The WKB Approximation

I hope that this is a clear explanation of what the important WKB approximation in elementary quantum mechanics is all about.

History of the WKB Approximation

The WKB, or BWK, or WBK, or BWKJ, or adiabatic, or semiclassical, or phase integral approximation or method, is known under more names than any confidence man. It approximates a real Schrödinger wave function by a sinusoidal oscillation whose phase is given by the space integral of the classical momentum, the phase integral, and whose amplitude varies inversely as the fourth root of the classical momentum. This approximation was already known for the physical waves of optics and acoustics, and was quickly applied to the new Schrödinger "probability" waves.

The new quantum theory that was introduced by Planck, Bohr and Sommerfeld after the turn of the 20th century, now the "old" quantum theory, was based on classical (Newtonian) mechanics supplemented by arbitrary postulates. These new postulates were completely absurd, but nevertheless explained a great deal of surprising phenomena, such as the black-body thermal radiation spectrum, the line spectra of hydrogen, Compton scattering, Landé's vector model, and the principal aspects of the photoelectric effect. The principal tool of this analysis was the "correspondence principle" that the large-scale behavior of quantum systems agreed with classical analysis. Much use was also made of advanced mechanics, such as Hamiltonian theory. Nevertheless, arbitrariness of the quantum postulates, and lack of progress in understanding more complex systems, created a hunger for a better theory.

The better theory arrived with Heisenberg's matrix mechanics (1925), de Broglie's matter waves (1924), and the Schrödinger wave equation (1926), soon elaborated by Born, Jordan and others. It was the introduction of waves that gave a necessary mental model to support the abstract algebra of Heisenberg matrices. It is necessary to repeat that the Schrödinger waves are not waves in the sense of electromagnetic waves, but include the essence of quantum behavior in combining phase and amplitude in their description. The waves can, in fact, be complex (consist of real and imaginary parts). Born's interpretation of the absolute value squared as the probability density for the coordinate x of the system described is the best intimation of its meaning. The wave function is not a wave in physical space, but a mathematical device.

Any model that combines classical waves and particles to explain wave mechanics is essentially incorrect, and leads to erroneous predictions. States are described in new ways in wave mechanics, of which the Schrödinger wave function is only one example, based on the position coordinate. The wave function includes information on position and momentum simultaneously. Heisenberg's uncertainty relations express this fact concisely. There is no surprise in uncertainty relations for waves, only in the application to what may be interpreted as particles. Heisenberg's famous thought experiments on the process of measurement do not show that measurement causes the quantum behavior, but rather that quantum behavior is implicit in any such measurement. There is no disagreement whatever between the predictions of wave mechanics and experimental observations.

The new wave mechanics gave complete explanations for the arbitrary postulates of the old quantum theory. There was no necessity for postulating quantum behavior--the behavior fell out of the theory naturally. The quantization of oscillators was quite analogous to the normal modes and proper frequencies of electromagnetic waves in a resonator. However, although the wave picture of the new mechanics gave many such wavelike properties naturally, there is always more to the picture, and essential differences with classical waves. These differences are hard to find in optics, but are certainly there, as in the photoelectric effect. The fascinating thing was the existence of quantum effects in particles. Wave mechanics is now called simply quantum mechanics, as it is no longer necessary to emphasize the difference from the Newtonian mechanics of the old quantum theory.

No sooner was wave mechanics abroad, than a method of applying it to the most important problems of the day was devised. These problems were the new phenomenon of tunnelling through a potential barrier, and the energy eigenstates of a potential well, either of an oscillator or the radial problem in atomic spectra. Solving problems in wave mechanics generally meant the solution of differential equations, for which even in one dimension there were no analytic solutions, except in a few special cases. By approximating the wave function as a oscillatory wave depending on a phase integral, many useful problems could be solved by a mere quadrature. Almost simultaneously, G. Wentzel [Zeits. f. Phys. 38, 518 (1926)], H. A. Kramers [Zeits. f. Phys. 39, 828 (1926)] and L. Brillouin [Comptes Rendus 183, 24 (1926)] published applications of this theory to the Schrödinger equation. Their initials give the term WKB approximation. The developments were independent, and it would not be fair to recognize one author rather than another. Why the inverse alphabetic order was chosen, I do not know. BWK and WBK are also found. The latter may be accurately chronological by publication date.

The question of the relation of quantum and classical mechanics is a large and important one, of which the WKB approximation is only one part. I think the term should be restricted to the one-dimensional approximation that is so valuable in applications, and not to the general subject of semiclassical approximations, as is done by P. J. E. Peebles (1992), who relegates it to an historical chapter and does not mention the important applications at all. Abbreviated references to authors in this page refer to well-known quantum mechanics or spectra texts of the dates given. The name adiabatic or semiclassical may also be applied with justice. Ehrenfest showed that the quantum expectation values of mechanical quantities behaved like their classical analogues. Also, Schrödinger's equation can be approximated in a form resembling Hamilton's principal function theory. We won't be concerned with these more general interesting questions here.

The method used by WKB was very similar to the theory developed by H. Jeffreys [Proc. London Math. Soc. (2)23,428 (1923)], who apparently went into a huff at not being mentioned. Now, this was before wave mechanics, so he could not have applied the theory to the Schrödinger equation, which is really what is in question. Whether WKB ever read his paper or used it I do not know. My suspicion is that they did not, since the theory, as needed by them, is not very difficult, as we shall see. Some authors, such as Condon and Odobasi, give Jeffreys his due by calling it the WBKJ approximation.

However, the only difficult part of the theory, the connection of solutions on opposite sides of a turning point, was published much earlier by Rayleigh [Proc. Roy. Soc. A86, 207 (1912)], so possibly WBKJR approximation would be fairer. The general matter was actually treated by J. Liouville [Jour. de Math. 2, 168, 418 (1837)], and the function used by Rayleigh was invented by G. B. Airy [Trans. Cambr. Phil. Soc. 6, 379 (1849)] in connection with the theory of the rainbow.

Anyone who thinks Physics is advancing at the present time should carefully consider the short period 1925-1930, and what was done in these few years. Later contributions were made by J. L. Denham [Phys. Rev. 41, 713 (1932), R. E. Langer [Phys. Rev. 51, 669 (1937)] and W. H. Furry [Phys. Rev. 71, 360 (1947)]. This would make it the WBKJRLDLF approximation, I suppose.

The WKB approximation appears in most quantum mechanics texts, with the notable exception of Dirac's. Pauling and Wilson (1935) have a short account (pp 198-201), E. C. Kemble (1937) (pp 90-112) with his own contributions, W. V. Houston (1951) (pp 87-90), N. F. Mott (1952) passim, A. Messiah (1962) (pp 194-202), A. S. Davydov (1963) (pp 73-86), L. I. Schiff (1968) (pp 268-279), E. U. Condon and H. Odobasi (1980) (pp 130-135) and P. J. E. Peebles (1992) (pp 44-47). These accounts vary in comprehensibility, and some are unnecessarily mathematical. Unfortunately, I do not have a copy of Rojansky's text with its exceptionally understandable treatments of many topics. The WKB was probably included. I borrowed the copy I studied from the Billings public library long ago, and have never been able to find a copy to buy.

Explanation of the WKB Approximation

The Schrödinger equation results if we make the operator substitution p = -i(h/2π)d/dx in the eigenvalue equation Hψ = Wψ, where H is the Hamiltonian p2/2m + V(x), and W is the energy eigenvalue. We are dealing with a single coordinate x. Recall that ψ may be chosen real in this case, and has (unexpressed) time dependence exp(iWt). The result is ψ" + (8π2m/h2)[W - V(x)]ψ = 0, or ψ" + k2(x)ψ = 0, which is simply Helmholtz's equation, very familiar in wave theory.

The "wave vector" k = 2π/λ, and the momentum p = hk/2π (Internet Explorer does not support h bar, so I have to insert the cumbersome 2π explicitly). This is de Broglie's relation. Remember that low momentum means a long wavelength. Let's try for a solution in the form ψ = A(x)exp[iS(x)]. To make things a little simpler, use ψ = exp[iS(x) + T(x)]. This merely expresses the amplitude in a different way, and puts the functions S(x) and T(x) on a more equal basis. Substitute this expression in the Schrödinger equation, and write the real and imaginary parts separately. What you find is: -S'2 + T" + T'2 + k2 = 0 and S" + 2S'T' = 0.

Now suppose we are considering cases where the oscillation of the wave function is very rapid compared to changes in amplitude. This will be true when the momentum p is large (and so λ is small) and does not change very rapidly--that is, when V(x) varies slowly enough. The condition is something like p'/p << 1. In this case, T" and T' will be very much smaller than S', and if they are neglected in the first equation, we have the happy result that S'2 = k2, or S' = ±k. This gives S(x) = ±∫ kdx, which is our phase integral, the fundamental quantity in this approximation. Because we neglected T" and T', it is indeed an approximation, which we must not forget, however good the results.

The second derivative S" is just ±k', so the imaginary equation gives 2kT' = -k', from which T = -(1/2)ln k plus a constant of integration. Then exp (T) = A/k1/2. This factor makes the amplitude vary with momentum so that the particle flux is constant, and is essential to a reasonable answer. Finally, therefore, the approximate solution is ψ = (A/k1/2) cos [S(x) + δ], where there are two arbitrary constants, A and δ

The function S was originally expressed as a series of powers of h: S0 + hS1 + h2S2 + ..., where the first term gave the classical result (Hamiltonian theory), the second term a quantum correction, and so on. In what we did above, S0 corresponds to S(x) and S1 to T(x), and the separation of real and imaginary parts was the separation of the zeroth and first powers of h. Unfortunately, this series is not convergent, merely asymptotic, and taking higher order terms into account is unprofitable. A lot of effort seems to have been wasted along this line.

Let's try to apply the approximation to a particle in a potential well. If we first consider an infinite square well, the well has hard, infinite walls at, say, x = 0 and a, and the boundary condition is that the ψ = 0 there. Then, (A/k1/2)sin[S(x)] satisfies the left-hand boundary condition, and we must have sin[S(a)] = 0, or S(a) = nπ. Hence, [(8π2/h2)W]1/2a = nπ, or W = n2h2/8a2, the energy eigenvalues for the infinite square well. This is the exact result, not surprising since our approximation is exact in this case (the amplitude is constant). So the WKB approximation works here.

Thus encouraged, let's study the harmonic oscillator, for which V(x) = mω2x2/2. For a given energy W, the particle bounces back and forth classically between turning points at which the momentum is zero and reversing. In wave mechanics, we have a region to the left of the turning point where the total energy is negative, and the wave function decreases to zero exponentially as we go deeper into this forbidden region. If we carry out the WKB approximation in this region, we find an exponential solution instead of the oscillatory solution, but everything else is the same. There is a similar region to the right of the right-hand turning point. Between the turning points we have the positive-energy region in which the wave function is oscillatory, as we have seen.

The great difficulty that now arises is that at the turning points, our approximation fails utterly. We have approximate solutions except in small regions at the turning points, and do not know how to connect the solutions is the different regions. Connection, however, is necessary for the success of the approximation.

One way to get a connection is to consider the x-axis as the real axis in a complex plane. Our solutions are analytic functions of the complex variable z = x + iy. Perhaps we can somehow start with a function on the positive x-axis (considering the turning point at x = 0) and move around to the other side of the origin by the principles of analytic continuation to find out what kind of solution it connects to there. This works, but not easily. Stokes discovered that asymptotic series can, disconcertingly, jump to represent different solutions as arg z (the angle θ in z = re) changes. This is known as Stokes's Phenomenon. A lot of work was done along these lines, but however interesting and arcane it is, it is also unnecessary for practical applications.

An easier way, due to Rayleigh, is simply to find an exact solution valid in the neighborhood of the turning point, and to fit it to the asymptotic solutions on either side. If the potential is replaced by a linear variation at the turning point, then the desired solution is the Airy function Ai(z) from the theory of the rainbow. Let's assume that V(x) = -ax near the turning point x = 0, so that k2 = (8πma/h2)x, which gives us S(x) = (2/3)(x/α)3/2, where α = (h2/8πma)1/3 is a characteristic distance near the turning point. The dimensionless variable s = x/α is very convenient to use. In terms of s, our approximate solution is (A/s1/4)cos[(2/3)s3/2 + δ].

The exact solution Ai(s), sketched at the right in the neighborhood of the turning point, has the asymptotic form (1/π1/2s1/4)cos[(2/3)s3/2 - π/4] to the right of the turning point. To the left, it decreases exponentially as required. The net effect is that the WKB approximate solution is pushed away from the turning point by an eighth of a wavelength, or phase π/4, in the asymptotic region. The Airy function can be expressed in terms of Bessel functions of order 1/3.

Therefore, we can carry the phase integral from turning point to turning point, as in the case of the infinite square well, and subtract the π/2 from the two ends to allow for the connection. This gives us S = (n + 1/2)π. The phase integral for a harmonic oscillator with energy W is S = Wπ/hν (the integral is easily done with the aid of a table of integrals), so we find W = (n + 1/2)hν. Surprisingly, this is the exact result, in spite of the fact that our method is approximate. The connection relations supply the 1/2 that implies a zero-point energy, which is not present in the old quantum theory.

To apply the WKB approximation to an arbitrary potential V(x), simply find the phase integral S as a function of the energy W. This can always be done by computer integration, where all the tedium is eliminated by the automatic calculations. Then the energy eigenvalues are the values of W for which S/π = n + 1/2. This can be done for anharmonic oscillators, such as vibrating diatomic molecules, or for the radial function in atomic spectra or in atomic collision problems, where the centrifugal potential is added to the self-consistent potential V(r). A large number of such calculations were done in the 30's and 40's, before great computing power was available for more elaborate schemes, and the results were quite satisfactory. The WKB approximation is second only to perturbation theory as a fruitful method of calculation.

One interesting problem to which the WKB approximation is convenient is tunnelling through a region in which W < V(x), and the approximate wave function is, therefore, exponential. I'll not work out the details here, except to state the resulting formula, as given in Mott (1951). The tunnelling probability is P = exp{-2∫ [(8π2m/h2)(V(x) - W)]1/2dx}, which is the absolute value squared of the ratio of the amplitudes on the two sides of the barrier. This result was used by Gamow, Gurney and Condon in the theory of alpha decay. The integral is extended from turning point to turning point, and is analogous to the phase integral S.

The subject of asymptotic expansions is treated in H. and B. S. Jeffreys, Mathematical Physics (Cambridge, 1956), Chapter 17, and in E. T. Whittaker and G. N. Watson, A Course in Modern Analysis (Cambridge, 1927), Chapter VIII. Such expansions are of great utility in physics. The rainbow problem is treated in R. A. R. Tricker, Atmospheric Optics (American Elsevier, 1970), Chapter VI.

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Composed by J. B. Calvert
Created 17 December 2001
Last revised