THE 1-DIMENSIONAL WKB EQUATION
Although the name of the approximation bears the initials of three authors that published, each separately, a corresponding paper (all in 1926) in the field of Quantum Mechanics, the history of the WKB approximation goes further back in time. Most of the essential ideas and methods had been worked out by Lord Rayleigh as far back as 1912, if not earlier in 1896, with the publication of his book “Theory of Sound”. As the name of the book implies, the mathematical method was aimed at the physical problem of sound propagation. Some authors refer to this method also as the JWKB approximation after an earlier publication (in 1923) by H. Jeffreys. Before tackling the 1-dimensional case in depth, we devote a few lines to show the connection between the classical wave equation and that of Schroedinger.
Lord Rayleigh tackled the problem of a scalar wave Ф propagating in a slowly varying medium characterized by a refractive index n(r). This is equivalent to solving the wave equation
If the refractive index n(r), which is slowly varying, were a constant equal to nlh the solutions of equation 6.1 would be plane waves of the form
It made sense therefore to try a solution in at least the 1-dimensional but position dependent case of the form
where a(r) is a slowly varying function of r.
The Schroedinger equation can be put into the form of 6.1 by simply writing it as
Then the -^-(E-V) term above is the equivalent of k2n2. Alternatively we may divide and multiply the second term in 6.4 by the energy E to get
where p = momentum and X is wavelength. Hence equation 6.4 may be put again into the form of 6.1 if we assign
The analogy between the Schroedinger equation and the classical wave equation does not seem at first sight to be universal because in the case of the Schroedinger equation the term E — V may be either negative or positive, whereas the term k2n2 is positive for wave propagation in vacuum. We remind the reader, however, that metals have an almost imaginary refractive index so that n2 in the case of metals is negative and the analogy is complete. It is worthwhile noting that in both types of equations the solutions are travelling waves if the
quantity —^-(E-V), or equivalently kn2, is positive or decaying functions of space if the n
quantity (E-V) is negative. Figures 6.1a and 6.1b portray these two cases.
FIGURE 6.1 Transmission of a wavefunction (a) through a potential barrier and (b) over a potential barrier.
We now return in detail to the 1-dimensional case. We mentioned in the introduction to this section that the WKB method is valid for slowly varying potentials. We need to define what is meant by “slowly varying” more accurately. An obvious mathematical expression of this statement would be
where X is the de Broglie wavelength of electrons. Let the average of the absolute quantity in 6.6 be denoted by
Then a better expression than 6.6 will be
Substituting equation 6.3 in 6.1 we have in 1 dimension by successive differentiation
Denoting by prime the derivative we get after a further differentiation
Actually the above equation holds for 3 dimensions also, as one can verify by going through the steps leading to equation 6.8, but we will remain within the 1-dimensional case. A zeroth order approximation to 6.8 can be readily obtained: if we assume that the slowly varying o(x) is almost linear (remember that if n(x) is constant o(x) is just x) then the second derivative c" is nearly zero and we get
where C is the constant of integration.
Now assume first that n(x)> 0 (or £> V(x)) so that the square root above is real. The opposite inequality will be examined next. We can get a better approximation than equation 6.9 as follows: we expand a in powers of
Note that we retain the factor 2n because к = 2ж/Х. We have
If we keep the first two terms now, as a better approximation to equation 6.9, and substitute these terms into equation 6.8 we get
Great care should be exercised in approximating the above equation. The term a2 |ai j can
be dropped as it is to second order in the parameter a. A further step is to note that a0 2 and n2 cancel out if use of equation 6.9 is made. The last step in our approximations is to neglect cti , just as in our zeroth order approximation we neglected Go • But in this order the second derivative g0 should be kept. We then get
In the equation above we have used equation 6.11, so that
Substituting for Go from equation 6.10 and for a<3 from 6.13 into equation 6.12 and keeping the first two terms we have
where the constant in 6.13 has been absorbed in the constant C. Substituting 6.14 in our basic equation 6.3 we get
Now we multiply and divide the pre-exponential factor by *Jk and absorb the latter in the numerator into C. Then using equations 6.5a and 6.5b to transform back to the domain of Quantum Mechanics (where now the field is T^x)) we have
In equation 6.16 both solutions with the (+) or (-) sign are acceptable solutions, so the general solution may be written as
What happens if E < V(x) or equivalently kn2 < 0? Then the solutions are real exponentials and not imaginary exponentials, i.e. travelling waves. They can be written in the form
An exactly parallel analysis to what we have presented so far, leads to
Both equations 6.17 and 6.19 present a mathematical problem at points x such that V = E, i.e. where the potential energy is equal to the total energy. Then the kinetic energy is zero and hence the classical velocity is zero. From both equations 6.17 and 6.19 we deduce that the wavefunction T goes to infinity. In this case the WKB approximation breaks down. The points in space at which this happens are shown in figure 6.1a and are designated “turning points” because classically the velocity would have changed direction at these points. These points are crucial because it is at these points we have to calculate the wave- function to obtain the transmission of a wave through a barrier such as the one shown in figure 6.1a. The transmission coefficient defined by
is divergent at these points. In this particular case one has to expand the potential around the turning points xx and x2 as V(x) = V(x;)+$tx and seek the solutions of the Schroedinger equation in terms of Airy functions. The outcome of such a procedure is the transmission coefficient given by the formula we have already presented in chapter 1. We have given a simplified proof there. The more rigorous proof is quite lengthy with no further physical insight so we omit it. For completeness we repeat the equation for the transmission coefficient of chapter 1 here (equation 1.65)
where in 6.20 we have written explicitly Ex to denote the energy along the x-direction for future use, when more dimensions will be involved.
This completes our exposition of the WKB method. Although we had presented the above formula in chapter 1 it was necessary to go into more detail in order to obtain the wavefunc- tions as these will be necessary in our quantum description of nanoscale transistors.