JWKB approximation: contributions to the theory
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The turning points are the points where the classical particle changes direction. We now consider the behavior of the wave function near the turning points. For this, we need a different method. This differential equation is known as the Airy equation , and the solution may be written in terms of Airy functions , . For the wave function to be square-integrable, we must take only the exponentially decaying solution in the two classically forbidden regions. These must then "connect" properly through the turning points to the classically allowed region.
The requirement that the two functions agree imposes a condition on the energy E , which will give an approximation to the exact quantum energy levels. Given the two coefficients on one side of the classical turning point, the 2 coefficients on the other side of the classical turning point can be determined by using the Airy function to connect them.
This relationship is obtained using known asymptotic of the Airy function. The relationship can be found to be as follows often referred to as the "connection formulas" : .
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Now the global approximate solutions can be constructed. The same can be done at the other turning points; assume there is just another one, x 2. The expression there, however, will appear different than the one determined above at x 1 by a difference in the argument of these trigonometric functions. The matching condition, needed to get a single-valued, square-integrable approximate solution, takes the following form:. Here n is a non-negative integer. This condition can also be rewritten as saying that.
It is possible to show that after piecing together the approximations in the various regions, one obtains a good approximation to the actual eigenfunction. One can then compute the probability density associated to the approximate wave function. The probability that the quantum particle will be found in the classically forbidden region is small. More about Perturbation Quantum dynamics. Faculty library sciences Open print view.
WKB approximation - Wikipedia
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