Statistics and Probability in Mechanics

  1. The intrinsic differences between the Maxwell velocity distribution and the law of radioactive decay [Wu, p.26, l.1-l.-6].

  2. From probability theory to quantum mechanics
    1. 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.
    2. In terms of measurement, the concept of eigenstates is a refinement of the the concept of probability in the quantum (discrete) level [Coh, chap I, §A.3].
    3. Consider Young's double-slit experiment. If there are only a few photons, we have no way 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 theory.
    4. 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.

  3. 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].