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Chapter 3. Wave optics
interference is manifest as alternating bright and dark intensity
bands called fringes. This is a purely wave-optical phenomenon
with no analog in classical mechanics. Its origin dates back several hundred years to the pioneering work of da Vinci, Grimaldi,
and Huygens.
In modern terms, the motion of a single charged particle in an
electromagnetic potential is described quantum mechanically by
a wave function, for which the absolute square is the probability
density that a single measurement will find the particle at precise space-time coordinates. A state which is a superposition of
two or more eigenstates with identical energy, but differing directions of momentum exhibits interference. Diffraction and interference are fundamental to a complete description of charged particle
optics.
The purpose of this section is to place the concept of diffraction
on a firm conceptual and mathematical basis, and then, based
on this, to describe several useful examples. Before embarking on
this, it is worthwhile to convey an intuitive feel for the subject by
considering a simple thought experiment, which was described by
Feynman, et. al. [30, Chapter 1, Volume 3]. This is shown schematically in Figure 3.8. We imagine a single charged particle with precisely known momentum and energy, incident perpendicularly on
an opaque screen S with two parallel slits. We assume that the
transverse position of the particle is completely unknown. Consequently, the particle could be stopped by the screen, or it could
pass through one of the two slits. Assuming it passes through one
of the slits, it is impossible to know which slit the particle passed
through. Having passed through one of the slits, the particle drifts
to a phosphor screen P at the bottom of the figure, where a flash of
light is emitted on impact. This represents a measurement of the
transverse position of the single particle on the phosphor screen.
By itself, this measurement does not reveal much information,
since the particle could land practically anywhere. This is corroborated by the fact that a second particle generally lands at a
different place from the first particle.
Chapter 3. Wave optics
interference is manifest as alternating bright and dark intensity
bands called fringes. This is a purely wave-optical phenomenon
with no analog in classical mechanics. Its origin dates back several hundred years to the pioneering work of da Vinci, Grimaldi,
and Huygens.
In modern terms, the motion of a single charged particle in an
electromagnetic potential is described quantum mechanically by
a wave function, for which the absolute square is the probability
density that a single measurement will find the particle at precise space-time coordinates. A state which is a superposition of
two or more eigenstates with identical energy, but differing directions of momentum exhibits interference. Diffraction and interference are fundamental to a complete description of charged particle
optics.
The purpose of this section is to place the concept of diffraction
on a firm conceptual and mathematical basis, and then, based
on this, to describe several useful examples. Before embarking on
this, it is worthwhile to convey an intuitive feel for the subject by
considering a simple thought experiment, which was described by
Feynman, et. al. [30, Chapter 1, Volume 3]. This is shown schematically in Figure 3.8. We imagine a single charged particle with precisely known momentum and energy, incident perpendicularly on
an opaque screen S with two parallel slits. We assume that the
transverse position of the particle is completely unknown. Consequently, the particle could be stopped by the screen, or it could
pass through one of the two slits. Assuming it passes through one
of the slits, it is impossible to know which slit the particle passed
through. Having passed through one of the slits, the particle drifts
to a phosphor screen P at the bottom of the figure, where a flash of
light is emitted on impact. This represents a measurement of the
transverse position of the single particle on the phosphor screen.
By itself, this measurement does not reveal much information,
since the particle could land practically anywhere. This is corroborated by the fact that a second particle generally lands at a
different place from the first particle.