3.2. Particle motion in a general electromagnetic potential 179
The intensity measured in the plane at z b again depends on the
difference between these two phases. The phase shift between the
cases with the potential V (t) switched on and off is
q
Δθ =
V (t) dt.
(3.204)
h ¯
This is independent of the electron energy, and is therefore the
same for all constituent energies in the wave packet.
The phase shifts (3.198, 3.204) result in a measurable lateral shift
in the fringe pattern on the screen at z b in principle. The solutions
(3.191, 3.201) for the wave function have the striking property
that they depend only on the magnetic vector potential A(x, t),
and the electrostatic scalar potential φ(x, t). Nowhere do the magnetic field B or the electric field E appear. This is distinctly different from the classical description, in which these fields appear
explicitly in the Lorentz force law (2.15). Indeed no Lorentz force
is present in this quantum mechanical description. The reader is
referred to [3, 87] for further elaboration, including a description
of experimental results.
3.2.4 The Klein–Gordon equation and the covariant wave function
The effects of special relativity become important when the kinetic energy of the particle is comparable to, or greater than mc
2 ,
where m is the rest mass. A correct treatment must also include
the effects of spin. The reader is referred to the book by Bjorken
and Drell [6].
In many practical instruments, spin does not play an important
role in the optics, however. A useful approximation is available in
the Klein–Gordon equation which ignores spin, but retains Lorentz
covariance. As before, we confine our attention to the lab frame
The intensity measured in the plane at z b again depends on the
difference between these two phases. The phase shift between the
cases with the potential V (t) switched on and off is
q
Δθ =
V (t) dt.
(3.204)
h ¯
This is independent of the electron energy, and is therefore the
same for all constituent energies in the wave packet.
The phase shifts (3.198, 3.204) result in a measurable lateral shift
in the fringe pattern on the screen at z b in principle. The solutions
(3.191, 3.201) for the wave function have the striking property
that they depend only on the magnetic vector potential A(x, t),
and the electrostatic scalar potential φ(x, t). Nowhere do the magnetic field B or the electric field E appear. This is distinctly different from the classical description, in which these fields appear
explicitly in the Lorentz force law (2.15). Indeed no Lorentz force
is present in this quantum mechanical description. The reader is
referred to [3, 87] for further elaboration, including a description
of experimental results.
3.2.4 The Klein–Gordon equation and the covariant wave function
The effects of special relativity become important when the kinetic energy of the particle is comparable to, or greater than mc
2 ,
where m is the rest mass. A correct treatment must also include
the effects of spin. The reader is referred to the book by Bjorken
and Drell [6].
In many practical instruments, spin does not play an important
role in the optics, however. A useful approximation is available in
the Klein–Gordon equation which ignores spin, but retains Lorentz
covariance. As before, we confine our attention to the lab frame