Bounded Quasi-Classical Motion

For a particle moving in a potential, we can partition the entire space into various regions considering the energy of the particle. We examine in detail the case with only one turning point q₀ that separates the classical accessible region from the inaccessible region. Consider a particle with energy V = E depicted in Figure 8.1. One turning point, q₀, separates the classical accessible region from the inaccessible region. The classical accessible region q < q₀ (I) lies left of the turning point while the classical inaccessible region q > q₀ (II) is to the right of the turning point.

We write the wave functions for regions (I) and (II):

1. For q < q₀, we have the wave function which is the superposition of two waves:

FIGURE 8.1 One turning point q₀ separating the classical accessible region from the inaccessible region. The classical accessible region q < q„ lies left of the turning point while the classical inaccessible region q > q„ is to the right of the turning point.

2. For q > q₀, which is the classical inaccessible region E < V, the validity of the WKB solution remains true and the momentum p is purely imaginary:

and

From the condition of boundedness of the wave function, C₂ = 0. In this case we have a damped wave function that vanishes at q -> ± oo. So, the condition of boundedness of the wave function permit the achievement of an evanescent wave that permits the particle to have a non-zero probability of the presence in classical inaccessible region.

It is instructive to note that from the solution of the Schrodinger equation, the wave function is supposed to be continuous across a boundary. This continuity, in addition to giving the physical conservation of particles, leads to quantization of energy levels for the case of bound states. For the case of WKB, matching i//_t(q) and щ,(q) at the turning point q = q₀ separating the classical accessible and inaccessible regions is impossible since at that point the quasi-classical approximation does not work. At the turning point the wave function becomes infinite. The solution of the Schrodinger equation is a univalent function while that via the WKB are multivalent functions. A difficulty arises when approximating a single-valued function by multivalent functions. So, matching the functions at the turning points is critical as we have to resolve the divergence of the wave functions. Hence, it is important that the solutions to be matched in the two regions should correspond to the same branch. So, the necessity to give an exact solution to the Schrodinger equation in a small region around the turning point. This permits to find the right matching prescription.

By assumption, the potential is smoothly or slowly varying at the boundaries between the transition region and the classical accessible and inaccessible regions. This implies that at the neighborhood of the turning point q = q₀, which is the domain of applicability of the quasi-classical approximation we assume the potential to be smoothly or slowly varying. So, based on this analysis, we thus obtain matching conditions where the wavefunction and its derivative must be continuous. It permits as well to Taylor series expand the potential energy V(q) about the turning point q = q₀:

It is instructive to note that though we still need to solve the Schrodinger equation explicitly or numerically in the transition region, the WKB is still imperative in that it gives a simple form and much better intuition for what the wave function should look like outside the transition regions ±<5. Besides, a generic quantization condition can be derived without having details of the transition region, as long as the potential can be linearly approximated.

To find a more explicit solution for the WKB wave function and, consequently, the quantization condition on the energy, it is essential to model the transition region wave function explicitly. So, we assume a linear approximation to the potential is appropriate and is, of course, consistent with the essence of the quasi-classical approximation and so we consider

and select q„ to be the origin of the coordinate system and q measures the distance from the turning point. So, at the neighbourhood of the turning point q = q₀ then

and

So, from equation 7.65 we have

The matching of the equations may then be done via the given turning points. For brevity, the onedimensional motion should be bounded. We consider the wave function non-degenerate and real. This is applicable as well to the wave function i//j(

So,

The expression for y/_tl(q) should then be automatically rewritten:

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Matching the two wave functions gives ф = — . The idea of matching involves the following: Without the time dependence, we examine the quasi-classical complex function on the real axis on which lies the turning point q = q₀. So, the quasi-classical approximation is applicable for all points on the plane except the turning point q = q₀ on the circle of radius p with center at q = q„ . In this case, for the motion described by the circle:

That is, we have the transition to the two terms in So, since then

From the excursion from the point q = q₀, the argument changes from 0 to n and so there arises the factor exp This implies that ф = -^-.

So,

If now we consider the case then

and for q < q₀ - S, then and for q > q₀ + <5, then