Stephen K. Knudson College of William & Mary, Williamsburg, VA 23185 D. W. Noid Chemistry Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831 and University of Tennessee, Knoxville, TN 37928 For many years, one of the standard methods for determining the eigenvalues of the Schroedinger equation when the potential energy function does not have a simple form has been the WKB aonroximation ( I ) . also called the semiclassical method. ~he'iheoreticaltreatment of the method is found in almost all standard textbooks on auantum mechanics, which discuss the method as applicable to systems with large mass or large quantum number. A principal limitation of the WKB method as presented is its restriction to the treatment of one-dimensional problems only. During the past 10 years, this situation has changed and a number of methods have been developed to calculate WKB eigenvalues for multidimensional nonseparable Hamiltonian systems (2).One of the most useful and practical of the exactmultidimensional WKB approximations is the surface of section (SOS) method (3), which computes the needed integrals from Poincar6 surfaces of the section. The purpose of this article is to acquaint a larger audience with this new method. For analytic potentials, i t is simple enough that i t can be implemented on microcomputers to find the eigenvalues of coupled systems. We anticipate that semiclassical methods will become more prevalent in chemistry for several reasons. One is that the method offers an efficient route to the calculation of eiaenvalues of hi~hlhlvexcited states. In addition, for the exc-ited states the ;emiclassical approach works well numerically, while quantum calculations become prohibitively expensive or impossible. Perhaps even more importantly, the semiclassical method offers conceptual insights into the dynamics of manv.svstems: this insieht mav he verv difficult to , extract from the quantum mechanical treatment. For exam. ole.. the semiclassical method clearlv shows differences be. tween resonant and nonresonant vibrational motion. In addition. the semiclassical method has been widelv used in the analy& of chemical dynamics (4). Finally, these same systems can exhibit the relativelv recentlv discovered heno omenon of chaos, first discussed in this journal several years ago (5). In this paper, we outline the WKB method for both onedimensional and two-dimensional systems. The method is described in the next section and a simple application given in the section Application. Method One-Dimensional Semiclassical Theory For completeness, we discuss here a derivation (6) of the one-dimensional WKB method, since i t has not been previously described in this Journal. The WKB method estahlishes the relationship between classical and quantum mechanics and is important not only in its own right but also for its influence in the development of quantum mechanical concepts. T o develop the method, we need merely to consider the use of a wave function of the form
in the (one-dimensional) Schroedinger equation with potential V(x),
where m is the mass of the particle, E is an eigenvalue, and h is Planck's constant h divided by 27r. Substituting eq 1into eq 2, we obtain the following equation for the unknown function S(x):
If the second term is much smaller than the first, we can disregard it; the condition justifying this is which is the same as (see eq 9 below)
where Xis the local deBroglie wavelength: X = Z*hlp(x) = 2irh/[2m(E - V(x)ll"
(6)
and IF(x)~is the magnitude of the force. The expansion thus corresponds to a short-wavelength approximation and as such is suitable for converting wave mechanics (or optics) to classical mechanics (or geometrical optics). In the region where eq 5 holds, eq 3 for S is just the HamiltonJacohi equation of classical mechanics (7):
T o implement a short-wavelength approximation, we take
S to be a series expansion in powers of h: Substituting this into eq 3 and equating like powers of h, we find that Sosatisfies S o b )=
*
I
pm(E
-
~(x'))'"dx' =
*
(9)
and that St satisfies so, from eq 1, we find the form of the semiclassical wave function including terms to order h is
where c is a constant and the sign has not yet been determined. T o establish the specific form, consider the wavefunction Volume 66 Number 2
February 1989
133
near a turning point, a point a t which the potential energy equals the total energy; an example of a turning point is the highest position reached by a pendulum. In a nonclassical region (aregion where V(x) > E ) t o theleft of a turning point x = a, the wavefunction must have the form of a decaying exponential, which is found from eq 11to be
while in the classical region to the right of a, the solution has
an oscillatory form
In the region where x is close to a and p vanishes, no valid solution of eq 3 is possible because eq 5 cannot be satisfied. Nonetheless, it remains possible to complete the solution by obtaining the "connection formula", which relates the as-yetunspecified constants el, cp, and cz to each other. In one way of doing this, the potential near the turning point is expanded linearly as V(x) = E - G(x - a), where G = (aV/ax)l,=.. The solution to the resulting differential equation is well known in mathematics as the Airy function (8).Comparison of the wave functions in eqs 12 and 13 with asymptotic' forms of the Airy function then provides the connection formula. In particular, we find that in the classically forbidden region the wave function may be expressed as
which connects to a particular wave function in the classically allowed region: +&)
[
= 2p(x)-'" sin h-'
I
p(xr)dx'
+ n/4
For a right-hand turning point "b" a similar treatment gives
In order for the wavefunction to be valid, it must be continuous and single-valued; applying these constraints to eqs 15 and 17, we find that the condition for a valid bound-state semiclassical wave function is
where n is an integer, the well-known Bohr-Sommerfeld result from old quantum theory (9).This is a fundamental result of the method, and i t applies in essentially the same form to the multidimensionalmethod as well. I t specifies the conditions a trajectory must satisfy if i t is to obey quantum rules. Multldlmenslonal Methd
In this section, the multidimensional semiclassical method will be presented, with special reference to the two-dimensional case for which it is best developed. The onedimensional case above focused on the development of the wave function; in two dimensions the development of the method focuses on the classical trajectories ohtained by solv-
' The asymptotic form of afunction is its limiting expression as the independent variable becomes very large; see reference 8. Newton's eauations. but usually it is more convenient to start with Hamilton's version. 134
Journal of Chemical Education
Figure 1. A "box" tralectoly, obtained by numerical solution of Haminn's equations. In two dlmenslons (x, yl. the particle follows the path drawn In the figure.The heavy line indicates the path of the particles for approximatelyone cycle of the motion. ing the classical equations of motion2.For concreteness, consider two Cartesian coordinates x and y, so that the trajectory may be visualized as a plot of y(t) versus x(t), where t is the time; an example of a quasiperiodic trajedory (quasiperiodic is defined below) is shown in Figure 1. The elliptical line in the figure is the boundary V(x, y) = E , which separates the inner, energetically allowed region from the outer, classically forbidden region. In the particular example shown in the figure the trajectory starts from (x(t = O), y(t = 0)) = (0,O) and traces out a series of distorted figure-eight loops or cycles, which precess, expand, and contract in time. The trajectory in Figure 1 is drawn with a heavy line for approximately one cycle to help the eye follow a typical path. After a period of time corresponding to a large number of cycles, the quasiperiodic trajectory can be seen to fill only a portion of the energetically allowed region. What appears to he an "edge" of the figure so generated is the locus of points called a caustic, where successive loops of the trajectory cross each other. The caustics roughly correspond to twodimensional turning points, but the total momentum vanishesat only four points, Iabeled A-D in the figure, where the caustics touch the energy boundary. The caustics are not necessarily straight lines, but for many cases they have been proven to intersect a t right angles (10). Kolmogorov, Arnold, and Maser (11) have proven an important theorem for celestial and nonlinear mechanics of systems in which the variables are not separable: under certain conditions, the motion is quasiperiodic rather than ereodic. In contrast to the auasioeriodic traiectorv. an ereodic irajectory has a very different appearanc; i t oft& look: like a taneled web of soaehetti rather than the reeular nath of a quas&eriodic orbit. ?he ergodic trajectory gas nd obvious caustics and no restriction from visitine.. anv. e. or ti on of the allowed space. Nevertheless, nature does not seem tomake a fundamental distinction between the two t w e s of traiectories: at some energies, both quasiperiodic and ergodirirajectoriescan occur, differina only in the remaining initial conditions used to kick off
