The intrinsic differences between the Maxwell velocity distribution and the law of radioactive decay [Wu, p.26, l.1-l.-6].
From probability theory to quantum mechanics
When considering a beam of incident light, we can use the concept of percentage to discuss
how much of it is reflected [Hec,
p.139, lc, l.-12]. When considering a single photon,
we must use the concept of probability to determine if it will reflect from or
transmit through the interface.
In terms of measurement, the concept of eigenstates is a refinement of the the concept of probability in the quantum
[Coh, chap I, §A.3].
Consider Young's double-slit experiment. If there are only a few photons, we have
to predict their behavior because the individual impacts are distributed in a
random manner [Coh, p.13, l.15]. However, if there are a large number of photons passing
through the slits, we can predict the interference pattern using probability
In the microscopic regime, we must discard the deterministic concepts such
as the trajectory (positions with time as the parameter) and initial conditions
(simultaneous measurement of the position and momentum) of a particle. In
the microscopic regime, a particle is characterized by its energy and momentum [Coh,
p.11, (A-1)]. According to the uncertainty principle, Dt
= +¥ and
Dx = +¥. The deterministic
concepts like trajectories and initial conditions are allowed in classical
mechanics because the Planck constant is small in the macroscopic scale. The
uncertainty principle forbids us from applying these deterministic concepts to
the microscopic regime.
Remark. Modern textbooks emphasize how we use Schrödinger's
equation to successfully explain quantum phenomena, but fail to address why the
basic methodology in classical mechanics cannot be applied to the quantum
regime. Consequently, their discussion about quantum mechanics is incomplete.
Frustrated total internal reflection [Fur, §2.2E4].
Remark 1. The reason that the tunnel effect is difficult to understand is not
because the effect is contrary to the classical predictions [Coh, p.74, l.3; In
fact, optics does provide examples of total internal reflection and evanescent waves. Therefore, one cannot
say that the tunnel effect contradicts classical mechanics.], but because modern
textbooks on quantum mechanics fail to use concrete examples to highlight the
essence of the effect [whose magic is similar to awakening a dormant animal].
[Fur, p.86, Fig. 2.9] provides exactly what we need: rescuing the transmitted
evanescent wave at a close range before it dies out;
the range must be in the order of the skin depth so that the wave may jump over
the barrier and pass through another medium. Conclusion: In [Fur, p.86, Fig.
2.9], the transmittance from medium n to the medium n' is zero [Coh, p.71,
(24)]; the transmittance from medium n, passing through medium n', to medium n
is nonzero [Coh, p.73, (30); Fur, p.86, (2.84)]; the transmittance changes from
zero to nonzero because the transmittance counts only the normal component of
the Poynting vector [Wangs, p.421, (25-71); because the evanescent wave travels
along the interface [Wangs, p.418, l.-2], its
existence does not contribute to the transmittance] and because the photon has a nonzero probability [consider the wave
function in the gap rather than the normal component of the Poynting vector on
the interface] of presence in the
gap of medium n' [Coh, p.71, l.-13].
Remark 2. Born's view [Born, p.48, l.-14-p.49, l.16].