The dawn of the twentieth century heralded a new era in intellectual and experimental pursuits of understanding the nature of the universe around us. Not only did man achieve a long cherished dream of flying in the air, but nature was seen from viewpoints previously unimagined. In 1901, Max Plank in connection with "black body" radiation, proposed the quantum theory of radiation, which was definitely formulated by Einstein in 1905. According to the Quantum theory radiation occurs, not continuously permitting all possible values as assumed by the wave theory, but in a discrete quantified form, as integral multiples of an elementary quantum of energy. This means that energy should be considered atomic in nature like matter. The concept was verified by Compton in connection with the scattering of X-rays and Plank's hypothesis was extended to the extent that radiation is considered corpuscular in nature, made up of discrete quanta, which are shot out in space with the velocity of light. This, however, did not nullify the previous proofs that light and other radiations are waves. The quantum is now known as photon, and its energy is given by hf or more accurately (1/2+n)hf, where n is an integer, h the Plank's constant having an accurately determined value of 6.624x10-34 joule-second, and f the frequency of radiation. (Looking at the construction of the energy expression and the unit of the Plank's constant this author is tempted to suggest that it could be an integral of energy with respect to time. It is dimensional problems like this that have resulted in the introduction of dimensionless expressions in thermodynamics and fluid mechanics.) The photon is said to have no mass of its own for reasons to be discussed later. The quantum theory has been instrumental in explaining the photoelectric effect, commonly observed in solar cells which would have worked just as well even without the quantum theory, though their development may not have been as rapid.
It may be stated here that zeros are basically of two kinds -- the subtractional zero which can be obtained by repeated subtractions and represents nothingness, and the divisional zero which is obtained by repeated divisions and represents a negligible fraction. The quantum theory, in a way, defined the ultimate fraction or arithmetical atom that could be treated as a basic unity; thus doing away with the divisional zero that could have diverse negligible values. Mathematical propriety dictates that if a=0 and also b=0 then you cannot write a=b since it would lead to a/b=1 or 0/0=1 which amounts to making something out of nothing. Indeed, many such situations arise due to taking x^0, whose value is unity, as a function or part of a function of x. In fact, a constant numerical term in a function definition indicates origin relocation.
In 1905, Einstein theorized the constancy of the speed of light irrespective of the motion of the source or the observer by putting forward his famous theory of Special Relativity. The two fundamental postulates used in the theory are:
a) Measurement of absolute motion is impossible to an observer stationed on a moving system.
Comment: The measurement of absolute motion requires reference to a permanently fixed or unmoving point within the field of observation or perhaps the universe which we do not seem to have discovered yet because we have not looked for it. Equally it is impossible to conceive relative motion without assigning absolute motion to the observer or the observed or both.
b) The velocity of light is constant, independent of the relative motion of the source and observer.
Comment: It would apply perfectly to waves generated in a vast medium such as water (the ether is undone) in which two persons may be rowing their boats, but when you consider a travelling source throwing corpuscular projectiles being observed by another moving observer it becomes complicated to say the least. Waves possess velocity of propagation which can not be compared with the velocity of transportation possessed by particles.
He then went on to set up the constitutive equations of motion for two sets of reference frames having a uniform relative translatory motion with respect to an event visible from both reference frames, using independent time and space variables for each system with the speed of light as a constant. He used a unique technique of implied division by masked zeros developing equations of the form ax0=bx0 which, depending on progressive manipulations, can give various solutions of the type a=bxc where the sign and value of the constant c would depend on other constants and intermediate mathematical operations. By dexterous manipulation of these equations he deduced the interrelationships of the time and space variables in the two systems, which turned out to be identical to the Lorentz's transformation equations. Later scientists have used different mathematical tools to arrive at the same conclusions. In a nutshell, these equations mean that simultaneous time and distance measurements made from two vantage points moving relative to each other will have different values, and events that appear simultaneous to one observer may not seem so to the other. The two should agree to disagree. However, they do not contradict the fact that a particular event should instantaneously appear exactly the same to any observer irrespective of his speed and direction if occupying a particular location in space at a particular time, e.g. pictures taken with a high speed camera.
Since the variables are related to each other by a numerical quantity given by (1-v^2/c^2)^1/2 where v is the relative speed of the systems and c the speed of light, and since it is inconceivable for the term to be an imaginary quantity, v cannot be faster than c; or in other words, nothing can move faster than light. A corollary of the theory also gives an equation for the addition of velocities which ensures that the resultant will not exceed the velocity of light, so that two photons linearly approaching one another, each travelling with the speed of light have a relative velocity equal to the velocity of light, not twice. Such a postulate can only be satisfied by adjusting or redefining the unit of time as a half of the normal second with reference to the specific situation. This compression of the second would, of course, be only a mathematical manipulation and not affect any natural phenomena. The basic flaw in the time dilation concept seems to be that time lag or observation delay due to the time taken by light to reach an observer, which is an aberration and changes with distance irrespective of speed of travel, has been confused with real time.
Another expression derived from the principles of relativity is of the form m = m0/(1-v2/c2)1/2; where m0 is the rest mass i.e the mass of a body measured when at rest relative to the observer, m the "effective mass" i.e. mass of the body measured when it is moving with a velocity v relative to the same observer, and c is the velocity of light. Since v is always less than c, m will always be greater than m0 except when v itself has an imaginary value which is regarded as impossible. It can be shown that the increase in mass is approximately equal to the kinetic energy divided by c squared.
This brings us to the most famous of Einstein's deductions -- the mass-energy relationship given by E = mc2, where E is the energy equivalent of mass m. In other words, one kilogram of any substance is equal to 9x10^16 joules or 2.51x10^10 kilowatt-hours of energy and vice versa. This means that even if all the earth's known resources of energy production were utilized, no more than a few tons of matter would be produced in a year. However, it is this conversion of matter into energy that produces all the heat in the nuclear fission reactors. But of course, the reactors would have worked the same even if the equation was not known. The mass-energy relationship was verified by Kaufman and Bucherer using the electrons of Beta rays from radium with widely different velocities, ranging up to as high as 0.99c. The mass-energy relationship carries interesting implications for the elastician. If the velocity of electromagnetic waves or light c is taken as the equivalent of distortional waves in a solid elastic medium, and the energy equivalent of mass as strain energy, then mass becomes a function of strain in the medium.
The confirmation of special relativity and its corollaries ushered in a new era of natural philosophy. The classical theory of mechanics in which the sums of forces and moments at a point had to be separately zero and matter and energy had to be conserved individually was superseded. It was obvious that accurate results could only be obtained by balancing the accounts of the energy equivalent of mass as well as the other forms of energy such as kinetic, potential, inertial, deformational etc., or the mass equivalent of the energies as the case might be.
Rutherford carried out experiments on the scattering of alpha particles, which are the positively charged bare helium nuclei, by thin foils of matter and found that although most of the alpha particles suffered only small deflection due to multiple scattering, yet there were a certain number that were scattered through much larger angles. To accommodate this phenomenon he proposed, in 1911, the nuclear atom model in which most of the mass and positive charge was concentrated at the center forming the nucleus, and the electrons revolved in circles around the nucleus in a manner similar to the planets revolving around the Sun. Although this model solved Rutherford's immediate problem, it was found to be in conflict with the electromagnetic theory, as revolving electrons must emit radiation at all times, and constantly consume energy which would compromise the stability of the atomic structure. Twelve years later in 1923, Niels Bohr applied the quantum theory to the Rutherford atom model and developed his theory of atomic structure which is now widely accepted in a further modified form as it resolves many of the dilemmas faced by earlier physicists. The theory is based on two postulates reproduced below as stated by Rajam:
i) The first postulate referring to the electronic structure, states the electrons cannot revolve in all possible orbits as suggested by the classical theory, but only in certain definite orbits satisfying quantum conditions. These orbits may, therefore, be considered as privileged orbits, non radiating paths of the electron.
ii) The second postulate referring to the origin of spectral lines states that radiation of energy takes place only when an electron (instantaneously) jumps from one permitted orbit to another (without existing in any intermediate location - hence the phrase quantum leap). The energy thus radiated which is equal to the difference in the energies of the two orbits involved must be a quantum of energy hv.
The mathematical deduction based on the above postulates produced quantitative results which agreed with available experimental data, and thus the assumptions were accepted as physical laws.
Bohr's simple theory of circular orbits, in spite of its many successes, was unable to explain certain fine details of the hydrogen spectrum such as the Balmer series, suggesting that for each quantum number there might be several orbits of slightly different energies. Somerfeld, in 1915, modified Bohr's theory by introducing the ideas of motion of electrons in elliptical orbits and of the consequent relativistic variation of the mass of the electron. By the application of the relativistic mass energy relationship, he found the path of the electron to be a complicated curve, known as a rosette -- a precessing ellipse, doubly periodic. In interpreting the observed fine structure of spectral lines, he was forced to introduce a selection rule to preclude some of the orbits permitted by his mathematics. The Rutherford-Bohr-Somerfeld atom model was only a partial success as it could only predict three out of five components of the H-alpha line. Nevertheless, Somerfeld was able to underscore the concept of "Spatial Quantization" i.e not only distances and forms, but also directions must have discrete values with permissible conditions. Nature's limitations were getting more and more exposed. The void was beginning to develop holes. In 1923, Uhlenbeck and Gouldsmit in order to explain satisfactorily the intricate spectral phenomena such as the fine structure and the Zeeman effect i.e the splitting of spectral lines under the influence of an applied magnetic field, put forward the hypothesis of the spinning electron, with a spin quantum number which is always 1/2 for an electron although most other quantum numbers are integral. Plank's joule-seconds had to be multiplied with something having time in the denominator to give straight forward energy.
Thus came into being the Quantized vector atom model which was further developed by Pauli, Stern and Gerlach with the help of a host of others. Although this is not the last word in atom models, we will stop here and take a look at some of the parallel developments.
Minkowsky, using the principles of special relativity and the four dimensional geometry of Rieman was able, in 1908, to present to the world a new and unique concept of four dimensional space time continuum which is represented mathematically by the second order differential equation ds2 = dx2 + dy2 + dz2 + c2dt2. The square of time has no practical significance by itself, but multiplied by the square of velocity it represents an area like the other terms of the equation. Similarly, in the Cartesian coordinate system x*y, y*z and z*x represent area but x*x means nothing. In terms of 20th century physical science this means that the minute displacement represented by ds is not completely described by the three coordinate dimensions dx, dy and dz, as in classical Euclidean geometry; but in addition by the time dimension dt also, forming the fourth coordinate of relativistic geometry. The quantity ds is renamed as a "point event" and is not exactly a distance in space, but an element in the four dimensional Minkowsky space-time continuum which is `simultaneously finite and boundless.' However, in this continuum, matter is able to assert itself as "the pressure of matter distorts the curvature of the four dimensional space-time continuum which is the physical universe." Einstein was able to correlate the idea with the balancing of centripetal and centrifugal forces in a string and stone respectively, which are attached and whirled giving the stone a curvilinear motion. In 1915, he extended his special theory of relativity that was limited to systems with uniform linear motion, to encompass systems moving in any way, even with accelerated velocity, and in particular to a special case of accelerated motion that is involved in the most common phenomenon of gravitation. Using, in turn, Minkowski's model, Einstein in his General Theory of Relativity worked out the law governing the motion of a body in a distorted and curved space-time continuum using the advanced mathematical tool of tensor calculus. He was able to show that Newton's inverse square law of gravitation is a first approximation of his relativistic law which is claimed to have been successfully tested in several astrophysical phenomena, such as the advance of perihelion of the planet mercury, the shift of spectral lines in the light received from the companions of Sirius etc. Now, if space can be pulverized (quantized) and curved under pressure, it can hardly be a void; it seems more like a personality.
With the advent of the quantum theory, physicists were obliged to admit a dual nature, wave and particle, for radiant energy for the simple reason that the Plank energy equation contains a frequency term for the photon that is assumed to be a massless particle. In 1924, Louis de Brogli took a bold step forward and suggested that matter, like radiation, has dual nature i.e particles believed to possess discrete rest mass such as molecules, atoms, protons, electrons and the like might exhibit wave-like properties under appropriate circumstances. There was a certain amount of initial pessimism about the theory of matter waves which later came to be known as de Brogli waves because of their difference from electromagnetic waves as their velocity of propagation is given by u = c2/v, where v is the velocity of the moving matter and c the velocity of light. The de Brogli waves have a wavelength L=h/mv, where h is Plank's constant and m the mass of the matter particle. However, the discovery of the diffraction of electrons in 1927 by Davisson and Gremer provided experimental confirmation of the theory. Dempster (1927) and Esterman (1930) obtained diffraction effects with hydrogen and helium atoms, thus lending support to the matter wave theory. The only snag is that u can be greater than c since v is less than c which, oddly enough, Einstein did not mind. So, in view of the rule that wave velocity is the product of wave-length and frequency, it may be said that every particle of matter has a characteristic frequency given by f = mc2/h, for notations defined earlier or a characteristic time period of 1/f.
