LECTURE 
15 November 2002 

Language consultation: J.B., Ontario  Canada. 

QUANTISATION.
PERIODIC PERTURBATIONS IN A KEPLER PROBLEM. 

Maxwell guilty  Newton punished. Wavecorpuscular duality puzzle, a solution of which I presented in the previous lecture, was a chief reason for the departure from deterministic concepts of classical physics and introducing a misty formalism of Quantum Mechanics that took place at beginning of the Twentieth century. Another puzzle, which gave origin to Quantum Mechanics and resulted in rejection of the concept of a localized electron, was the controversy between Maxwell's theory of radiation and the evident stability of the atom. According to Maxwell's theory, an electron moving in the Coulomb field of the nucleus emits electromagnetic energy and must finally fall down on the nucleus. Physicists, however, investigating rich spectroscopic material have arrived at the conclusion that


at some specific conditions atomic electrons may not emit radiation.


They discovered rules (Bohr, Sommerfeld)  called quantum rules, specifying conditions at which the electron moving on the given orbit in the atom, molecule or in a crystal lattice may not emit electromagnetic radiation. One of the principal rules specifying radiationless states of the electron has the form:


(1) 
p(l) dl = nh, n = 1, 2, 3, ... ,


where h is Planck constant, p is momentum of the moving electron and the path integral is taken along a whole closed orbit of the electron.
Although various quantum rules have been successfully used for many years, efforts undertaken to identify dynamics hidden by rules remained without a success. As a consequence, frustrated physicists have questioned the applicability of Newtonian dynamics to the description of atomic systems, however, it was Maxwell's theory of radiation, which failed to describe emission of the photon from the atom.
Thus, on the basis of an entirely false argument, Newtonian dynamics has been eliminated from atomic theory and the concept of the localized electron has been remored from atomic physics for good. Quantum rules have been replaced by Schrödinger equation and theoretical considerations have been, in principle, limited to global characteristics of the atom without any invoking the notion of the electron trajectory. A discrete spectrum of integer numbers n of the old quantum theory have been replaced by a discrete spectrum of periodic solutions represented by the variable y of the wave equation of quantum theory  there are resons to think that n and y may be formally related in a following way:


(2) 

Linear spectrum of atomic hydrogen  beginning of the quantum puzzle. 
To remove the controversy between Classical theory and Quantum mechanics (which in fact is apparent) one must have in view that quantum considerations are, as a rule, concerned with energy and angular momentum  quantities, which in classical dynamics stay at the beginning of the whole calculation procedure, as they define initial conditions for equations of motion for the electron. In view of the above, one can formulate a thesis:
Classical dynamics and Quantum mechanics represent two complementary procedures of the atomic theory.
This rather surprising thesis, however, finds support in the fact that quantum rules of the old quantum theory were formulated within a classical concept of the localized electron. Moreover, sometime ago, in 1968, Russian physicist Chetaev has shown that the differential equation describing the stability of mechanical systems in the presence of periodic perturbations may have, as long as the perturbations satisfy some specific conditions, the form of Schrödinger equation. Thus, orbital motion of electrons is determine by equations of classical mechanics while quantum relations describe conditions of stacionary motion in the presence of some, undefined at the moment, periodic perturbations. One can suppose that this perturbations may have their origin in spin properties of the electron, as the quantisation rules are inherently related with Planck constant h.
Now, we will try to decipher dynamics hidden by quantum formalism to show the quite artificial nature of quantumclassical controversy. Undertaking this task, let us come back for a moment to early days of atomic spectroscopy, when the quantum puzzle was born. Balmer, de Broglie and Bohr  the birth of the quantum puzzle. The quantum puzzle takes its origin in Balmer's discovery, that is, in the numerical rule describing a discrete spectrum of the light emitted from hydrogen atoms. The discovered rule has the form: 

(3) 
v_{ nm} = v_{ 0} (1/n^{ 2}  1/m^{ 2}) , n = 1, 2, 3, ...m > n ,


where v_{ 0} represents the extrapolated limit of the observed spectrum.
Bohr, by multiplying both sides of the Balmer formula by the Planck constant h linked to aech other two principally different thinks: the photon emitted and by atom, an the atom, which has emitted this photon. In this way energy of the photon h v_{ n m} has been attributed to transition of the atom from some energetic state W_{ m} to some other energetic state W_{ n}: 

(4) 
h v_{ nm} = W_{ n}  W_{ m} .


Thus, discrete spectrum of light has its origin in a discrete spectrum of atomic energy states:


(5) 
W_{ n} = W_{ 0} / n^{ 2} and W_{ m} = W_{ 0} / m^{ 2} .


The observed limit of spectral series, h v_{ 0}, coresponds to the lowest, groundenergy state of the atom. W_{ 0}, therefore represents the final stage of radiative evolution of the atom.
When the ground, energy state of the atom, appeared to be equal to the binding energy of the electron in the atom measured in the ionization process: 

(6) 
W_{ 0} = 13.6 eV ,


Bohr could interpret the discrete spectrum of atomic energy levels in terms of electron energy levels  stops that the electron makes during radiative evolution of the atom. At the assumption that the electron is a point like particle moving according to laws of classical dynamics in the Coulomb field of nucleus, he could relate the discrete spectrum of energy levels with dimensions of eliptic orbits of the electron. Taking into account the well known relation of the Kepler problem:


a = Z e^{ 2} / 2 W ,


where a is the semi major axis of the ellipse and W is the binding energy of the electron, he could calculate dimensions of the atom. For the atom in the ground energy state he obtained:


a = Z e^{ 2} / 2 W_{ 0} = 0,538 • 10^{ 8} cm .


Although, Bohr calculated in fact the lenght of the semimajor axis of the elipse it has been called the Bohr radius, as he arbitrarily made the assumption (quite erroneous!!!) that the orbit of electron is a circular orbit (in fact the electron stays on a zero angular momentum freefall orbit, which we will show in the next lecture). Accidentally, a wrong assumption on the form of the electrons orbit enabled him to make one of the most important discoveries of atomic physics:
Bohr discovered that for a discrete spectrum of energetic levels of atomic spectroscopy holds a following relation:


(7) 
n l = 2 p r , the origin of the quantum puzzle


where l is de Broglie wavelength of the electron moving with a speed corresponding to the radius of the circular orbit r.
The result obtained by Bohr shocked a physical community: energy is quantised, electron orbits in the atom are quantised, wave properties of the electron are responsible for quantisation, the electron moves around the nucleus along a circular orbit  like the planet around the Sun. It was the origin of the quantum puzzle and of the circular (erroneous!!!) model of the atom. One can put here the question: how could the erroneous assumption on the form of electron orbits result in the correct spectrum of atomic energy levels? The answer has been given in fact by A. Sommerfeld some few years later, when he formulated this famous quantum rules like the one given by Eq.(1). He has shown that
the spectrum of atomic energy levels does not depend, in the first approximation, on the shape of the elliptic orbit.
The dependence of the spectrum on the eccentricity of the elliptic orbit may be observed, when the atom is placed in an electric or magnetic field (Stark and Zeeman). In spite of the fact that already Stark and Zeeman have shown that angular momentum of the electron in the ground energy state is equal to zero, a wrong, circular model of the atom is still presented in popular, as well as, in scientific literature. Leaving a discussion on the problem on the model of the atom for a later time, it will be a topic of next lectures, we will confine ourselves at the moment to search the solution of the quantum puzzle.
Undertaking the problem one must have in view the experimental fact: period of motion of the electron on the orbit is by few orders of magnitude, by a^{ 2} !!!, smaller then emission time of the photon from the atom. This implies that during the radiation process forces influencing orbital motion of the electron are much smaller then the Coulomb force and, therefore, analysis of quantum rules may be undertaken on the grounds of perturbation calculus developed by Gauss for analysis of slow evolutions of our solar system. Small perturbations in a Kepler problem. In principle, equations of motion for the electron moving in the Coulomb field of the nucleus and for the planet moving in gravitational field of the Sun are identical and, therefore, treatment of the evolution of the system in the presence of the perturbation forces is identical. Let us denote the perturbation force by df. In such a case differential equation describing the motion of the electron has the form: 

(8) 

Multiplying both sides of the equation (8) by v we obtain the following relation:


(9) 

which in view of the fact that the term in brackets represents a constant of motion in the Kepler problem may be written as the differential equation describing evolution of the considered constant of motion in time:


(10) 

In a similar way, multiplying both sides of equation (8) by ( x r ) we arrive at the differential equation describing the evolution of angular momentum, a second constant of motion of the Kepler problem:


(11) 

Multiplying Eq.(8) by ( x L ) we obtain the differential equation describing evolution of the third constant of motion, which the eccentricity vector ', directed from the nucleus towards the periapsis of the ellipse, is:


(12) 

If the force d f is much smaller then the Coulomb force, then the three constants of motion W, L, ' in right hand parts of the three equations given above can be considered constant, and for not too large intervals of time we obtain a set of three differential equations describing slow evolution of the Kepler orbit in space:


(13) 

(14) 

(15) 

where v and r describe unperturbed motion of the electron on the Kepler orbit.
Now, to illustrate how the perturbation method works, and to show that a far rething analogy may exist between the motion of the electron in the atom and the motion of big macroscopic objects, we will consider two simple problems of astronautics. Satellite slowing down by the friction force. A satellite moving in a rarefied Earth atmosphere is slowing down by a drug force, which in the first approximation is proportional to velocity of the satellite. Thus, 

(16) 
d f =  k v . 

Introducing this force into Eq.(13) one arrives at the following differential equation for a constant of motion W


(17) 
d W / d t = 2 k / m W


which after integration yields:


(18a) 
W = W_{ 0} e^{ (2k / m) t} . 

As a result of analogous calculations carried out for the angular momentum L we obtain:


(18b) 
L = L_{ 0} e^{ (2k / m) t} . 

Introducing these, evolving in time, quantities into equations defining semimajor axis a and eccentricity ' of the ellipse, one gets: 

(19a) 
a = a_{ 0} e^{  (2k / m) t} , 

(19b) 
' = const = '_{ 0} .


Thus, for the perturbation force, as given by Eq.(16), linear dimensions of the elliptic orbit decrease with time while the shape of the ellipse remains unchanged. Although the two sets of solutions  (18a), (18b) and (19a), (19b)  are entirely equivalent to each other their practical use is quite different. The first one is, as a rule, used in atomic physics, as we have no possibility to watch the electron moving around the nucleus, while the second one is, as a rule, used in a satellite navigation.
"Quantized" orbits of a satellite with periodically working engines. Let us assume now that there is an engine on a satellite, which periodically accelerates and deaccelerates the satellite. The engine works in such a way that the force, which periodically changes the velocity of the satellite, is controlled by the formula: 

(20) 
d f = f (r) v sin y (t) ,


and y depends indirectly on time in the following way:


(21) 
d y / d t ~ v^{ 2} .


Since the engine accelerates the satelite for some time, and deaccelerates it for same time, one can expect that there must be a situation when the global effect of the working engine is equal to zero. Let us try to find, for instance, conditions at which the major axis of the orbit defined in a Kepler problem by the binding energy W will remain constant. As it follows from Eq.(13) W may be constant if:


(22) 

where t = 0 is a moment of time when the satellite crosses the major axis of the elliptic orbit.
Since the distance r, as well as, the speed of the satellite v are symmetric with respect to the major axis of the ellipse, therefore, the value of the above integral will be equal zero if:


(23) 
y ( t ) =  y ( t ) ,


and this means that y ( t ) must satisfy criterion


(24) 
y ( 0 ) = 0 and y ( 1/2 T ) = n p ,


where n is an integer number. The above requirement imposed in differential equation (21) and applied to a satelite of mass m and moving in the gravitational field of the Earth yields:


(25) 

where G is gravitational constant and M is the mass of the Earth. Performing integration one obtains a dependence identical with Bohr's quantisation rule:


(26) 
W_{ n} =  W_{ 0} / n^{ 2} .


Thus, there is a discrete set of stationary orbits on which a satellite with periodically working engines can stay for an infinitely long time. If the observer would not know that the satellite has a periodically working engines, he would be surprised while orbits of the satellite are "quantised".
Translational precession. Trying to decipher dynamics hidden by the path quantum integral (1) let us write this integral in the alternative form: 

(27) 

Formally, for the integral given above we can write the following differential equivalent:


(28) 

Since on the left hand side of the above equation is the kinetic energy, and Planck constant h represents kinetic momentum of a spinning body, therefore, the term d n/d t must represent angular velocity. Thus, the above equation may be written in the following way:


(29) 

It follows from the above that h in the path quantum rule (1) represents the spin of the electron and n an is total number of revolutions of the spin axis of the electron.
Denoting the angle describing the angular motion of the spin axis of the electron by y, the above equation may be written in a following way: 

(30) 
d y / d l = m v / h ,


where dl represents the element of the path traveled by the electron with a speed v. Thus,
translations of the electron are accompanied by spin axis precession. It follows from the above that de Broglie wavelength is a distance at which the electron spin axis rotates by 2 p: 

(31) 

If the electron moves with a constant speed then we obtain the famous de Broglie relation:


(32) 
l = h / m v .


Identification of the frequency factor w in Eq. 29 with spin axis motion has great philosophical aspects. It follows from the analysis that the path quantum integral (1) defines conditions at which the rotating spin axis of the electron at the encircling of the orbit performs a total number of revolutions. But we still do not know why?
"Wave" field of the electron. Deciphering a physical meaning of the natural number n present in quantum rules and in particular the discovery of spin translational precession opened the way to identification the dynamics hidden by the phenomenon of quantisation. Taking into account that with the angular kinetic momentum of the electron h, there is an associated spin magnetic field represented by the magnetic moment m , on the basis of the AmperFaraday's electrodynamics we arrive at a conclusion that the moving electron has a periodically varryig component of the electric field, as given below: 

(33) 
E_{ s} =  1 / c [ ( dm / d t x r ) / r^{ 3} ] ,


where


(34) 
d m / d t = m w ,


and


(35) 
w = 2 E_{ kin} / ( h / 2 p ) .


Thus, the electric field of the moving electron has in fact the following form:


(36) 

where s (t) is determined by spin translational relation as given by Eq.(30). Solving Newton equation of motion for the spinning electron one can show that the "wave" term of electric field of the electron is responsible for "difraction" phenomena we have discussed in the first lecture, as well as, the quantisation of electron orbits in the atom we have just considered. One can easily check that:


(37) 
d W = E_{ s} • d l = 0 ,


if only


(38) 
W_{ n} = W_{ 0} / n ^{ 2} .


It follows from the entire discution above that the electric field E_{ s} as given by equations (33, 34 and 35) may be interpreted as the "wave" field of the electron de'Broglie was looking for until the last days of his life.


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