Teresa Griffin, Physics and Astronomy
My research over the past six months has produced a method of understanding the differences between semiclassical approximation methods and the exact quantum results. By analyzing a particle in an infinite perturbed well, I have found that the discontinuities in the potential cause the semiclassical result to be inaccurate. This inaccuracy is predicted in the theory of the WKB approximation where it is not possible to account for the discontinuity in the first derivative of the wavefunction. I have also discovered that one can transform the semiclassical results into nearly exact results by taking the expectation value of the Hamiltonian of the semiclassical results (see Figure 2). In March I participated in the College of Mathematical and Physical Sciences Spring Research Conference where I gave a presentation on the source of inaccuracy in semiclassical methods. Figure 1 shows the energies ofWKB (semiclassical) wavefunctions, their expectation values, and the exact quantum energies for a particle in and infinite perturbed potential well. This graph is for the second energy level where an equation for the wavefunction is \jf(x) = ~211 sin (2nx/1). Note that at x = 1/4 the sin is 1 and the wavefunction is at its maximum while at x = 3114 the sin is -1 and the wavefunction is at its minimum.
Figure 1 shows that the first order WKB energy dramatically disagrees with the exact energy whenever the potential is discontinuous near one of the peaks in the probability.
Figure 2 shows the differences between the WKB zeroth and first order expectation values and the exact quantum energies. Note that for the zeroth order, the difference is never greater then 0.006, and, except near the discontinuities at 0.25 and 0.75, the first order is never off by more than 0.0002. The expectation value includes an additional term from the discontinuity in the first derivative of the wavefunction that is not included in the WKB energy in Figure 1. Thus, these results quantitatively show how the WKB method fails when the potential is discontinuous.
I am currently working on extending the expectation value analysis to a particle in a finite box which has similar discontinuities. I will then study a particle in a finite well with linear potential walls of finite slope; Airy functions describe the motion of the particle in the regions of linear slope. This research points the way toward an understanding of the correlation between classical and quantum mechanics. By understanding what happens at the discontinuities our understanding of the reasons for the success of quantum mechanics is strengthened, and we are able to use semiclassical methods as a highly accurate approximation to the exact quantum solutions.