9
Introduction: The optical nature of a charged particle beam
Classical mechanics regards a single particle as a hypothetical
point, with the position and velocity known in principle at any
given instant in time. In reality, a single particle also behaves like
a wave. The wavelength is is equal to Planck’s constant h divided
by the particle momentum, where h = 6.6261 × 10
−34 Joule · sec. A
faster particle thus has a shorter wavelength than a slower particle. This so-called wave-particle duality is a hallmark of quantum
mechanics, which is a more accurate description of nature than
classical mechanics on the atomic and subatomic scale of dimensions. Classical mechanics is sufficiently accurate for many purposes, however, so it is worth retaining. Quantum mechanics has a
very specific correspondence with classical mechanics for a charged
particle in the limit of high energy. This will prove to be a central
theme in the present study.
Quantum mechanics teaches that the absolute square of the wave
amplitude is equal to the probability that a single measurement
finds the particle at a given position at any given instant in time.
Because this probability is described by a propagating wave, it is
not possible to know the position and momentum simultaneously
with perfect precision. This is known as the Heisenberg uncertainty principle, after the physicist who first elucidated it in the
1920s. A remarkable consequence of quantum mechanics, and one
which may appear counterintuitive at first, is that a single particle
can be described by two or more waves which interfere constructively or destructively with one another. Each wave corresponds to
a particular alternative path of motion of the particle, where the
actual path of motion is fundamentally unknowable. For example,
it is impossible to know which path in Figure 1.4 is the actual path
taken by the charged particle. Each possible path can be described
by a separate wave, where all of the waves corresponding to the
different paths propagate coherently, with a particular phase relationship to one another. They all interfere at the image plane to
cause a blurred spot (not depicted in the figure).
This interference is intimately related to diffraction, which results from the propagation, spreading, and interference of waves.
Introduction: The optical nature of a charged particle beam
Classical mechanics regards a single particle as a hypothetical
point, with the position and velocity known in principle at any
given instant in time. In reality, a single particle also behaves like
a wave. The wavelength is is equal to Planck’s constant h divided
by the particle momentum, where h = 6.6261 × 10
−34 Joule · sec. A
faster particle thus has a shorter wavelength than a slower particle. This so-called wave-particle duality is a hallmark of quantum
mechanics, which is a more accurate description of nature than
classical mechanics on the atomic and subatomic scale of dimensions. Classical mechanics is sufficiently accurate for many purposes, however, so it is worth retaining. Quantum mechanics has a
very specific correspondence with classical mechanics for a charged
particle in the limit of high energy. This will prove to be a central
theme in the present study.
Quantum mechanics teaches that the absolute square of the wave
amplitude is equal to the probability that a single measurement
finds the particle at a given position at any given instant in time.
Because this probability is described by a propagating wave, it is
not possible to know the position and momentum simultaneously
with perfect precision. This is known as the Heisenberg uncertainty principle, after the physicist who first elucidated it in the
1920s. A remarkable consequence of quantum mechanics, and one
which may appear counterintuitive at first, is that a single particle
can be described by two or more waves which interfere constructively or destructively with one another. Each wave corresponds to
a particular alternative path of motion of the particle, where the
actual path of motion is fundamentally unknowable. For example,
it is impossible to know which path in Figure 1.4 is the actual path
taken by the charged particle. Each possible path can be described
by a separate wave, where all of the waves corresponding to the
different paths propagate coherently, with a particular phase relationship to one another. They all interfere at the image plane to
cause a blurred spot (not depicted in the figure).
This interference is intimately related to diffraction, which results from the propagation, spreading, and interference of waves.