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Chapter 3. Wave optics
to be relatively small for a single particle in an unbound state.
This solution is known as the WKB approximation [59, 79]. It
has an important, and still very relevant history in quantum mechanics, in explaining the classical limit of large quantum numbers. For a bound system the approximation breaks down at
the classical turning points where the kinetic momentum p(x) =
2m(H − qφ) = 0. Physically, the WKB approximation applies
in the case where the fractional change in the electrostatic potential φ(x) is small over a distance comparable with the deBroglie
wavelength. For an unbound system at relatively high energy, the
approximation is excellent. In the free-particle case where the potentials are zero everywhere, this solution reverts to the familiar
plane wave solution as required.
It is important to remember that this solution applies to a multiplicity of paths, of which the classical trajectory is just one. These
must be summed to obtain the overall wave function. This involves
a procedure similar to (3.139), adapted to three dimensions. In
most charged particle optical systems, the action integrals in the
solutions (3.169, 3.186) are very large relative to h ¯. We showed
in the preceding section that only trajectories infinitesimally separated from the classical trajectory, together with the classical
trajectory itself, contribute appreciably to the overall wave function. In this case it is a very good approximation to assume that
the action integrals are applied only along the classical trajectory.
This can be further understood by applying the operator for the
canonical momentum P to the wave function (3.186). This gives
−ih ¯v ψ(x, t) = P ψ(x, t).
(3.187)
Geometrically, this means that the canonical momentum vector P
is perpendicular to the surfaces of constant phase. The kinetic momentum vector p is everywhere tangent to the classical trajectory.
In the presence of a magnetic vector potential A, this gives rise to
a geometrical interpretation as shown in Figure 3.4.
Chapter 3. Wave optics
to be relatively small for a single particle in an unbound state.
This solution is known as the WKB approximation [59, 79]. It
has an important, and still very relevant history in quantum mechanics, in explaining the classical limit of large quantum numbers. For a bound system the approximation breaks down at
the classical turning points where the kinetic momentum p(x) =
2m(H − qφ) = 0. Physically, the WKB approximation applies
in the case where the fractional change in the electrostatic potential φ(x) is small over a distance comparable with the deBroglie
wavelength. For an unbound system at relatively high energy, the
approximation is excellent. In the free-particle case where the potentials are zero everywhere, this solution reverts to the familiar
plane wave solution as required.
It is important to remember that this solution applies to a multiplicity of paths, of which the classical trajectory is just one. These
must be summed to obtain the overall wave function. This involves
a procedure similar to (3.139), adapted to three dimensions. In
most charged particle optical systems, the action integrals in the
solutions (3.169, 3.186) are very large relative to h ¯. We showed
in the preceding section that only trajectories infinitesimally separated from the classical trajectory, together with the classical
trajectory itself, contribute appreciably to the overall wave function. In this case it is a very good approximation to assume that
the action integrals are applied only along the classical trajectory.
This can be further understood by applying the operator for the
canonical momentum P to the wave function (3.186). This gives
−ih ¯v ψ(x, t) = P ψ(x, t).
(3.187)
Geometrically, this means that the canonical momentum vector P
is perpendicular to the surfaces of constant phase. The kinetic momentum vector p is everywhere tangent to the classical trajectory.
In the presence of a magnetic vector potential A, this gives rise to
a geometrical interpretation as shown in Figure 3.4.