

Volume 247, number 2,3 PHYSICS LETTERS B 13 September 1990

S U S Y - W K B is neither exact nor never worse than WKB for all solvable potentials

D a v i d D e L a n e y and Michae l M a r t i n N ie to Theoretical Division, T-8, MS B285, Los Alamos National Laboratory, Los Alamos, NM 87545, USA

Received 28 May 1990; revised manuscript received 19 June 1990

The supersymmetric WKB (SUSY-WKB) approximation has been shown to be exact for wide classes of solvable potentials. This has led to folklore that SUSY-WKB is (i) exact for all exactly solvable potentials, and (ii) is never worse than the ordinary WKB approximation for these potentials. We emphasize that both statements are false.

For a dimensionless Schr6dinger equat ion o f the form

a d z 2 + V ( z ) ~ 'n (z )=enu /~ (z ) , (1)

the regular WKB approx imat ion for the eigenvalues states that

x / ~ f [E _ v ( z ) ] , / 2 d z = n + ½ , (2) ZL

where the ZL,R are de te rmined by the zeros o f the integrand for a given value of e,. This approx imat ion has a basis going back to the Bohr -Sommer fe ld quant iza t ion condi t ion in the old quan tum theory [ 1 ], and has been widely s tudied [2,3 ]. The eigenvalues ob ta ined from eq. (2 ) are exact for the harmonic oscillator, and are a good approx imat ion for many solvable potentials . Fur thermore , i f one adds a specific Rosenzweig and Krieger ( R K ) [ 4 ] correct ion term to v in eq. (2) , an exact set o f eigenvalues can be obta ined for many potent ials v [ 4 ]. Then, in 1984, Comtet , Bandrauk, and Campbel l (CBC) [5] made a startl ing discovery based on supersymmetry (SUSY) in quan tum mechanics.

The results of SUSY-QM can be stated thusly [ 6-8 ]: I f one has a potent ia l Vwhose ground-state eigenvalue has been shifted to zero, then the potent ia l can be wri t ten as

1 { [W, ( z ) ]2_w , , ( z ) } , (3) v= v+ (z) = S

where W is related to the ground-state eigenfunction by

go : e x p [ - W(z) ] . (4)

There is a compan ion potent ia l V_,

v_ (z) = 1 { [ w ' (z) ] 2 + w" (z) }, (5) OL

which has the same eigenvalues as V+, but with the ground state (Eo = 0) removed. V+ are the supersymmetr ic par tner potentials .

0370-2693/90/$ 03.50 © 1990 - Elsevier Science Publishers B.V. ( North-Holland ) 301


CBC discovered that if you made the following new "SUSY-WKB" approximation,

x/-~f [E,-Vs(z)l/2]dz=n, Z L

1 [ w , ( z ) ] 2 V s ( z ) = a

(6)

(7)

it was exact for many solvable potentials. The difference between eqs. (2) and (6) is related to the fact that eq. (6) comes from a different expansion in h than does eq. (2) [5,9].

In a sense, from supersymmetry, CBC obtained a method to systematically obtain an RK-type correction [4 ] that makes WKB exact for many potentials, which are shifted so that the ground-state energy is zero (the ½ in eq. (2) is removed).

The statement that Eo= 0 for SUSY-WKB is exact for all potentials. Consider eq. (4). Since this represents the ground state, ~Uo has no nodes between the endpoints at _+ co, so W( _+ ~ ) = + ~ . Thus, Wmus t have at least one local min imum (and possibly more extrema). This in turn implies that W' must go through zero at these extrema. Thus, at these extrema, Vs = 0 and is itself at a local minimum, meaning Eo = 0 ~.

The method of CBC was applied to wide classes o f solvable potentials [ 10,11 ], always yielding that SUSY- WKB is exact in these cases. From these observations, some authors came to believe that SUSY-WKB is (i) exact for all known solvable potentials, which would mean that for these potentials (ii) SUSY-WKB is never worse than WKB [ 12,13 ]. We here point out that the above beliefs are incorrect. We do this using three exactly solvable potentials which were in the literature before the CBC paper appeared.

In fact, a counterexample to claim (i) has been known for some time. Soon after the paper of CBC appeared, one of us ( M M N ) proposed to those authors that the inverse harmonic oscillator (our first example below) would provide a different test o f " h o w exact is SUSY-WKB?". One of CBC's collaborators reported that he had found a deviation from exactness [ 14 ], but to our knowledge this finding was never published. Also, Khare and Varshni [ 15 ] recently demonstrated the nonexactness o f SUSY-WKB for particular examples of the Natanzon potential and potentials related to, but not the same as, the inverse harmonic oscillator.

The three exactly solvable potentials we use are obtained using the Gel ' fand-Levi tan inverse method [ 16 ], as espoused by Abraham and Moses [ 17 ]. Among other things, the inverse method allows one to obtain a new class of potentials from a given original potential, by removing any specific state. In particular, one can obtain a potential with the ground state removed ~2.

Restricting ourselves to the case of removing the ground state from an exactly solvable potential v with eigenvectors q/j and eigenvalues ej, j = 0, 1, 2, ..., the inverse method yields a new, exactly solvable, potential V~, with eigenvectors Z, and eigenvalues e,, n = 0, 1,2 .... , given by

Vl=U"t-Vl ,

lfl 1 =2 d K(z,z) ,

K(z,Y)=~o(Z)~Vo(Y) ( i ~(x) dx)-' z

(8)

(9)

(10)

~ For finite endpoints, this result still holds, but the argument is more complicated. ~2 The reader might wonder if this procedure is not the same thing as SUSY. In fact, it is not, although it is mathematically related:

SUSY and the inverse method for removing the ground state are different solutions to a Ricatti equation [ 18,19 ].

302


Z,,(z)=~',,+I(Z)+ i K(z,y)v,,+l(y) dy, ( I 1 )

e. =~n+] • (12)

By now subtracting the ground state energy eo from VI, one obtains a potential which satisfies the criterion to be a SUSY bosonic potential V+,

V+ = Vi-eo =v+ vl -eo . (13)

We now study three potentials obtained in this manner. The inverse harmonic oscillator. Using the inverse method on the harmonic oscillator potential

V : IZ2

with a = 2 , one obtains [ 17,20]

v~ =4~(~-z ) ,

oo --1

@(z)=exp( - z2 ) [~z l /2er f c ( z ) ] - l=exp( - z2 ) (2 Iexp ( - t2 )d t ) , z

yielding V~ = v+ v~ of fig. 1. The eigenvectors are

•n(z) : ~¢n+ 1 ( Z ) - - [ 2 / ( n + 1 ) ]'/2f~(z)~.(z) ,

where the ~u. are the ordinary harmonic oscillator wave functions

V.(z) = (lrl/22"n!)-1/2 exp( - ½z2)H.(z) .

Now subtracting the ground state energy of VI, 3, one obtains

3__1 ) 2 V+=v+v , -~= ~(W' ½(W")=½z2+4@(@-z)-~,

E . = n = 0 , 1, 2 . . . . .

Therefore,

(14)

(15)

(16)

(17)

(18)

(19)

(20)

I - 3

V '-4

n= 3

n : 2

_

n - I

/ . / /

~ n=O

- 2 - I 0 I 2 3

Fig. 1. The harmonic-oscillator potential, v(z), is the thin curve. The additional potential, vl (z), is the dashed curve. The complete inverse harmonic oscillator potential, V](z), is the thick curve. The number eigenvalues are also indicated.

303


Z o ( z ) ~ e x p ( - W )

implies

W ' = ( z - 2 0 ) + 1 / ( ~ - z ) ,

W " = 1 - 4 0 ( O - z ) - [ 2 O ( ~ - z ) - 1 ] / ( ~ - z ) 2 .

(21)

(22)

(23)

In table 1 we show the SUSY-WKB and ordinary WKB approximations to the energy levels for 0 ~< n ~< 10 ~3. First note that SUSY-WKB is not exact. More surprisingly, for n >/3, ordinary WKB is a better approximation than SUSY-WKB. (This demonstrates our two assertions. ) However, in our second and third examples, SUSY- WKB will be better than ordinary WKB. (The fundamental potentials of these examples are defined only on [0, oo) and are not symmetric.) This emphasizes that the answer to which expansion in h is superior depends on the potential.

The inverse isotonic oscillator. The isotonic oscillator can be thought of as the radial equation for the three- dimensional harmonic oscillator, with non-integer angular momen tum allowed. Its dimensionful potential is h a a Z v 2 ( a x - 1 / a x ) 2 / 2 m [21]. Taking units where the energy eigenvalues are ho~n, 2 u a 2 h / m e o = l , and y = z 2 = ua 2x2, one obtains a dimensionless Schr6dinger equation of the form (2) with c~- 4 and u 2 = it (it + 1 ):

v ( z ) = ~ ( z - u / z ) 2, (24)

~'n = Nn exp( - ½y)y~X+')/2L~+'/2) (y) ,

N, = [ 2 u ' / 2 F ( n + 1 ) / F ( 2 + ~ + n ) ] '/2 , (25)

G = n + ½ ( i t + 3 - v) . (26)

Applying the inverse method to eq. (24) yields a family of potentials as a function of v, with the incomplete gamma function, F(it, y), playing a role similar [21 ] to that of the error function in the inverse harmonic oscillator case [ 17,20 ]. However, for the special value

v = ½x/3, 2 = ½ , (27)

Table 1 Inverse harmonic oscillator and inverse isotonic oscillator potentials.

En=n Inverse harmonic oscillator Inverse isotonic oscillator

SUSY-WKB WKB SUSY-WKB WKB

0 0.0 0.000416 0.0 -0.10078 1 1.001399 1.001528 1.00590 0.92551 2 2.000143 2.000466 2.00486 1.92988 3 2.999822 3.000116 3.00386 2.93135 4 3.999788 4.000019 4.00312 3.93200 5 4.999813 4.999993 5.00257 4.93234 6 5.999844 5.999987 6.00217 5.93254 7 6.999871 6.999986 7.00186 6.93267 8 7.999892 7.999988 8.00161 7.93275 9 8.999909 8.999989 9.00142 8.93281

10 9.999922 9.999990 10.00126 9.93285

~3 The numerical calculations were done with two independent programs, using Mathematica and Fortran, respectively.

304


the incomple te g a m m a func t ion reduces to a po lynomia l , a n d one has the s imple result ~4

V 1 = (3"1"1"22)22/( 1 "t-2" 2 ) 2,

V x = v + v l •

(28)

(29)

(See fig. 2. ) The e igenvectors a n d e igenvalues are [ 21 ]

Z,, = ~6,+ l + [ N , , + l / ( n + 1 ) ] exp( - ½z 2) [ z 2 / ( 1 + z 2) ]z7/2L(,,2)(z 2) ,

G = n + Z - ¼ v / 3 , n = 0 , 1 , 2,. . .

By subt rac t ing the g round state energy ( 2 - ~w/3), one ob ta ins a SUSY poten t ia l o f the form

V+ = 1 (z2.1. 3/4z 2) - 2 , 1 , (3Z2+Z4) / ( 1 + Z 2) 2,

2Z6+3Z4- -Z2- - 6

2 z ( z 2 + 1 ) ( z 2 + 2 ) '

2 z l ° + 1 5 z 8 + 3 2 z 6 + 5 1 z 4 + 5 2 z 2 + 12 W" ~-- 222 (Z2"[ - 1 )2(2"2"1"2)2

(30 )

(31)

(32 )

(33)

(34)

In table 1 we show the S U S Y - W K B an d regular W K B a p p r o x i m a t i o n s to the exact e igenvalues En = n, for 0 ~< n ~< 10. Once again S U S Y - W K B is n o t exact, a l though this t ime it is be t te r t han regular WKB.

Inverse hydrogen a tom. As wi th the last example , we cons ider a radia l equa t ion . Th i s t ime it is o f the hydrogen a tom, with integral angu la r m o m e n t u m , l. The equa t i on can be pu t in an a = 1 d imens ion less form [ 22 ] wi th

Note that the value v = ½ ~/3, which makes the incomplete gamma function a polynomial, corresponds to a half-integer angular momentum, 2= ½, in the radial equation.

v

5

o

- - n = 4

- - n = 3

- - n = 2

n=l

I 2 5

Fig. 2. For the special case 2 = ½, the thin curve is the isotonic oscillator potential, v(z). The additional potential, v~ (z), is the dashed curve. The complete inverse isotonic oscillator potential, Vr(z), is the thick curve. The number eigenvalues are also indicated.

0 . 5 0

0.25

> 0.00

-0 .25

-0.50 0.0

/ \ / \ / \ / \

/ "-

1.0 2.0 3 .0 4 .0 5 .0

t = Z vz

Fig. 3. For the case l= 0, the short-dashed curve is the hydrogen atom potential, v(z) versus t=z ~/2. The additional potential, v~(z), is the long-dashed curve. The complete inverse hydrogen atom potential, V~ (z), is the continuous curve. The ground state eigenvalue of the hydrogen atom is indicated by the dotted line going to the open circle. The ground state of the inverse hydrogen atom is indicated by the dotted line going to the closed circles.

305


v= - 1 / z + l ( l + 1 ) / Z 2 ,

~n=Nntp 1+1 exp( ! . ~ L ( 2 / + l ) , - 2 y J . - I - l ( P ) , p = z / n ,

1 N.,~ = ~5 [ F ( n - l ) / 2 F ( n + l+ 1 ) ],/2,

E . = - l / 4 n 2, n = l + l , l + 2 , . . . .

Applying the inverse method to this potential yields [22 ] #5

v, (l) = 2 ~ [ 0 ~ - 1 / (1+ 1 ) + 2 ( l+ 1 ) / z ] ,

( 2/+2 zj J~. )-1 ~ t = z 2(t+l) ( 2 l+2 ) ! j~o ( /+1)2l+3-J "

Now taking the special case l= 0, one obtains the potential (already discussed in ref. [7 ] )

ul = z ( z + 2 ) / ( 1 + z + ½ z 2 ) 2 ,

V~ = - l / z + v~ = V J [ - ~ ) I .

(See fig. 3. ) The eigenvectors are once again defined by eqs. (10) and ( 11 ). Most importantly,

Zo = [exp( - ~z) /96 ( 1 + z + ½z 2) ] ( 2 4 z + 18z2+ 16z3+z 4) ,

and the eigenvalues are

e , , = l / 4 ( n + 2 ) 2, n = 0 , 1 , 2 , . . . ,

where we have renumbered so that the ground state has n = 0. By now subtracting the ground state energy, - 1 , from V,, one finds

V+ = - l l z + z ( z + 2 ) / [ 1 + z + ½z 212+~6,

1 6 E , , = n ( 4 + n ) / ( 2 + n ) 2, n = 0 , 1 , 2 ....

From eq. (41 ) one has

z 6 - 16z4- 56z 3 - 1 0 8 z 2 - 2 4 0 z - 192 W ' =

4 z ( z 2 + 2 z + 2 ) ( z 3 + 6 z Z + 1 8 z + 2 4 ) '

z 1o+ 12z 9 + 64z8 + 208z 7 + 584z 6 + 1632Z5 + 3936Z 4+ 6720Z 3 + 7056Z 2 + 4032Z+ 1152 W " = 2

(35)

(36)

(37)

(38)

(39)

(40)

(41)

(42)

(43)

(44)

(45)

(46) Z2(Z2+2Z+2)2(Z3+6Z2+ 1 8 Z + 2 4 ) 2

which agrees with eq. (43). In table 2 we give the SUSY-WKB and regular WKB approximations for 0 ~< n ~< 10. Once again, SUSY-WKB is not exact, but is superior to regular WKB.

In fact, regular WKB is quite bad for n = 0 , 1. This is due to the singularity of the Coulomb potential. The Langer correction [25 ] to the angular momentum term,

l ( l+ 1 )/2 "2-t' ( l+ ½ )2/z2 , (47)

is meant to improve WKB agreement. Here that would mean one adds

~s This potential was obtained to elucidate the difference between SUSY and the inverse method in a study of supersymmetry in atomic systems [ 22-24 ]. It comes from integer angular momenta, and does not involve any transcendental functions, which arose previously, However, in the latter part of the atomic supersymmetry study [24 ], effectively non-integer angular momenta were used to parame- trize quantum defects in the Sehr6dinger equation.

306

Volume 247, number 2,3 PHYSICS LETTERS B

Table 2 Inverse hydrogen atom (l= 0 ). For ease of study, the eigenvalues are multiplied by 16.

13 September 1990

1 6 E , , = n ( 4 + n ) / ( 2 + n ) 2 SUSY-WKB WKB WKB+ Langer term

0.0 0.0 -7.52025 -0.10562 0.55556 0.55356 0.26301 0.52865 0.75 0.74908 0.64309 0.73917 0.84 0.83952 0.78896 0.83458 0.88889 0.88861 0.86058 0.88579 0.91837 0.91819 0.90105 0.91643 0.9375 0.93738 0.92613 0.93621 0.95062 0.95053 0.94276 0.94971 0.96 0.95994 0.95434 0.95935 0.96694 0.96690 0.96273 0.96645 0.97222 0.97219 0.96901 0.97184

ULanger : 1//422 (48 )

to eq. (43 ) . Then , as is shown in table 2, W K B wi th the Langer co r rec t ion i m p r o v e s o v e r regular W K B . H o w -

ever, W K B wi th Langer co r rec t ion does no t y ie ld the exact answer, as it does wi th the o rd ina ry C o u l o m b po ten-

tial, and it still is no t as good as S U S Y - W K B .

In conc lus ion we r e e m p b a s i z e tha t there are en t i re classes o f new, exact ly solvable po ten t ia l s which can be

genera ted f r o m the " s t a n d a r d " solvable po ten t ia l s using the inverse me thod . Fur ther , there are c o m p l e m e n t a r y

classes to the inverse po ten t ia l s which can be ob t a ined using the D a r b o u x m e t h o d espoused by Luban and

Pursey [26] . As we h a v e shown here, and as has been no t ed in o the r contex ts [20,21,27 ], these new exact ly

solvable po ten t ia l s of ten have d i f fe ren t p roper t i es than the s t andard potent ia ls . There fore , one mus t be wary o f

m a k i n g general s t a t emen t s on exact ly so lvable po ten t ia l s which do not take these new solut ions in to account .

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SUSY-WKB is neither exact nor never worse than WKB for all solvable potentials