WKB Approximation

Inhomogeneous media, i.e., media with variable index of refraction or potential, also have eigenfunctions, although the calculation of these functions is more difficult than for the stepwise homogeneous case considered in the previous section. Analytical solutions can be found only for very special cases; thus, in general, one must employ numerical techniques such as the shooting method described by Ian Johnston elsewhere in this series. An approximation, the Wentzel-Kramer-Brillouin (WKB) method, can often be employed when the properties of the medium vary slowly on a scale comparable to the wavelength. This approximation can be used for both the classical wave and Schrödinger equations and is a very useful guide for estimating solutions to these equations. It is based upon the idea that we can estimate the wavelength based upon the properties of the medium. The wavelength in the classical-wave or Schrödinger equations will be shorter if the medium is optically more dense or the kinetic energy is larger, respectively.

WKB wavefunctions can be obtained by assuming that the wavefunctions are oscillatory--- these are, after all, hyperbolic PDEs--- and that the total phase of the oscillations in a medium with fixed (periodic) boundaries must be half-integer (integer) multiples of . Since the wavenumber, , gives the number of radians per meter at x, we require that a medium with fixed boundaries at a, b satisfy

The unnormalized WKB wavefunction is

Clearly, the function will be zero when x=a, since the integral is zero, and again when x=b, since the integral will be by the previous condition, eq:WKBMode.

• Classical WKB
    The classical WKB approximation can be credited to Lord Rayleigh who used it in 1912 to analyze the propagation of an optical disturbance in a nonuniform medium. It is discussed on page 310 by Elmore and Heald.[2] Rayleigh's approximation is equivalent to assuming that the wavenumber is proportional to the index of refraction where . Substituting this expression into eq:WKBMode we see that there are an infinite number of values of (or ) that satisfy the condition. This is actually a trivial eigenvalue equation

  

  that requires the evaluation of a single indefinite integral. Once the values of are known, the WKB wavefunctions can be calculated using equation eq:WKBFun.

  a.

  Enter the following linear position-dependent index of refraction using [Medium]|[Parser] .

  

  Since the parser adds its values to the values already existing in the medium, the actual index of refraction will be . Select mode 32 and run the program. Notice that the antinodes do not have the same amplitude and are not uniformly spaced. Are the overtones harmonics?

  b.

  Using pencil and paper, evaluate eq:WKBMode and eq:WKBFun and compare with the results obtained from [a.].

  c.

  Construct an index of refraction that does not satisfy the condition that changes occur slowly on the scale of one wavelength and demonstrate that the WKB wave functions constructed by WAVE are not eigenfunctions.

  d.

  (Advanced) The exact analytical solution to the classical wave equation with a linear variation in the medium can be written in terms of Airy functions.[5] Compare this analytical solution to the WKB solution obtained in [b.].

• Quantum WKB
    Obtaining WKB solutions to the Schrödinger equation is more difficult than for the classical case. The wavenumber is not a linear function of potential, , and the wavefunction changes from an oscillatory to an exponential function if the potential is larger than the energy E. In addition, the WKB approximation fails in the transition region where changes sign because the wavelength becomes large and encompasses appreciable variation in potential. We have therefore chosen to limit our WKB calculation to systems where the wavefunction is oscillatory, i.e., the classically-allowed regions where , and have set the wavefunction equal to zero in the classically forbidden region. The eigenvalue equation for , eq:WKBMode, is first solved by numerical methods.

  

  and each eigenvalue is substituted into eq:WKBFun to obtain the wavefunction.

  a.

  Select the Schrödinger equation and fixed boundary conditions. Enter the following potential using [Medium]|[Parser]

  

  and then select an initial condition of mode 6. Notice that WKB eigenfunctions are very similar to the WKB solutions of the classical wave equation from the previous exercise.

  b.

  Change the potential to a ramp.

  

  and study the time evolution of the following WKB eigenfunctions, . For what modes is the WKB approximation appropriate? Why is there a spike in the wavefunction for mode 3?