Abstract
The chapter tackles the difficult problem of bridging the concepts of Classical Mechanics and Electromagnetism with those of Quantum Mechanics. The subject, which is fascinating per se, is illustrated within a historical perspective, covering the years from 1900, when Planck’s solution of the black-body radiation was given, to 1926, when Schrödinger’s paper was published.
At the end of the 1800s, the main branches of physics (mechanics, thermodynamics, kinetic theory, optics, electromagnetic theory) had been established firmly. The ability of the physical theories to interpret the experiments was such, that many believed that all the important laws of physics had been discovered: the task of physicists in the future years would be that of clarifying the details and improving the experimental methods. Fortunately, it was not so: the elaboration of the theoretical aspects and the refinement in the experimental techniques showed that the existing physical laws were unable to explain the outcome of some experiments, and could not be adjusted to incorporate the new experimental findings. In some cases, the theoretical formulations themselves led to paradoxes: a famous example is the Gibbs entropy paradox [70]. It was then necessary to elaborate new ideas, that eventually produced a consistent body generally referred to as modern physics. The elaboration stemming from the investigation of the microscopic particles led to the development of Quantum Mechanics, that stemming from investigations on high-velocity dynamics led to Special Relativity.
The chapter starts with the illustration of the planetary model of the atom, showing that the model is able to justify a number of experimental findings; this is followed by the description of experiments that can not be justified in full by the physical theories existing in the late 1800s: stability of the atoms, spectral lines of excited atoms, photoelectric effect, spectrum of the black-body radiation, Compton effect. The solutions that were proposed to explain such phenomena are then illustrated; they proved to be correct, although at the time they were suggested a comprehensive theory was still lacking. This part is concluded by a heuristic derivation of the time-independent Schrödinger equation, based upon the analogy between the variational principles of Mechanics and Geometrical Optics.
In the final part of the chapter the meaning of the wave function is given: for this, an analysis of the measuring process is carried out first, showing the necessity of describing the statistical distribution of the measured values of dynamical quantities when microscopic particles are dealt with; the connection with the similar situations involving massive bodies is also analyzed in detail. The chapter is concluded with the illustration of the probabilistic interpretation of the wave function.
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Notes
- 1.
These particles are obtained by ionizing helium atoms.
- 2.
The meaning of term “nucleus” in this context is clarified in Sect. 7.8.1.
- 3.
The estimate means that for scale lengths equal or larger than those indicated in ({7.1}), an atom or a nucleus can be considered as geometrical points having no internal structure. The electron’s radius can be determined in a similar way using X-ray diffusion.
- 4.
The combination of the number of such electrons with other factors also determines whether the material is a conductor or a semiconductor (Chap. 18).
- 5.
The description is qualitative; for instance, it does not consider the band structure of the solid (Sect. 17.6).
- 6.
When the material is a conductor, \(E_\beta\) coincides with the Fermi level (Sect. 15.8.1), and W is called work function; in a semiconductor, \(E_\beta\) coindices with the lower edge E C of the conduction band (Sect. 17.6.5) and the minimum extraction energy (typically indicated with a symbol different from W) is called electron affinity (Sect. 22.2).
- 7.
The ratio \(R =\nu_R / c \simeq 1{.}1 \times 10^{5}\) cm-1 is called Rydberg constant. The formula was generalized in the 1880s to the hydrogenic-like atoms by Rydberg: the expression (7.4) of the frequencies must be multiplied by a constant that depends on the atom under consideration.
- 8.
The units of η and ν are \([\eta] =\mathrm{J~cm^{-2}}\) and \([\Uppi] = \mathrm{J}\), respectively.
- 9.
The concentration of electrons in the vacuum tube is small enough not to influence the electric field; thus, the latter is due only to the value of V AK and to the form of the electrodes.
- 10.
The most energetic electrons succeed in overcoming the effect of the reverse bias and reach the vicinity of the anode; they constantly slow down along the trajectory, to the point that their velocity at the anode vanishes. Then, their motion reverses and they are driven back to the cathode.
- 11.
Compare with the general definition (5.10) of w em, where the assumption of equilibrium is not made.
- 12.
This unphysical outcome is also called ultraviolet catastrophe.
- 13.
As the particles’ velocities that occur in solid-state physics are low, Special Relativity is not used in this book; the only exception is in the explanation of the Compton effect, illustrated in Sect. 7.4.3.
- 14.
The detailed calculation leading to (7.18) is shown in Prob. 6.1.
- 15.
Einstein’s hypothesis is more general than Planck’s: the latter, in fact, assumes that energy is quantized only in the absorption or emission events.
- 16.
This result shows that the physical constants appearing in ({7.29}) are not independent from each other. Among them, ν R is considered the dependent one, while q, \(m=m_0\), ϵ0, and h are considered fundamental.
- 17.
The wavelength associated to the particle’s momentum is called de Broglie’s wavelength.
- 18.
- 19.
From this interpretation it follows that \(\vert \psi \vert^2 \,\, {\rm d}^3 r\) is proportional to an infinitesimal probability, and \(\vert \psi \vert^2\) to a probability density.
- 20.
A more detailed discussion about the units of the wave function is carried out in Sect. 9.7.1.
- 21.
This issue is further discussed in Sect. 8.2.
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Rudan, M. (2015). From Classical Mechanics to Quantum Mechanics. In: Physics of Semiconductor Devices. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-1151-6_7
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DOI: https://doi.org/10.1007/978-1-4939-1151-6_7
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