Chapter 3. On Quantum Theory

3.1 Planck's Quantum Postulate

The quantum theory of light was initiated by Max Planck (1858-1947) in 1900. Until then, the electromagnetic theory and thermodynamics yielded important laws for the electromagnetic radiation from a cavity of a hot 'blackbody', the so-called 'thermal' or blackbody radiation. Stefan-Boltzmann's law, stated that the radiated electromagnetic energy is proportional to the fourth power of the absolute temperature $T$ of the blackbody. Wien's displacement law defined the wavelength of maximum radiation energy, and Wien's distribution law governed the energy distribution among the different wavelengths of the blackbody radiation spectrum. The latter fitted very well the short-wavelength part of the experimental distribution and its regions not far from the peak. The long-wavelength part of the experimental distribution was fitted well only by the Rayleigh-Jeans law. However, the Rayleigh-Jeans law was 'catastrophically' unfit for short wavelengths.

At first, Planck interpolated between the equations of Wien and Rayleigh-Jeans and obtained his equation, which fitted the whole experimental distribution curve and became the Planck distribution law. However, this law was half-empirical. Planck saw that he could theoretically derive his equation by introducing a postulate that

the radiation energy of the blackbody is emitted in discrete quantities or quanta of energy; the energy $E_q$ of a radiation quantum is proportional to the frequency $f$ of the radiation.

Thus,

E_q = h \cdot f,

where $h$ is a proportionality factor. The value of $h$ was laboriously derived by Planck from the blackbody radiation with his distribution law. It is the Planck constant,

h = 6.63 \cdot 10^{-34} \ \mathrm{J \cdot s} = 4.14 \ \mathrm{feV \cdot s}

Planck attributed great importance to his constant, naming it 'the quantum of physical action' (Section 7.11).

Planck's postulate contradicted established physical concepts. First, it was and still is not understood what in the nature of emission of the thermal radiation, with its continuous spectrum, could cause the energy to be quantized. Then, it was well-known that vibrational and wave energy is proportional to the square of the amplitude. Again, it could not be understood what in the physical nature of radiating oscillators made their emitted energy proportional to the frequency of the radiation. Therefore, in the presentation at the meeting of the Prussian Academy (December 14, 1900), Planck had to ask his colleagues to 'just let him through' with his postulate, without necessarily accepting it, and then see to where it leads. They did, first, because of the importance of the spectral distribution law. Then, also because they considered Planck's postulate as a device, applicable only to the derivation of the distribution law. It turned out, just in spite, that the postulate itself became a cornerstone of modern science, known as Planck's law.

Letting Planck 'through' with his postulate was not a nonnal procedure, and it would certainly not happen nowadays. If nowadays a statement contradicts "established scientific knowledge", then the regular procedure of referees is to stop reading or listening and to reject the presentation outright, without bothering to find out if there is anything of value. Any results achieved on the basis of statements which contradict established scientific knowledge are beyond consideration, no matter how important they appear.

3.2 The photoelectric effect

The photoelectric effect was discovered by Heinrich Hertz in 1887. He observed that shining ultraviolet light on metal electrodes caused their electrical charges to decrease. Then, in 1899, P. Lenard found that the discharge is due to the emission of free electrons out of the illuminated surfaces of metals. Lenard had established that the energy of the emitted electrons depends on the frequency of the radiation, and not on the intensity. The radiation intensity influences the number of emitted electrons. Also, there was no storing of radiation energy. When the intensity of radiation was very low, but the frequency was above the threshold frequency for the particular metal, the photoelectric response was as fast as under high intensity. On the other hand, when the frequency of the radiation was below the threshold, the effect could not be observed, no matter how high the light intensity or how long the irradiation. These features of the photoelectric effect could not be explained by a wave theory of light.

The quantum theory of the photoelectric effect was developed by A. Einstein in 1905. Einstein saw that Planck's postulate can be used for the photoelectric effect, if light is not only emitted in discrete quanta of energy, but also interacts with electrons as quanta. For the photoelectric effect to occur, the radiation quanta should interact individually, each with one free electron in the metal. The energy of the radiation quantum is fully transferred to the electron. The electron then spends part of the obtained energy to overcome the energy barrier on the surface of the metal; this is the 'exit energy', called (unfortunately) the 'work function' of the metal. The work function $W$ determines the threshold for the photoeffect, because if the energy of the quantum does not exceed the work function, then the electron (its thermal energy disregarded) cannot leave the metal. The energy $E_e$ of the emitted electron is then

E_e = hf - W

This is Einstein's equation for the photoelectric effect. It represents a simple balance of energy and yet is the work of a genius.

3.3 The obscurity of the photon concept

We should point out that Einstein's theory of the photoeffect does not explain the effect. The same questions which were asked regarding Planck's postulate, as to why is radiation quantized and why is the quantum energy proportional to the frequency of the radiation, remain unanswered, and new unanswered questions arise. If light is a flux of energy quanta or photons, as Einstein called them, thus a flux of corpuscules, then how could the definite wave features of light be explained? The answer to this question is the 'particle-wave duality principle', stating that in some phenomena light exhibits corpuscular properties, while in other phenomena it presents the wave features. Again, this is not an explanation, just a reminder about an ad hoc postulated principle, and the leaning on it.

Another arising question regards the physical nature of the photon, i.e., the quantum of radiation energy. This question arose first in Planck's postulate, but there it was not so acute, being hidden in the mass-performance of these quasi-particles. In the photo-electric effect the photons perform solo, and the question of what they are is burning. If they are particles, then of what? Perhaps of the vacuum, which is considered as carrier of the electromagnetic radiation. Then, the knocking of an electron out of the metal by such a vacuum particle sounds like a new problem.

The ability of photons to knock out electrons is also revealed in a number of phenomena, discovered after the photoelectric effect. Photons of energies exceeding $10 \ \mathrm{eV}$ can knock atomic valence electrons out of their atoms. This effect is known as the photoionization of atoms. Photons of the far ultraviolet, of soft and bard X-rays, can knock out electrons from deeper and deeper atomic orbits, corresponding to their energies. Such processes cannot be really understood without a clear knowledge of what the photons are.

The corpuscular nature of electromagnetic radiation is not fully manifested in the photoelectric effect and in photoionization, because in them the photon energy is completely absorbed and the momentum ofthe photon is not revealed. These effects correspond to inelastic collisions between the electron and the photon. The momentum of the photon is disclosed in the Compton effect.

3.4 The Compton Effect

A.H. Compton (1892-1962) published, in 1923. the results of his experiments on 'elastic' collisions between X-ray photons and free electrons in graphite and in other materials. In these experiments, an X-ray (or, later, a $\gamma$ -ray) photon of frequency $f$ collides with, e.g., a resting free electron. Due to the collision, the photon is scattered by an angle $\theta$ and has frequency reduced to $f'$ , while the electron moves away with velocity $v$ , at an angle $\varphi$ from the direction of the incident photon.

Compton considered that, according to the energy conservation law, the energy lost by the photon, $hf - hf'$ , should be equal to the kinetic energy of the recoil electron. Thus,

\begin{align} hf = hf' - m_ec^2\left[\left(1-\frac{v^2}{c^2}\right)^{-1/2}-1\right] \end{align}

Einstein's relativistic formula for the kinetic energy of the electron is used here, because the velocity $v$ of the 'recoil' electron is very large; here, $m_e$ is the rest mass of the electron.

The momentum $M$ of the photon can be found, as the momentum of any particle, by dividing its energy, $hf$ , by its velocity, $c$ ,

M = \frac{hf}{c}

Replacing $f / c = \lambda^{-1}$ , we have

M = \frac{h}{\lambda}

Due to the conservation of momentum, the momentum $m\vec{\textbf{v}}$ gained by the electron,

\begin{align} m\vec{\textbf{v}} = m_e\vec{\textbf{v}}\left(1-\frac{v^2}{c^2}\right)^{-1/2} \end{align}

should be equal to the change in the momentum of the photon, which is $h / \vec{\lambda} - h / \vec{\lambda'}$ . Hence,

\begin{align} \frac{h}{\vec{\lambda}} - \frac{h}{\vec{\lambda'}} = m_e\vec{\textbf{v}}\left(1-\frac{v^2}{c^2}\right)^{-1/2} \end{align}

Solving equations 1 and 3, Compton obtained an expression for the increase of the wavelength $\lambda - \lambda'$ , of the radiation in the collision,

\begin{align} \lambda - \lambda' = \frac{h}{m_ec}\cdot(1-cos\theta) \end{align}

It follows from this equation that the maximum increase in wavelength occurs when the photon is scattered backwards.
Then, $\theta = 180\degree$ , $cos\theta = -1$ , and

\lambda - \lambda' = \frac{2h}{m_ec}

Substituting the known values of $h$ , $m_e$ and $c$ , we obtain

\lambda - \lambda' = 4852 \ \mathrm{fm}

When the direction of the scattered photon forms a right angle with the direction of the incident photon, $\theta = 90\degree$ , $cos\theta = 0$ , the wavelength increases by

\lambda - \lambda' = 2426 \ \mathrm{fm}

independent of the wavelength of the incident radiation. This value of wavelength-increase is the Compton wavelength $\lambda_c$ ; thus,

\begin{align} \lambda_c = \frac{h}{m_ec} = 2426 \ \mathrm{fm} \end{align}

In other words, if an X-ray photon is scattered from a free electron by an angle of $90\degree$ , then the wavelength of the photon is lengthened by $\lambda_c$ or 2426 fm, the frequency of the photon decreases by $c / \lambda_c = m_ec^2 / h$ and the energy of the photon is reduced by $m_ec^2$ .

The Compton effect demonstrates fully the particle properties of the photon, thus strengthening the dissatisfaction with our not knowing the nature of the photon.

3.5 The de Broglie Waves

Louis Victor de Broglie (born 1892), in his dissertation (1924), came out with a hypothesis that electrons should have wave properties. He proceeded from the fact that a photon has momentum $M_p$ , which is the photon energy $hf$ , divided by its velocity, the velocity of light. Hence,

M_p = \frac{hf}{c} = \frac{h}{\lambda}

This formula should therefore be considered as a linkage between the particle property - the momentum, and the wave property - the wavelength $\lambda$ . For the photon we have, therefore, $\lambda = h / M_p$ . Assuming that the same relation should also be valid for electrons, de Broglie found that, for an electron having momentum $mv$ , there should be a corresponding wavelength $\lambda_b$ ,

\lambda_b = \frac{h}{mv}

The formula for the de Broglie wavelength $\lambda_b$ of the electron was proved experimentally in 1927, by C.J. Davisson and L.H. Germer, and also by G.P. Thomson. They showed that electron beams scattered from crystals produce diffraction and interference patterns, as do X-rays. The electron wavelengths measured from these patterns were in full agreement with the de Broglie formula.

3.6 On waves of matter

The success of de Broglie's presentation resulted in the development of a concept that matter itself has wave properties, and that the de Broglie wavelengths of particles are the wavelengths of "waves of matter", which represent the particles and can even replace them. It should be emphasized that the de Broglie waves are connected with the motion of the particles and due to this motion and do not represent the particles themselves. When the particle rests, there are no waves to describe it.

The de Broglie formula was derived for electrons. It was also proven right for neutrons, other "elementary" particles and atomic nuclei. These are what we call 'dense' particles. The use of the de Broglie formula to calculate the de Broglie wavelengths of atomic bodies, especially of jetliners and ships, so beloved in textbooks, is an unjust extrapolation, as in Aristotle's four-horse rule.

Contrary to what some texts expound, there is no correspondence between wave and particle properties of the electron and the photon. The resting and slowly moving electron has stable mass and no wave properties; the electron may also move with any velocity, smaller than the velocity of light. On the other hand, the photon must propagate with the velocity of light, at which it has energy, mass and momentum. At any other velocity, there are just no photons. Therefore, the photon cannot have a separate independent existence. It must be considered as a corollary effect of the propagation process of the electromagnetic wave. The particle properties of the photon may thus belong to the particles of the unknown carrier of the electromagnetic waves. The de Broglie electron wave should then be a wave, caused in that carrier by the motion of the electron (Section 8.1). Only then will the answer be given to the question of why should an electron exhibit wave properties (Sections 8.5, 8.6).

3.7 Rutherford's model of the atom

The 'nuclear' or 'planetary' model of the atom was created by Ernest Rutherford (1871-1937) in 1911. Based on his experiments on the scattering of $\alpha$ -particles, Rutherford found that almost all the mass of an atom is in its nucleus. This positively-charged nucleus has a radius in the order of 1 fm. The atomic electrons are orbiting around the nucleus on orbits of radii 10 to 100 pm and their total mass is 0.05 percent of the mass of the atom. Altogether, only a $10^{-15}$ part of the volume of the atom is filled by 'dense' particles. The rest of the volume, i.e., practically all the volume of atoms and atomic bodies, including ourselves, is 'empty', as empty as the vacuum space, and penetrable to particles of 'dense' matter: elementary particles and nuclei (and even to light atoms). The Rutherford model of the atom can be used to calculate the energies and velocities of the orbital electrons as functions of the radii of the orbits. However, the orbiting electrons are centripetally accelerated. Therefore, according to the electromagnetic theory, they have to radiate energy as long as they are accelerated and, losing energy, they would have to move on spirals, coming closer and closer to the nucleus. Thus, electron orbits and atoms in whole could not be stable and should collapse. Hence, the Rutherford model could not account for the stability of atoms. It was also unable to interpret the radiation spectra of atoms.

3.8 Bohr's orbits of the hydrogen atom

The first model of the hydrogen atom which agrees with the observed spectrum was proposed by Niels Bohr (1885-1962) in 1913. Bohr adapted the Rutherford model of the atom as the basis for his model. To overcome the collapse in Rutherford's model, Bohr introduced his postulate that

in the atom, there are certain allowed orbits; when the electron is on one of the allowed orbits, it does not radiate, and the energy ofthe atom is stable.

For the calculation of the allowed orbits, Bohr postulated, that the allowed orbits are such that the angular momentum $mvr$ of the electron on these orbits consists of an integral number of Planck's constants $\hbar \equiv h / 2\pi$ . Thus, Bohr's condition for orbit stability is

\begin{align} mvr \equiv n \cdot \hbar \end{align}

where n is an integer, $n = 1,2,3,...$

For the energies of emitted radiation, Bohr postulated that

the electron can make a transition from one allowed orbit, of energy $E$ , to another allowed orbit, of energy $E'$ , with the emission (or absorption) of a photon, the energy of which, $hf$ , is equal to the energy difference between the two orbits.

Thus,

\begin{align} hf = E' - E \end{align}

Considering the Coulomb attraction of the electron by the nucleus (proton) as the centripetal force for the orbit yields

\begin{align} \frac{ke^2}{r^2} = \frac{mv^2}{r} \end{align}

where $k$ is an SI unit coefficient, $k = 9 \mathrm{Gm / F}$ , and $e$ is the absolute value of the charges of the electron and the proton.

Editorial note:
The SI unit coefficient, $k$ , is "Coulomb's Force Constant".

k = \frac{1}{4\pi\epsilon_0} = 8.988 \cdot 10^9 \ \mathrm{m/F}

Where $\epsilon_0$ is the vacuum permittivity, also called the electric constant.

In Bohr's condition for orbit stability, equation 6, the velocity $v$ can now be eliminated, to obtain

\begin{align} r = n^2 \cdot \frac{\hbar^2}{kme^2} = n^2\cdot \mathrm{(53 \ pm)} \end{align}

In the ground state, $n = 1$ , the radius of the orbit is $53 \ \mathrm{pm}$ , which also is the Bohr radius of the hydrogen atom. The radii of the allowed orbits increase as the squares of the integer $n$ , i.e., as $1, 4, 9,...$ As the radii increase, the energy $E_n$ of the atom decreases at the same rate, i.e.,

E_n = \frac{E_1}{n^2}

In the ground state ( $n = 1$ ), the energy of the atom is $E_1 = 13.6 \ \mathrm{eV}$ , so that

E_n = \frac{(13.6 \ \mathrm{eV})}{n^2}

With his theory, Bohr calculated all the then-known series of lines in the spectra of hydrogen atoms (the Balmer, Lyman and Paschen series) as electron transitions between allowed orbits. When techniques were later developed to investigate the far-infrared end of the spectrum, the discovered new series of lines completely agreed with Bohr's theory.

In spite of the success of Bohr's theory, it is not right to see in it an explanation of the hydrogen atom and spectra. The theory is based on Bohr's postulates, which were introduced without explanation. It is not understood why the electron does not radiate, while centripetally accelerated in its motion on the "allowed" orbit, and why should the radii and the energies of the orbits be as postulated. It is true that wonderful results were achieved with these postulates, as also with the postulates of Planck and Einstein. However, results are not the only concern of science.

3.9 Linkage between Bohr's orbits and de Broglie waves

The physical explanation of Bohr's postulate might be linked with the relation between the radii of the Bohr orbits and the de Broglie electron waves. We can rewrite the two expressions as

mv = \frac{n \cdot h}{2\pi r}

and,

mv = \frac{h}{\lambda_B}

This yields a united "Bohr-de Broglie" condition for orbit stability

2\pi r = n \cdot \lambda_B

circumferences of allowed atomic orbits must contain integral numbers of de Broglie electron wavelengths.

In this form, the postulate still does not explain why the electron on a stable orbit does not radiate, but it prepares the basis for a future explanation. We see that in the ground state, $n = 1$ , the circumference of the orbit is equal to the de Broglie wavelength of the orbital electron,

2\pi r_1 = \lambda_1

The velocity of this electron is

v_1 = \frac{h}{2\pi r_1 m} = 2.2 \cdot 10^6 \ \mathrm{m/s}

For the next allowed orbit, $n = 2$ ,

2\pi r_2 = 2\lambda_2

there are two de Broglie electron wavelengths on the circumference of the orbit. However, the radius of this orbit is 4 times that of the ground state orbit. Thus, the electron wavelength is twice that of the ground state, meaning that the velocity of the electron is half that of the ground state, and the energy is $E_1/4$ .

With the united Bohr-de Broglie orbit-stability condition, we have the following values for the allowed orbits:

Orbit	Radius, wavelength			Speed	Energy
$n = 1$	$2 \pi r_1 = \lambda_1 = 333 \ \mathrm{pm}$			$v_1 = 2.2 \cdot 10^6 \mathrm{m/s}$	$E_1 = 13.6 \ \mathrm{eV}$
$n = 2$	$2 \pi r_2 = 2\lambda_2$	$r_2 = 4 r_1$	$\lambda_2 = 2 \lambda_1$	$v_2 = v_1 / 2$	$E_2 = \ \ 3.4 \ \mathrm{eV}$
$n = 3$	$2 \pi r_3 = 3 \lambda_3$	$r_3 = 9 r_1$	$\lambda_3 = 3 \lambda_1$	$v_3 = v_1 / 3$	$E_3 = \ \ 1.5 \ \mathrm{eV}$
$n = 4$	$2 \pi r_4 = 4 \lambda_4$	$r_4 = 16 r_1$	$\lambda_4 = 4 \lambda_1$	$v_4 = v_1 / 4$	$E_4 = 0.85 \ \mathrm{eV}$

3.10 On quantum mechanics

The Bohr theory, so successful for the hydrogen atom, was unable to account for the spectra of other atoms, not even for the spectrum of helium. Modifications were introduced to allow elliptical orbits in this model, but the results were not fully satisfying. By 1925, Max Born, W. Heisenberg and P. Jordan, then E. Schroedinger, in 1926, founded and later actively developed the quantum mechanics. This was a priori a very sophisticated mathematical theory and, in years and decades of its successful march through physics, it became more and more sophisticated. Quantum mechanics can account for the complexities of atomic spectra, the structure of atoms, molecular binding, crystal structure and what not. There is not a single branch of physics and chemistry which did not gain from the "invasion" of quantum theory into its domain.

However, the quantum theory did not contribute towards the understanding of physical phenomena. For each difficulty, it introduced ad hoc postulates, principles, forbiddals and exclusions, without even a trial of explanation. Moreover, the inability of this mathematical theory to explain physical phenomena is acquitted by postulating that there are no explanations and will never be; that all there is to it are the mathematical equations and their solutions. With such attitudes, the seeking of understanding became a sign of ignorance. As a result, quantum mechanics is not understandable. R. Feynman, in his 1967 paper "The Character of Physical Laws", expressed it by saying:

I think I can safely say that nobody understands quantum mechanics.

The postulate "there is no explanation and cannot be" reminds one of the usual responses of a merchant, when asked for a piece of merchandise which he does not have. These are: "there is no such thing... ", "you will never find it", "they quit making it". If you believe him, then he is right. If you do not, you find the merchandise in another store.

Chapter 2: On Relativity Chapter 4: Bound Electrons and Positrons in the Vacuum Space