Understanding the Limitations of WKB Approximation in One-Dimensional Problems

[InGaAsP]

Hi all
I have a question about WKB approximation
Why is it that WKB method can be applied only to problems that are one dimensional or those which can be reduced to forms that are one dimensional ones?
any help is deeply appreciated

 

[explain]

Hi all
I have a question about WKB approximation
Why is it that WKB method can be applied only to problems that are one dimensional or those which can be reduced to forms that are one dimensional ones?
any help is deeply appreciated

This is not quite correct. The one-dimensional WKB is widely used because you can have explicit approximate solutions. In many dimensions, there are no such explicit solutions. The analog of WKB approximation in many dimensions is the approximation of geometric optics (or eikonal approximation). This approximation can be applied to a wave equation in many dimensions, and you get a Hamilton-Jacobi equation for the "eikonal" function.

It works like this. Suppose you have a wave equation, let's write it symbolically as Wave($$\psi(x)$$)=0, where "Wave" is a differential operator and $$\psi(x)$$ is a function describing the wave; x is a multidimensional position vector. You would like to consider solutions $$\psi(x)$$ that look like quickly oscillating waves. So then you write the ansatz, $$\psi=\exp(\imath \lambda \Omega(x))$$, where $$\Omega$$ is a new unknown function, representing the phase of the oscillation, and $$\lambda$$ is a large constant (it's large because you want the wave to oscillate quickly). Then you substitute this ansatz into the wave equation and keep only terms of highest order in $$\lambda$$. You get a nonlinear equation for $$\Omega(x)$$, which looks like the Hamilton-Jacobi equation. The function $$\Omega$$ is called the 'eikonal' but it's just the phase of the wave. This function can be interpreted as the action of a particle, as in the usual Hamilton-Jacobi formalism in mechanics. So in this way you can visualize propagation of waves as motion of particles along "rays".

In one dimensions, you do this and get an ordinary differential equation of first order for the eikonal $$\Omega$$. So this can be solved and you get an explicit formula for the eikonal. In many dimensions, you can't solve it symbolically, so you just kind of say "grrr..." but at least the approximate solution exists and looks like a wave propagating in straight lines.

 

[denian]

The WKB approximation is a method used to solve differential equations in quantum mechanics. It is based on the assumption that the wave function can be expressed as a product of an exponential function and a slowly varying function. This approximation works well for one-dimensional problems because it allows us to ignore the higher order derivatives in the differential equation, reducing it to a simpler form.

However, in higher dimensions, the wave function cannot be expressed in this simple form, and the higher order derivatives become significant. As a result, the WKB approximation is not accurate and can lead to incorrect solutions. Therefore, it is limited to one-dimensional problems or problems that can be reduced to one-dimensional forms.

In addition, the WKB approximation also assumes that the potential energy is slowly varying, which is often not the case in higher dimensions. This further limits its applicability to only one-dimensional problems.

It is important to understand the limitations of the WKB approximation and use it only in cases where it is appropriate. In higher dimensional problems, other methods such as perturbation theory or numerical methods may be more suitable.