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The superposition principle can be illustrated with an analogy from simple mathematics. Add two numbers and then take the square of their sum. As opposed to just adding the squares of the two numbers. Obviously, (2 + 3)2 is not equal to 22 + 32. The former is 25, and the latter are 13. In the language of quantum probability theory


| ø1 + ø2 | 2   | ø1 | 2 + | ø2 | 2

Where ø1 and ø2 are the individual wave functions. On the left - hand side, the superposition principle results in extra terms that cannot be found on the right - hand side. The left - hand side of the above relations is the way a quantum physicist would compute probabilities, and the right - hand side is the classical analogue. In quantum theory, the right - hand side is realized when we know, for example, which slit through which the electron went. Heisenberg was among the first to compute what would happen in an instance like this. The extra superposition terms contained in the left - hand side of the above relations would not be there, and the peculiar wave - like interference pattern would disappear. The observed pattern on the final screen would, therefore, be what one would expect if electrons were behaving like a bullet, and the final probability would be the sum of the individual probabilities. But when we know which slit the electron went through, this interaction with the system causes the interference pattern to disappear.

In order to give a full account of quantum recipes for computing probabilities, one has to examine what would happen in events that are compound. Compound events are ‘events that can be broken down into a series of steps, or events that consists of a number of things happening independently.’ The recipe here calls for multiplying the individual wave functions, and then following the usual quantum recipe of taking the square of the amplitude.

The quantum recipe is | ø1 • ø2 | 2, and, in this case, it would be the same if we multiplied the individual probabilities, as one would in classical theory. Thus, the recipes of computing results in quantum theory and classical physics can be totally different. The quantum superposition effects are completely nonclassical, and there is no mathematical justification per se why the quantum recipes work. What justifies the use of quantum probability theory is the coming thing that justifies the use of quantum physics  -it has allowed us in countless experiments to extend our ability to co-ordinate experience with the expansive nature of unity.

A departure from the classical mechanics of Newton involving the principle that certain physical quantities can only assume discrete values. In quantum theory, introduced by Planck (1900), certain conditions are imposed on these quantities to restrict their value; the quantities are then said to be ‘quantized’.

Up to 1900, physics was based on Newtonian mechanics. Large - scale systems are usually adequately described, however, several problems could not be solved, in particular, the explanation of the curves of energy against wavelengths for ‘black-body radiation’, with their characteristic maximum, as these attemptive efforts were afforded to endeavour upon the base-cases, on which the idea that the enclosure producing the radiation contained a number of ‘standing waves’ and that the energy of an oscillator if ‘kT’, where ‘k’ in the ‘Boltzmann Constant’ and ‘T’ the thermodynamic temperature. It is a consequence of classical theory that the energy does not depend on the frequency of the oscillator. This inability to explain the phenomenons has been called the ‘ultraviolet catastrophe’.

Planck tackled the problem by discarding the idea that an oscillator can attain or decrease energy continuously, suggesting that it could only change by some discrete amount, which he called a ‘quantum.’ This unit of energy is given by ‘hv’ where ‘v’ is the frequency and ‘h’ is the ‘Planck Constant,’ ‘h’ has dimensions of energy ‘x’ times of action, and was called the ‘quantum of action.’ According to Planck an oscillator could only change its energy by an integral number of quanta, i.e., by hv, 2hv, 3hv, etc. This meant that the radiation in an enclosure has certain discrete energies and by considering the statistical distribution of oscillators with respect to their energies, he was able to derive the Planck Radiation Formulas. The formulae contrived by Planck, to express the distribution of dynamic energy in the normal spectrum of ‘black-body’ radiation. It is usual form is:

8ðchdë/ë 5 (exp[ch/këT]   1.

Which represents the amount of energy per unit volume in the range of wavelengths between ë and ë + dë? ‘c’ = the speed of light and ‘h’ = the Planck constant, as ‘k’ = the Boltzmann constant with ‘T’ equalling thermodynamic temperatures.

The idea of quanta of energy was applied to other problems in physics, when in 1905 Einstein explained features of the ‘Photoelectric Effect’ by assuming that light was absorbed in quanta (photons). A further advance was made by Bohr(1913) in his theory of atomic spectra, in which he assumed that the atom can only exist in certain energy states and that light is emitted or absorbed as a result of a change from one state to another. He used the idea that the angular momentum of an orbiting electron could only assume discrete values, i.e., was quantized? A refinement of Bohr’s theory was introduced by Sommerfeld in an attempt to account for fine structure in spectra. Other successes of quantum theory were its explanations of the ‘Compton Effect’ and ‘Stark Effect.’ Later developments involved the formulation of a new system of mechanics known as ‘Quantum Mechanics.’

What is more, in furthering to Compton’s scattering was to an interaction between a photon of electromagnetic radiation and a free electron, or other charged particles, in which some of the energy of the photon is transferred to the particle. As a result, the wavelength of the photon is increased by amount Äë. Where:

Äë = ( 2h / m0 c ) sin 2 ½.

This is the Compton equation, ‘h’ is the Planck constant, m0 the rest mass of the particle, ‘c’ the speed of light, and the photon angle between the directions of the incident and scattered photons. The quantity ‘h/m0c’ and  is known to be the ‘Compton Wavelength,’ symbol ëC, which for an electron is equal to 0.002 43 nm.

The outer electrons in all elements and the inner ones in those of low atomic number have ‘binding energies’ negligible compared with the quantum energies of all except very soft X- and gamma rays. Thus most electrons in matter are effectively free and at rest and so cause Compton scattering. In the range of quantum energies 105 to 107 electro volts, this effect is commonly the most important process of attenuation of radiation. The scattering electron is ejected from the atom with large kinetic energy and the ionization that it causes plays an important part in the operation of detectors of radiation.

In the ‘Inverse Compton Effect’ there is a gain in energy by low-energy photons as a result of being scattered by free electrons of much higher energy. As a consequence, the electrons lose energy. Whereas, the wavelength of light emitted by atoms is altered by the application of a strong transverse electric field to the source, the spectrum lines being split up into a number of sharply defined components. The displacements are symmetrical about the position of the undisplaced lines, and are prepositional of the undisplaced line, and are propositional to the field strength up to about 100 000 volts per. cm. (The Stark Effect).

Adjoined alongside with quantum mechanics, is an unstretching constitution taken advantage of forwarded mathematical physical theories - growing from Planck’s ‘Quantum Theory’ and deals with the mechanics of atomic and related systems in terms of quantities that can be measured. The subject development in several mathematical forms, including ‘Wave Mechanics’ (Schrödinger) and ‘Matrix Mechanics’ (Born and Heisenberg), all of which are equivalent.

In quantum mechanics, it is often found that the properties of a physical system, such as its angular moment and energy, can only take discrete values. Where this occurs the property is said to be ‘quantized’ and its various possible values are labelled by a set of numbers called quantum numbers. For example, according to Bohr’s theory of the atom, an electron moving in a circular orbit could occupy any orbit at any distance from the nucleus but only an orbit for which its angular momentum (mvr) was equal to nh/2ð, where ‘n’ is an integer (0, 1, 2, 3, etc.) and ‘h’ is the Planck’s constant. Thus the property of angular momentum is quantized and ‘n’ is a quantum number that gives its possible values. The Bohr theory has now been superseded by a more sophisticated theory in which the idea of orbits is replaced by regions in which the electron may move, characterized by quantum numbers ‘n’, ‘I’, and ‘m’.

Properties of [Standard] elementary particles are also described by quantum numbers. For example, an electron has the property known a ‘spin’, and can exist in two possible energy states depending on whether this spin set parallel or antiparallel to a certain direction. The two states are conveniently characterized by quantum numbers + ½ and   ½. Similarly properties such as charge, Isospin, strangeness, parity and hyper-charge are characterized by quantum numbers. In interactions between particles, a particular quantum number may be conserved, i.e., the sum of the quantum numbers of the particles before and after the interaction remains the same. It is the type of interaction - strong electromagnetic, weak that determines whether the quantum number is conserved.

The energy associated with a quantum state of an atom or other system that is fixed, or determined, by given set quantum numbers. It is one of the various quantum states that can be assumed by an atom under defined conditions. The term is often used to mean the state itself, which is incorrect accorded to: (i) the energy of a given state may be changed by externally applied fields (ii) there may be a number of states of equal energy in the system.

The electrons in an atom can occupy any of an infinite number of bound states with discrete energies. For an isolated atom the energy for a given state is exactly determinate except for the effected of the ‘uncertainty principle’. The ground state with lowest energy has an infinite lifetime hence, the energy, in principle is exactly determinate, the energies of these states are most accurately measured by finding the wavelength of the radiation emitted or absorbed in transitions between them, i.e., from their line spectra. Theories of the atom have been developed to predict these energies by calculation. Due to de Broglie and extended by Schrödinger, Dirac and many others, it (wave mechanics originated in the suggestion that light consists of corpuscles as well as of waves and the consequent suggestion that all [standard] elementary particles are associated with waves. Wave mechanics are based on the Schrödinger wave equation describing the wave properties of matter. It relates the energy of a system to wave function, usually, it is found that a system, such as an atom or molecule can only have certain allowed wave functions (eigenfunction) and certain allowed energies (Eigenvalues), in wave mechanics the quantum conditions arise in a natural way from the basic postulates as solutions of the wave equation. The energies of unbound states of positive energy form a continuum. This gives rise to the continuum background to an atomic spectrum as electrons are captured from unbound states. The energy of an atom state sustains essentially by some changes by the ‘Stark Effect’ or the ‘Zeeman Effect’.

The vibrational energies of the molecule also have discrete values, for example, in a diatomic molecule the atom oscillates in the line joining them. There is an equilibrium distance at which the force is zero. The atoms repulse when closer and attract when further apart. The restraining force is nearly prepositional to the displacement hence, the oscillations are simple harmonic. Solution of the Schrödinger wave equation gives the energies of a harmonic oscillation as:


En = ( n + ½ ) h.


Where ‘h’ is the Planck constant,  is the frequency, and ‘n’ is the vibrational quantum number, which can be zero or any positive integer. The lowest possible vibrational energy of an oscillator is not zero but ½ h. This is the cause of zero-point energy. The potential energy of interaction of atoms is described more exactly by the ‘Morse Equation,’ which shows that the oscillations are anharmonic. The vibrations of molecules are investigated by the study of ‘band spectra’.

The rotational energy of a molecule is quantized also, according to the Schrödinger equation, a body with the moment of inertial I about the axis of rotation have energies given by:

EJ = h2J ( J + 1 ) / 8ð 2I.

Where J is the rotational quantum number, which can be zero or a positive integer. Rotational energies originate from band spectra.

The energies of the state of the nucleus are determined from the gamma ray spectrum and from various nuclear reactions. Theory has been less successful in predicting these energies than those of electrons because the interactions of nucleons are very complicated. The energies are very little affected by external influence but the ‘Mössbauer Effect’ has permitted the observations of some minute changes.

In quantum theory, introduced by Max Planck 1858-1947 in 1900, was the first serious scientific departure from Newtonian mechanics. It involved supposing that certain physical quantities can only assume discrete values. In the following two decades it was applied successfully by Einstein and the Danish physicist Neils Bohr (1885-1962). It was superseded by quantum mechanics in the tears following 1924, when the French physicist Louis de Broglie (1892-1987) introduced the idea that a particle may also be regarded as a wave. The Schrödinger wave equation relates the energy of a system to a wave function, the energy of a system to a wave function, the square of the amplitude of the wave is proportional to the probability of a particle being found in a specific position. The wave function expresses the lack of possibly of defining both the position and momentum of a particle, this expression of discrete representation is called as the ‘uncertainty principle,’ the allowed wave functions that have  described stationary states of a system

Part of the difficulty with the notions involved is that a system may be in an indeterminate state at a time, characterized only by the probability of some result for an observation, but then ‘become’ determinate (the collapse of the wave packet) when an observation is made such as the position and momentum of a particle if that is to apply to reality itself, than to mere indetermincies of measurement. It is as if there is nothing but a potential for observation or a probability wave before observation is made, but when an observation is made the wave becomes a particle. The wave-particle duality seems to block any way of conceiving of physical reality - in quantum terms. In the famous two-slit experiment, an electron is fired at a screen with two slits, like a tennis ball thrown at a wall with two doors in it. If one puts detectors at each slit, every electron passing the screen is observed to go through exactly one slit. But when the detectors are taken away, the electron acts like a wave process going through both slits and interfering with itself. A particle such an electron is usually thought of as always having an exact position, but its wave is not absolutely zero anywhere, there is therefore a finite probability of it ‘tunnelling through’ from one position to emerge at another.

The unquestionable success of quantum mechanics has generated a large philosophical debate about its ultimate intelligibility and it’s metaphysical implications. The wave-particle duality is already a departure from ordinary ways of conceiving of tings in space, and its difficulty is compounded by the probabilistic nature of the fundamental states of a system as they are conceived in quantum mechanics. Philosophical options for interpreting quantum mechanics have included variations of the belief that it is at best an incomplete description of a better-behaved classical underlying reality ( Einstein ), the Copenhagen interpretation according to which there are no objective unobserved events in the micro - world (Bohr and W. K. Heisenberg, 1901-76), an ‘acausal’ view of the collapse of the wave packet (J. von Neumann, 1903-57), and a ‘many world’ interpretation in which time forks perpetually toward innumerable futures, so that different states of the same system exist in different parallel universes (H. Everett).

In recent tars the proliferation of subatomic particles, such as there are 36 kinds of quarks alone, in six flavours to look in various directions for unification. One avenue of approach is superstring theory, in which the four-dimensional world is thought of as the upshot of the collapse of a ten-dimensional world, with the four primary physical forces, one of gravity another is electromagnetism and the strong and weak nuclear forces, becoming seen as the result of the fracture of one primary force. While the scientific acceptability of such theories is a matter for physics, their ultimate intelligibility plainly requires some philosophical reflection.

A theory of gravitation that is consistent with quantum mechanics whose subject, still in its infancy, has no completely satisfactory theory. In controventional quantum gravity, the gravitational force is mediated by a massless spin-2 particle, called the ‘graviton’. The internal degrees of freedom of the graviton require hij (÷) represent the deviations from the metric tensor for a flat space. This formulation of general relativity reduces it to a quantum field theory, which has a regrettable tendency to produce infinite for measurable qualitites. However, unlike other quantum field theories, quantum gravity cannot appeal to renormalizations procedures to make sense of these infinites. It has been shown that renormalization procedures fail for theories, such as quantum gravity, in which the coupling constants have the dimensions of a positive power of length. The coupling constant for general relativity is the Planck length,

Lp = ( Gh/c3 )½   10  35 m.

Supersymmetry has been suggested as a structure that could be free from these pathological infinities. Many theorists believe that an effective superstring field theory may emerge, in which the Einstein field equations are no longer valid and general relativity is required to appar only as low energy limit. The resulting theory may be structurally different from anything that has been considered so far. Supersymmetric string theory (or superstring) is an extension of the ideas of Supersymmetry to one - dimensional string-like entities that can interact with each other and scatter according to a precise set of laws. The normal modes of super-strings represent an infinite set of ‘normal’ elementary particles whose masses and spins are related in a special way. Thus, the graviton is only one of the string modes - when the string-scattering processes are analysed in terms of their particle content, the low-energy graviton scattering is found to be the same as that computed from Supersymmetric gravity. The graviton mode may still be related to the geometry of the space-time in which the string vibrates, but it remains to be seen whether the other, massive, members of the set of ‘normal’ particles also have a geometrical interpretation. The intricacy of this theory stems from the requirement of a space-time of at least ten dimensions to ensure internal consistency. It has been suggested that there are the normal four dimensions, with the extra dimensions being tightly ‘curled up’ in a small circle presumably of Planck length size.

In the quantum theory or quantum mechanics of an atom or other system fixed, or determined by a given set of quantum numbers. It is one of the various quantum states that an atom can assume. The conceptual representation of an atom was first introduced by the ancient Greeks, as a tiny indivisible component of matter, developed by Dalton, as the smallest part of an element that can take part in a chemical reaction, and made very much more precisely by theory and excrement in the late-19th and 20th centuries.

Following the discovery of the electron (1897), it was recognized that atoms had structure, since electrons are negatively charged, a neutral atom must have a positive component. The experiments of Geiger and Marsden on the scattering of alpha particles by thin metal foils led Rutherford to propose a model (1912) in which nearly, but all the mass of an atom is concentrated at its centre in a region of positive charge, the nucleus, the radius of the order 10 -15 metre. The electrons occupy the surrounding space to a radius of 10-11 to 10-10 m. Rutherford also proposed that the nucleus have a charge of ‘Ze’ and is surrounded by ‘Z’ electrons (Z is the atomic number). According to classical physics such a system must emit electromagnetic radiation continuously and consequently no permanent atom would be possible. This problem was solved by the development of the quantum theory.

The ‘Bohr Theory of the Atom,’ 1913, introduced the concept that an electron in an atom is normally in a state of lower energy, or ground state, in which it remains indefinitely unless disturbed. By absorption of electromagnetic radiation or collision with another particle the atom may be excited - that is an electron is moved into a state of higher energy. Such excited states usually have short lifetimes, typically nanoseconds and the electron returns to the ground state, commonly by emitting one or more quanta of electromagnetic radiation. The original theory was only partially successful in predicting the energies and other properties of the electronic states. Attempts were made to improve the theory by postulating elliptic orbits (Sommerfeld 1915) and electron spin (Pauli 1925) but a satisfactory theory only became possible upon the development of ‘Wave Mechanics,’ after 1925.

According to modern theories, an electron does not follow a determinate orbit as envisaged by Bohr, but is in a state described by the solution of a wave equation. This determines the probability that the electron may be located in a given element of volume. Each state is characterized by a set of four quantum numbers, and, according to the Pauli exclusion principle, not more than one electron can be in a given state.