VOLUME60, NUMBER 1 P H Y S I C A L R E V I E W L E T T E R S 4JANUARY 1988

Supersymmetry and Double-Well Potentials

Wai-Yee Keung, Eve Kovacs, and Uday P. Sukhatme Department of Physics, University of Illinois at Chicago, Chicago, Illinois 60680

(Received 24 July 1987)

The ideas of supersymmetric quantum mechanics are applied to the tunneling problem for double-well potentials. We evaluate the tunneling by developing a systematic perturbation expansion whose leading term is an improvement over the standard WKB result. We find that the perturbation series converges rapidly.

PACS numbers: 73.50.Gr, 03.65.Ge, 11.30.Pb, 71.70.Gm

Recently, there have been many applications of super-symmetry to quantum-mechanical potential prob­lems,1"12 all stemming from one key observation: Given any potential of interest K-Gc), unbroken supersym­metry allows one to construct a partner potential, K+Oc), with the same energy eigenvalues (except for the ground state). Thus, in order to determine the energy eigenstates one can use either the original potential, K-(x) , or its supersymmetric partner, V+(x). This freedom of choice has been exploited to provide a deeper understanding of all known analytically solvable poten­tials,2""5 and to improve several approximation tech­niques such as the WKB method6,7 and large-TV expan­sions. 8

In this Letter we focus on double-well potentials, which have extensive applications in many branches of physics. Usually the quantity of interest is the energy difference t^Ex—Eo between the lowest two eigen­states, and corresponds to the tunneling rate through the double-well barrier. The quantity t is often small and difficult to calculate numerically, especially when the po­tential barrier between the two wells is large. Here, we reexamine carefully and extend considerably previous claims1112 that supersymmetry facilitates the evaluation of t. Indeed, using the supersymmetric partner potential K+Oc), we obtain a systematic, highly convergent per­turbation expansion for the energy difference t. The leading term is more accurate than the standard WKB tunneling formula, and the magnitude of the nonleading terms gives a reliable handle on the accuracy of the re­sult.

First, we briefly review the standard approach for determining t in the case of a symmetric, one-dimen­sional double-well potential, V-(x), whose minima are located a t x " ±JC0. We define the depth, D, of V-(x) by D = V-(0) — F-Gco). An example of such a poten­tial is shown in Fig. 1. For sufficiently deep wells, the double-well structure produces closely spaced pairs of energy levels lying below K-(0). The number of such pairs, n, can be crudely estimated from the standard WKB bound-state formula applied to V-(x) for x > 0:

nn=rc{V-{0)-V-{x)V,2dx, (1)

where xc is the classical turning point corresponding to energy K-(0). Throughout this paper we have chosen units where h —2m — I. We shall call a double-well po­tential "shallow" if it can hold at most one pair of bound states, i.e., n^l. In contrast, a "deep" potential refers to n ^ 2.

The energy splitting t of the lowest-lying pair of states can be obtained by a standard argument.13 Let X(x) be the normalized eigenfunction for a particle moving in a single well whose structure is the same as the right-hand well of V-(x) (i.e., x > 0). If the probability of barrier penetration is small, the lowest two eigenfunctions of the double-well potential V-(x) are well approximated by

vfciHx)-\X(x)±Z(-x)]/J2. (2)

By integration of Schrodinger's equation for the above eigenfunctions, it can be shown that13

t=Ei-E0-4Z(Q)Z'(O), (3)

where the prime denotes differentiation with respect to x. This result is accurate for "deep" potentials, but be­comes progressively worse as the depth decreases. Use

FIG. 1. A "deep" double-well potential, K-Gc), and its su persymmetric partner potential, V+(x).

© 1987 The American Physical Society 41

VOLUME 60, NUMBER 1 P H Y S I C A L R E V I E W L E T T E R S 4 JANUARY 1988

of WKB wavefunctions in Eq. (3) yields the standard result:

rwKB = U 2 K ' l ( x o ) 3 1 / 2 / ^ e x p ( - 2 j o " V - U ) - F - ( x 0 ) ] 1 / 2 ^ ) .

The same result can also be obtained via instanton tech- \ niques.14

Using the supersymmetric formulation of quantum mechanics for a given Hamiltonian, H- — —d 2/dx2

+ K-(x) , and its zero-energy ground-state wave func­tion yoCx), we can construct K+Cx), the supersymmetric partner potential of V-(x), as follows:

(4)

K + U ) - K - ( x ) - 2 W / r f x ) ( y o / y o )

= - K - ( x ) + 2(y^Vo)2 (5)

Alternatively, in terms of the superpotential W(x) given by

wGc) = -[^6/vo],

we can write

V± (x)=W2(x) ± dW/dx.

(6)

(7)

paired), then the spectrum of V+ is well separated. In this case, V+ is relatively structureless and simpler than V-. Previous papers11,12 have implicitly treated just the case of shallow potentials, and, not surprisingly, have found that the use of supersymmetry simplifies the evalu­ation of the energy difference t. In contrast, let us now consider the case of a deep double well. (See Fig. 1.) Here, the spectrum of V+ has a single unpaired ground state followed by doubled excited states. In order to pro­duce this spectrum, K+ has a double-well structure to­gether with a sharp "£-functionlike" dip at x =0. This central dip produces the unpaired ground state, and be­comes sharper as the potential V-(x) becomes deeper.

As a concrete example, we consider the class of poten­tials whose ground-state wave function is the sum of two Gaussians, centered around ± xo,

It is easy to show that the energy spectra of the poten­tials V+ and V - are identical, except for the ground state of V - which is missing from the spectrum of K+.1

Hence, for the double-well problem, we see that if V-(x) is "shallow" (i.e., only the lowest two states are

W(x) =2{x -x0tanh(2jcxo)},

V± (x) =4{x -x 0 tanh(2xx 0 )} 2 ± 2{l - 2XQ sech2(2xx0)}.

y/oKX)ae +e ° . (8)

Throughout this paper we have chosen the variables x and xo to be dimensionless. The corresponding superpo­tential W(x), and the two supersymmetric partner po­tentials V ± (x), are given respectively by

(9)

(10)

The minima of V-(x) are located near ± x o and the r well depth (in the limit of large xo) is Z)—4xo- We il­lustrate V-(x) and F+(x) in Figs. 2(a) and 2(b), re­spectively, for the choices xo = 1.0 and xo + 2.5. We see that in the limit of large xo, for both V-(x) and F+(x), the wells become widely separated and deep and that V+ (x) develops a strong central dip.

The asymptotic behavior of the energy splitting, t, in the limit xo—• °° can be calculated from Eq. (3), with X(x) given by one of the (normalized) Gaussians in Eq. (8). We find that

f~8x0(2//r)1/2e -2x$ (11)

The same result can be obtained by observing that K- (x)—*4( |x | — xo)2 — 2 as xo—* °°* This potential has a well known15 analytic solution with the lowest two energy levels located at Eo** — 4xo(2/^)1/2exp( — 2xo) and Ei - +4x0(2/;r) 1/2exp( - 2x§).

We now turn to the evaluation of t via the ground-state energy of the supersymmetric partner potential F+(x). In general, since V+(x) is not analytically solv­able, we must solve an approximate problem and calcu­late the corrections perturbatively. The use of supersym­metry, coupled with the observation that the magnitude

of t is in general small, allows us to construct a suitable unperturbed problem. Consider the Schrodinger equa­tion for K+(x) with E =0. From supersymmetry [Eq. (5)] we see immediately that 1/yo is a solution. Since t is small, we expect this solution to be an excellent ap­proximation to the correct eigenfunction for small values of x. However, 1/y o is not normalizable and hence is not acceptable as a starting point for perturbation theory. One possibility is to regularize the behavior artificially at large | x \.11 This procedure is cumbersome and results in perturbation corrections to the leading term which are substantial. Instead, we choose as our unperturbed prob­lem the second linearly dependent solution of the Schro­dinger equation,16

* ) - — f y/o(x')dx\ x > 0 , (12)

and ^(JC)3BE0( —x) for x < 0 . Clearly, 0(x) is well behaved at x a s s ± ° ° and closely approximates 1/Vo at small x—thus we expect it to be an excellent approxima­tion to the exact ground-state wave function of V+(x) for all values of x. The derivative of 0(x) is continuous except at the origin, where, unlike the exact solution, it

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VOLUME60, NUMBER 1 P H Y S I C A L R E V I E W L E T T E R S 4 JANUARY 1988

-10

- 2 0

i i i i i i i i i I i i i t i i i i i i | i i i i i i i i i

xo=1.0

x0=2.5

I I I 1 1 I I I I 1 I I I I 1 I I I I I I I I 1 I I I I I I I I I I I I I I

- 4 - 2 0 2 4 - 4 - 2 0 2 4 X X

(a) (b)

FIG. 2. (a) V-(x) and (b) V+(x) as given by Eq. (10) for xo — 1.0 and xoa"2.5.

has a discontinuity </> |0+ —^lo- "• —2yo(0). Hence 0tx) is actually a zero-energy solution of the Schro-dinger equation for a potential VQ(X) given by

V0(x)-V+ix)-4vi(0)8(x)9 (13)

where we have assumed that yoGc) is normalized. We calculate the perturbative corrections to the ground-state energy using AVsa*+4y/o(0)8(x) as the perturbation. Note that the coefficient multiplying the S function is quite small so that we expect our perturbation series to converge rapidly.

For the case of a symmetric potential such as K+Gc), the perturbative corrections to the energy arising from AV can be most simply calculated by use of the log­arithmic-perturbation-theory17 formulation of the usual Rayleigh-Schrodinger series. The first- and second-order corrections to the unperturbed energy E ""0 are

(°-^k' *C*)« fVt*')*; 2^(0)

Jo £ ( 1 )g0c')

dx\

(14a)

(14b)

For our example, we evaluate numerically these correc­tions in order to obtain an estimate of t.

The results are shown in Fig. 3 for values of xo = 2. Estimates of t correct to first, second, and third order calculated from logarithmic perturbation theory are compared with the exact result for K+, obtained by the Runge-Kutta method. The asymptotic behavior of t given by Eq. (11) is also shown. This asymptotic form can also be recovered from Eq. (14a) by a suitable ap­proximation of the integrand in the large-xn limit. Even

w i

II

f0(x) oc exp[-(x-x0)8]+exp[-(x+x0)

2]

1* order 2nd order 3 rd order; exact asymptotic form: 8x0(2A) l /2exp(-2x0

2)

i i — I i i i i _

0.5 1 i i i v i

FIG. 3. Various estimates of t as a function of xo for the po­tentials given by Eq. (10).

for values of xn^ lA/2 , in which case V-(x) does not exhibit a double-well structure, the approximation tech­nique is surprisingly good. The third-order perturbative result and the exact result are indistinguishable for all values of xo.

In the above example, the analytic expression for the ground-state wave function i oOO is known. Indeed, there are many physical examples, such as the approach to thermal equilibrium governed by the Fokker-Planck equation,11 where either yro(x) or equivalently the super-potential, W(x), are explicitly known. However, there are certain problems where this may not be the case and only V-(x) is known analytically. As an example of this type, we consider the widely discussed double-well

TABLE I. Estimates of t for the potential K-(JC) ™ — yx2

+x4. Note that numerical Runge-Kutta calculations are un­reliable for y> 10.

Runge-Kutta Third order

Second order

First order

0.5 1 2 3 4 5 6 7 8 9

10 11 12

2.464 2.177 1.575 0.9712 0.4624 0.1595 0.0414 8.65x10" 1.52x10" 2.28x10" 2.86x10"

2.451 2.168 1.573 0.9712 0.4624 0.1595 0.0414 8.65X10"3

1.52X10"3

2.28X10"4

2.98xl0~5

3.43X10"6

3.51X10"7

2.324 2.063 1.513 0.9520 0.4605 0.1594 0.0414 8.65x10" 1.52x10" 2.28x10" 2.98x10" 3.43 x 10 "6

3.51X10"7

3.355 2.923 2.025 1.160 0.5033 0.1632 0.0416 8.65x10" 1.52x10" 2.28x10" 2.98x10" 3.43x10" 3.51x10"

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VOLUME 60, NUMBER 1 P H Y S I C A L R E V I E W L E T T E R S 4 JANUARY 1988

potential given by12

V-(x) = -yx2 + x\ (15)

The ground-state wave function, y/o(x)9 is obtained by the Runge-Kutta method. The value of t can be calcu­lated directly by our solving Schrodinger's equation again for the first excited state and calculating the ener­gy difference, t=E\—Eo. As the value y increases, *, which is the difference between two approximately equal numbers, becomes progressively smaller. For y ^ 10, the numerical calculation becomes unreliable. However, the determination of the wave function y oOc) is still quite feasible. Hence our approximation technique provides a viable alternative method for calculating t. For values of Y so large that even the numerical evaluation of yo(x) fails, a combination of the WKB technique for determin­ing the tail of the wave function, coupled with the nu­merical evaluation, can be used to obtain a good estimate of y/o(x).

In Table I, we present the various estimates for t cal­culated from logarithmic perturbation theory, as well as the result obtained from a direct numerical evaluation. Note that our results extend over the entire range of values of y, unlike earlier treatments.12

In conclusion, we have examined how supersymmetry can be used to calculate f, the energy splitting for a double-well potential. We have shown that, rather than calculating this splitting as a difference between the lowest-lying two states of V-(x), one can instead devel­op a perturbation series for the ground-state energy t of the partner potential F+Oc). By choosing as an unper­turbed problem the potential whose solution is the nor-malizable zero-energy solution of F+Gc), we obtain a very simple ^-function perturbation which produces a rapidly convergent series for t. The procedure is quite general and is applicable to any arbitrary double-well potential. The numerical results are very accurate for both deep and shallow potentials.

We are grateful to the High Energy Physics Division,

Argonne National Laboratory, for its kind hospitality. This work was supported in part by the U.S. Department of Energy and the Research Corporation. We also thank the National Center for Supercomputing Applications at Urbana-Champaign for allocation of supercomputer time.

lE. Witten, Nucl. Phys. B185, 513 (1981); F. Cooper and B. Freedman, Ann. Phys. (N.Y.) 146, 262 (1983).

2L. Gendenshtein, Pis'ma Zh. Eksp. Teor. Fiz. 38, 299 (1983) [JETP Lett. 38, 356 (1983)].

3E. Schrodinger, Proc. Roy. Ir. Acad., Sect. A 46, 9, 183 (1940-1941); L. Infeld and T. E. Hull, Rev. Mod. Phys. 23, 21 (1951); P. Deift, Duke Math. J. 45, 267 (1978).

4R. Dutt, A. Khare, and U. Sukhatme, University of Illinois at Chicago Report No. UIC-86-12, 1986 (to be published).

5F. Cooper, J. Ginocchio, and A. Khare, Phys. Rev. D 36, 2458 (1987).

6A. Comtet, A. Bandrauk, and D. Campbell, Phys. Lett. 105B, 159 (1985); A. Khare, Phys. Lett. 161B, 131 (1985).

7R. Dutt, A. Khare, and U. Sukhatme, Phys. Lett. B 181, 295 (1986).

8T. Imbo and U. Sukhatme, Phys. Rev. Lett. 54, 2184 (1985).

9V. Kostelecky and M. M. Nieto, Phys. Rev. Lett. 53, 2285 (1984).

10C. Sukumar, J. Phys. A 18, 2917 (1985); D. Baye, Phys. Rev. Lett. 58, 2738 (1987).

nM. Bernstein and L. S. Brown, Phys. Rev. Lett. 52, 1933 (1984).

12P. Kumar, M. Ruiz-Altaba, and B. S. Thomas, Phys. Rev. Lett. 57, 2749 (1986).

13L. Landau and E. Lifshitz, Quantum Mechanics (Per-gamon, New York, 1977).

14S. Coleman, Aspects of Symmetry (Cambridge Univ. Press, New York, 1985).

15E. Merzbacher, Quantum Mechanics (Wiley, New York, 1970).

16See, for example, F. Hildebrand, Advanced Calculus for Applications (Prentice-Hall, Englewood Cliffs, 1976).

17T. Imbo and U. Sukhatme, Am. J. Phys. 52, 140 (1984).

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