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3.2. Particle motion in a general electromagnetic potential 177
the solenoid being excited to a specific value and the solenoid
current turned off. This is
q
Δθ =
A · ds.
(3.198)
h ¯
Applying Stokes’s theorem we write
q
Δθ =
(v × A) · dS
h ¯ S
=
B · dS,
(3.199)
S
where S is any surface bounded by the closed ray paths. Equivalently,
q Φ
Δθ =
,
(3.200)
h ¯
where Φ is the total magnetic flux enclosed by the ray paths. The
magnetic vector potential is nonzero in the vicinity of the classical trajectories, but the magnetic field is zero there. Therefore, no
magnetic Lorentz force acts on the electron. This result was first
predicted by Ehrenberg and Siday [25], and later expanded upon
by Aharonov and Bohm [2].
An electrostatic analog was first predicted by Aharonov and Bohm
[2]. This is shown schematically in Figure 3.6. An electron traversing the upper path passes through a conducting tube. When the
electron is inside the tube, near its center, an electrostatic potential V (t) is momentarily applied to the tube by an external source.
Assuming the length of the tube is much larger than its diameter,
the electron experiences no electric field during the time interval in
which V (t) is switched on. Consequently, no electrostatic Lorentz
force is exerted on the electron.
This is only possible in principle if the electron is represented by a
wave packet, rather than a monochromatic plane wave. The wave
packet must be sufficiently localized for the above condition to
be met, whereas a monochromatic plane wave has infinite extent.
This inevitably requires a spread in momentum as well, consistent
with the Heisenberg uncertainty principle. Measurable intensity at
3.2. Particle motion in a general electromagnetic potential 177
the solenoid being excited to a specific value and the solenoid
current turned off. This is
q
Δθ =
A · ds.
(3.198)
h ¯
Applying Stokes’s theorem we write
q
Δθ =
(v × A) · dS
h ¯ S
=
B · dS,
(3.199)
S
where S is any surface bounded by the closed ray paths. Equivalently,
q Φ
Δθ =
,
(3.200)
h ¯
where Φ is the total magnetic flux enclosed by the ray paths. The
magnetic vector potential is nonzero in the vicinity of the classical trajectories, but the magnetic field is zero there. Therefore, no
magnetic Lorentz force acts on the electron. This result was first
predicted by Ehrenberg and Siday [25], and later expanded upon
by Aharonov and Bohm [2].
An electrostatic analog was first predicted by Aharonov and Bohm
[2]. This is shown schematically in Figure 3.6. An electron traversing the upper path passes through a conducting tube. When the
electron is inside the tube, near its center, an electrostatic potential V (t) is momentarily applied to the tube by an external source.
Assuming the length of the tube is much larger than its diameter,
the electron experiences no electric field during the time interval in
which V (t) is switched on. Consequently, no electrostatic Lorentz
force is exerted on the electron.
This is only possible in principle if the electron is represented by a
wave packet, rather than a monochromatic plane wave. The wave
packet must be sufficiently localized for the above condition to
be met, whereas a monochromatic plane wave has infinite extent.
This inevitably requires a spread in momentum as well, consistent
with the Heisenberg uncertainty principle. Measurable intensity at