190
Chapter 3. Wave optics
preparation of the quantum mechanical state for each measurement. The remarkable result is that bright and dark fringes are
observed on the phosphor screen. This is represented by the intensity distribution as a function of transverse position x plotted
at the bottom of the figure. This result has been observed directly
for a variety of particle species, indicating that this is more than
just a thought experiment [91, Page 1068, Chapter 38].
Analysis reveals that the bright bands occur where the path length
difference d sin θ between the two possible paths equals an integral number of wavelengths λ. Dark bands occur where the path
length difference equals a half-odd number of wavelengths. The
wavelength is related to the particle momentum p by the deBroglie
relation
h
p = ,
(3.229)
λ
where h is Planck’s constant. According to Einstein’s hypothesis, light propagates in the form of discrete energy packets called
photons, where each photon obeys this same relationship between
momentum and wavelength. Indeed, the same two-slit interference
was observed much earlier for light by Young. This experiment
and many related topics are authoritatively described by Born
and Wolf [11]. This is one of many illustrations of the close correspondence between light optics and particle optics.
As a related intuitive concept, we next consider the propagation
of a wave front through space and time, as described by Huygens’ principle. This will prove to be indispensible to formulating
a mathematical description of diffraction, which is derived in the
following sections. It will not be necessary to specifically invoke
the discreteness of particles. Rather we will take a more traditional
approach, regarding the wave function as continuous in space and
time. We will develop a scalar theory, where the optical disturbance is adequately described by the scalar wave function. This is
permissible, because we consider only particle motion in a vacuum,
which is inherently isotropic. We will ignore the intrinsic spin, since
it is not needed for this discussion. The reader is referred in the
Chapter 3. Wave optics
preparation of the quantum mechanical state for each measurement. The remarkable result is that bright and dark fringes are
observed on the phosphor screen. This is represented by the intensity distribution as a function of transverse position x plotted
at the bottom of the figure. This result has been observed directly
for a variety of particle species, indicating that this is more than
just a thought experiment [91, Page 1068, Chapter 38].
Analysis reveals that the bright bands occur where the path length
difference d sin θ between the two possible paths equals an integral number of wavelengths λ. Dark bands occur where the path
length difference equals a half-odd number of wavelengths. The
wavelength is related to the particle momentum p by the deBroglie
relation
h
p = ,
(3.229)
λ
where h is Planck’s constant. According to Einstein’s hypothesis, light propagates in the form of discrete energy packets called
photons, where each photon obeys this same relationship between
momentum and wavelength. Indeed, the same two-slit interference
was observed much earlier for light by Young. This experiment
and many related topics are authoritatively described by Born
and Wolf [11]. This is one of many illustrations of the close correspondence between light optics and particle optics.
As a related intuitive concept, we next consider the propagation
of a wave front through space and time, as described by Huygens’ principle. This will prove to be indispensible to formulating
a mathematical description of diffraction, which is derived in the
following sections. It will not be necessary to specifically invoke
the discreteness of particles. Rather we will take a more traditional
approach, regarding the wave function as continuous in space and
time. We will develop a scalar theory, where the optical disturbance is adequately described by the scalar wave function. This is
permissible, because we consider only particle motion in a vacuum,
which is inherently isotropic. We will ignore the intrinsic spin, since
it is not needed for this discussion. The reader is referred in the