�
�
184
Chapter 3. Wave optics
This is known as the Klein–Gordon equation. This has nonnormalized plane wave solutions given by
i
ψ(x, t) = exp
(± p · x − H t) .
(3.227)
h ¯
The reader can verify that this is the correct solution direct substitution into the Klein–Gordon equation. One can also verify that
H
2
2
2
= p c
2 + m c
4 for the free-particle case with φ = 0 and A = 0.
The interpretation of |ψ(x, t)|
2 in terms of probability density is
more subtle than in the nonrelativistic approximation. At modest
energies, |ψ|
2 remains a very good approximation to the relativistic probability density, however. The reader is referred to Bjorken
and Drell [6] for a detailed discussion.
In summary, all relevant information about quantum mechanical
particle motion in general, time-independent potentials A(x) and
φ(x) is contained in the relativistic wave function (3.225).
3.2.5 Physical interpretation of the wave function and its practical application
The central problem in optics is to understand the intensity distribution in a given transverse plane of an optical system. This might
be the image plane of an electron microscope, the Fourier plane
of a diffractometer, or the dispersion plane of an energy-dispersive
charged particle spectrometer, to name a few examples. The intensity distribution is proportional to the probability distribution for
finding a single particle at a given position. This in turn is given
by the absolute square of the wave function, which we have called
ψ(x, t). In the preceding analysis we have focused on calculating
the wave function for a single charged particle moving in a general
electromagnetic potential. The aim of the present section is to understand how to translate this into an areal intensity distribution,
such as one would measure in a practical instrument.
�
184
Chapter 3. Wave optics
This is known as the Klein–Gordon equation. This has nonnormalized plane wave solutions given by
i
ψ(x, t) = exp
(± p · x − H t) .
(3.227)
h ¯
The reader can verify that this is the correct solution direct substitution into the Klein–Gordon equation. One can also verify that
H
2
2
2
= p c
2 + m c
4 for the free-particle case with φ = 0 and A = 0.
The interpretation of |ψ(x, t)|
2 in terms of probability density is
more subtle than in the nonrelativistic approximation. At modest
energies, |ψ|
2 remains a very good approximation to the relativistic probability density, however. The reader is referred to Bjorken
and Drell [6] for a detailed discussion.
In summary, all relevant information about quantum mechanical
particle motion in general, time-independent potentials A(x) and
φ(x) is contained in the relativistic wave function (3.225).
3.2.5 Physical interpretation of the wave function and its practical application
The central problem in optics is to understand the intensity distribution in a given transverse plane of an optical system. This might
be the image plane of an electron microscope, the Fourier plane
of a diffractometer, or the dispersion plane of an energy-dispersive
charged particle spectrometer, to name a few examples. The intensity distribution is proportional to the probability distribution for
finding a single particle at a given position. This in turn is given
by the absolute square of the wave function, which we have called
ψ(x, t). In the preceding analysis we have focused on calculating
the wave function for a single charged particle moving in a general
electromagnetic potential. The aim of the present section is to understand how to translate this into an areal intensity distribution,
such as one would measure in a practical instrument.