SUSY and Tunneling

Reference:

published in Phys. Rev. Lett. 60, 41 (1988)  UIC-87-26







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













Abstract


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.

Recently, there have been many applications of supersymmetry to quantum mechanical potential problems,(1-12) all stemming from one key observation: given any potential of interest V-(x), unbroken supersymmetry allows one to construct a partner potential, V+(x), 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, V-(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 method(6,7) and large-N 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 º E1 - E0 between the two lowest 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 re-examine carefully and extend considerably previous claims(11,12) that supersymmetry facilitates the evaluation of t. Indeed, using the supersymmetric partner potential V+(x), 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 non-leading 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 at x = ±x0. We define the depth, D, of V-(x) by D º V-(0) - V-(x0). 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 V-(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:
np = ó
õ 		xc

0 
Ö
 
V-(0)-V-(x)
 
 dx ,
(1)
where xc is the classical turning point corresponding to energy V-(0). Throughout this paper we have chosen units where (h/2p) = 2m = 1. We shall call a double well potential ``shallow'' if it can hold at most one pair of bound states i.e., n £ 1. In contrast, a ``deep'' potential refers to n ³ 2. The energy splitting t of the lowest lying pair of states can be obtained by a standard argument.(13) Let c(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 two lowest eigenfunctions of the double well potential V-(x) are well approximated by
y0,1(-)(x)=  1
Ö2
 [ c(x)±c(-x)]     .
(2)
By integrating Schrödinger's equation for the above eigenfunctions, it can be shown that(13)
t º E1 - E0 = 4 c(0) c¢(0)     ,
(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. Using WKB wavefunctions in Eq. (3) yields the standard result:
tWKB =

Ö
2V¢¢-(x0)
p
 exp ì
í
î 		- 2 ó
õ 		x0

0 
Ö
 
V-(x)-V-(x0)
 
 dx ü
ý
þ
(4)
The same result can also be obtained via instanton techniques.(14)

Using the supersymmetric formulation of quantum mechanics, for a given Hamiltonian, H- = - [(d2)/(dx2)] + V-(x), and its zero-energy ground state wavefunction y0(x), we can construct V+(x), the supersymmetric partner potential of V-(x), as follows:
V+(x) = V-(x) - 2  d
dx
 æ
è 		 y¢0
y0
 ö
ø 		= -V-(x) + 2 æ
è 		 y¢0
y0
 ö
ø 		2

 
  .
(5)
Alternatively, in terms of the superpotential W(x) given by
W(x) = - æ
è 		 y¢0
y0
 ö
ø 		 ,
(6)
we can write
V±(x) = W2(x) ±dW/dx     .
(7)

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 V+.(1) Hence, for the double well problem, we see that if V-(x) is ``shallow'' (i.e. only the two lowest states are paired), then the spectrum of V+ is well separated. In this case, V+ is relatively structureless and simpler than V-. Previous papers(11,12) 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, V+ has a double well structure together with a sharp ``d-function like'' 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 wavefunction is the sum of two Gaussians, centered around ±x0.
y0(x) µ e-(x-x0)2 + e-(x+x0)2        .
(8)
Throughout this paper we have chosen the variables x and x0 to be dimensionless. The corresponding superpotential W(x), and the two supersymmetric partner potentials V±(x) are given respectively by
W(x) = 2 {x - x0 tanh(2 x x0)}  ,
(9)

V±(x) = 4{x - x0 tanh(2 x x0)}2± {1 - 2x02 sech2 (2 xx0)}     .
(10)
The minima of V-(x) are located near x0 and the well-depth (in the limit of large x0) is D @ 4x02. We illustrate V-(x) and V+(x) in Figs. 2a and 2b respectively, for the choices x0 = 1.0 and x0 = 2.5. We see that in the limit of large x0, for both V-(x) and V+(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 x0®¥ can be calculated from Eq. (3), with c(x) given by one of the (normalized) Gaussians in Eq. (8). We find that
t ~ 8x0
Ö
 
2/p
 
 e-2x02     .
(11)
The same result can be obtained by observing that V-(x)® 4(|x|-x0)2 - 2 as x®¥. This potential has a well known(15) analytic solution with the two lowest energy levels located at E0 = -4x0Ö{2/p}e-2x02 and E1 = +4x0Ö{2/p}e-2x02. We now turn to the evaluation of t via the ground state energy of the supersymmetric partner potential V+(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 Schrdinger equation for V+(x) with E = 0. From supersymmetry (Eq. (5)) we see immediately that 1/y0 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/y0 is not normalizable and hence is not acceptable as a starting point for perturbation theory. One possibility is to artificially regularize the behavior 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 independent solution of the Schrdinger equation,(16)
f(x) =  1
y0
 ó
õ 		¥

x 
 y02(x¢) dx¢     ,        x > 0     ,
(12)
and f(x) = -f(-x) for x < 0. Clearly, f(x) is well-behaved at x = ±¥ and closely approximates 1/y0 at small x - thus we expect it to be an excellent approximation to the exact ground state wavefunction of V+(x) for all values of x. The derivative of f(x) is continuous except at the origin, where, unlike the exact solution, it has a discontinuity f|0+ - f|0- = -2y0(0). Hence, f(x) is actually a zero-energy solution of the Schr‡odinger equation for a potential, V0(x), given by
V0(x) = V+(x) - 4y02(0) d(x)     ,
(13)
where we have assumed that y0(x) is normalized. We calculate the perturbative corrections to the ground state energy using DV = + 4y02(0) d(x) as the perturbation. Note that the coefficient multiplying the delta function is quite small so that we expect our perturbation series to converge rapidly. For the case of a symmetric potential such as V+(x), the perturbative corrections to the energy arising from DV can be most simply calculated by using the logarithmic perturbation theory (LPT)(17) formulation of the usual Rayleigh-Schrdinger series. The first and second order corrections to the unperturbed energy E = 0 are
E(1) =  1
2x(0)
     ;       xi(x) = ó
õ 		¥

x 
 f2(x¢) dx¢,
(14)

E(2) = -2 ó
õ 		¥

0 
 é
ë 		 E(1)x(x¢)
f(x¢)
 ù
û 		2

 
 dx¢    ,
(15)
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 x0 £ 2. Estimates of t correct to first, second and third order calculated from LPT are compared with the exact result for V+, 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 xà limit. Even for values of x0 < 1/Ö2, 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 x0. In the above example, the analytic expression for the ground state wavefunction y0(x) is known. Indeed, there are many physical examples, such as the approach to thermal equilibrium governed by the Fokker-Planck equation,(11) where either y0(x) or equivalently the superpotential, 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 potential given by(12)
V-(x) = - gx2 + x4        .
(16)
The ground state wavefunction, y0(x), is obtained by the Runge-Kutta method. The value of t can be calculated directly by solving Schr‡odinger's equation again for the first excited state and calculating the energy difference, t = E1 - E0. As the value of g increases, t, which is the difference between two approximately equal numbers, becomes progressively smaller. For g[ > ||  ] 10, the numerical calculation becomes unreliable. However, the determination of the wavefunction y0(x) is still quite feasible. Hence, our approximation technique provides a viable alternative method for calculating t. For values of g so large that even the numerical evaluation of y(x) fails, a combination of the WKB technique for determining the tail of the wavefunction, coupled with the numerical evaluation can be used to obtain a good estimate of y0(x). In Table I, we present the various estimates for t calculated from LPT, as well as the result obtained from a direct numerical evaluation. Note that our results extend over the entire range of values of g unlike earlier treatments.(12)

In conclusion, we have examined how supersymmetry can be used to calculate t, the energy splitting for a double well potential. We have shown that, rather than calculating this splitting as a difference between the two lowest lying states of V-(x), one can instead develop a perturbation series for the ground state energy t of the partner potential V+(x). By choosing as an unperturbed problem, the potential whose solution is the normalizable zero- energy solution of V+(x), we obtain a very simple delta 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 NCSA at Urbana-Champaign for allocation of supercomputer time. 

                                REFERENCES

1.   E. Witten, Nucl. Phys. B185, 513 (1981); F. Cooper and B.
     Freedman, Ann. Phys. (N.Y.) 146, 262 (1983).  
2.   L. Gendenshtein, Pis'ma Zh. Eksp. Teor. Fiz. 38, 299 (1983)
     [JETP Lett. 38, 356 (1983)].  
3.   E. Schrdinger, Proc. Roy. Irish Acad. A46, 9 (1940) and A46,
     183 (1941); L. Infeld and T. E. Hull, Rev. Mod. Phys. 23, 21
     (1951); P. Deift, Duke Math. J. 45, 267 (1978).
4.   R. Dutt, A. Khare and U. Sukhatme, University of Illinois at
     Chicago preprint UIC-86-12 (1986); to be published in Am. J.
     Phys. (1987).  
5.   F. Cooper, J. Ginocchio and A. Khare, Phys. Rev. D36, 2458  (1987).
6.   A. Comtet, A. Bandrauk, and D. Campbell, Phys. Lett. 105B, 159
     (1985); A. Khare, Phys. Lett. 161B, 131 (1985). 
7.   R. Dutt, A. Khare and U. Sukhatme, Phys. Lett. 181B, 295 (1986). 
8.   T. Imbo and U. Sukhatme, Phys. Rev. Lett. 54, 2184 (1985).  
9.   V. Kostelecky and M. M. Nieto, Phys. Rev. Lett. 53, 2285 (1984).  
10.  C. Sukumar, Jour. Phys. A18, 2917 (1985); D. Baye, Phys. Rev.
     Lett. 58, 2738 (1987).
11.  M. Bernstein and L. S. Brown, Phys. Rev. Lett. 52, 1933 (1984).
12.  P. Kumar, M. Ruiz-Altaba and B. S. Thomas, Phys. Rev. Lett.
     57, 2749 (1986).
13.  L. Landau and E. Lifshitz, Quantum Mechanics (Pergamon Press,
     New York, 1977).  
14.  S. Coleman, Aspects of Symmetry (Cambridge Univ. Press, 1985).
15.  E. Merzbacher, Quantum Mechanics (Wiley, New York, 1970).
16.  See for example, F. Hildebrand, Advanced Calculus for Applica-
     tions (Prentice-Hall, Inc., 1976).
17.  T. Imbo and U. Sukhatme, Am. J. Phys. 52, 140 (1984).
FIGURE CAPTIONS


File translated from TEX by TTH, version 3.02.
On 8 Apr 2002, 19:53.