Quantum law

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Introduction

The term quantum (Latin, "how much") refers to discrete units that the theory assigns to certain physical quantities, such as the energy of an atom at rest (see Figure 1, at right). The discovery that waves could be measured in particle-like small packets of energy called quanta led to the branch of physics that deals with atomic and subatomic systems which we today call Quantum Mechanics.

The foundations of quantum mechanics were established during the first half of the 20th century by Max Planck, Albert Einstein, Niels Bohr, Werner Heisenberg, Erwin Schrodinger, Max Born, John von Neumann, Paul Dirac, Wolfgang Pauli and others. Some fundamental aspects of the theory are still actively studied.

Quantum mechanics is a more fundamental theory than Newtonian mechanics and classical electromagnetism, in the sense that it provides accurate and precise descriptions for many phenomena that these "classical" theories simply cannot explain on the atomic and subatomic level. It is necessary to use quantum mechanics to understand the behavior of systems at atomic length scales and smaller. For example, if Newtonian mechanics governed the workings of an atom, electrons would rapidly travel towards and collide with the nucleus. However, in the natural world the electron normally remains in a stable orbit around a nucleus -- seemingly defying classical electromagnetism.

Definition

Quantum theory is the theoretical basis of modern physics that explains the nature and behavior of matter and energy on the atomic and subatomic level. In 1900, physicist Max Planck presented his quantum theory to the German Physical Society. Planck had sought to discover the reason that radiation from a glowing body changes in color from red, to orange, and, finally, to blue as its temperature rises. He found that by making the assumption that energy existed in individual units in the same way that matter does, rather than just as a constant electromagnetic wave - as had been formerly assumed - and was therefore quantifiable, he could find the answer to his question. The existence of these units became the first assumption of quantum theory.

Quantum mechanics is a fundamental physical theory which extends and corrects classical Newtonian mechanics, especially at the atomic and subatomic levels. It takes its name from the term quantum (Latin for "how much") used in physics to describe the smallest discrete increments into which something is subdivided. The terms quantum physics and quantum theory are often used as synonyms of quantum mechanics. Some authors refer to "quantum mechanics" in the restricted sense of non-relativistic quantum mechanics. Quantum mechanics should however be taken to mean quantum theory in its most general sense when used in this article.

THE DEVELOPMENT OF QUANTUM THEORY

In 1900, Planck made the assumption that energy was made of individual units, or quanta.

In 1905, Albert Einstein theorized that not just the energy, but the radiation itself was quantized in the same manner.

In 1924, Louis de Broglie proposed that there is no fundamental difference in the makeup and behavior of energy and matter; on the atomic and subatomic level either may behave as if made of either particles or waves. This theory became known as the principle of wave-particle duality: elementary particles of both energy and matter behave, depending on the conditions, like either particles or waves.

In 1927, Werner Heisenberg proposed that precise, simultaneous measurement of two complementary values - such as the position and momentum of a subatomic particle - is impossible. Contrary to the principles of classical physics, their simultaneous measurement is inescapably flawed; the more precisely one value is measured, the more flawed will be the measurement of the other value. This theory became known as the uncertainty principle, which prompted Albert Einstein's famous comment, "God does not play dice.

DESCRIPTION OF THE THEORY

There are a number of mathematically equivalent formulations of quantum mechanics. One of the oldest and most commonly used formulations is the transformation theory invented by Cambridge theoretical physicist Paul Dirac FRS, which unifies and generalizes the two earliest formulations of quantum mechanics, matrix mechanics (invented by Werner Heisenberg) and wave mechanics (invented by Erwin Schrödinger).

In this formulation, the instantaneous state of a quantum system encodes the probabilities of its measurable properties, or "observables". Examples of observables include energy, position, momentum, and angular momentum. Observables can be either continuous (e.g., the position of a particle) or discrete (e.g., the energy of an electron bound to a hydrogen atom).

Generally, quantum mechanics does not assign definite values to observables. Instead, it makes predictions about probability distributions; that is, the probability of obtaining each of the possible outcomes from measuring an observable.

Naturally, these probabilities will depend on the quantum state at the instant of the measurement. There are, however, certain states that are associated with a definite value of a particular observable. These are known as "eigenstates" of the observable ("eigen" meaning "own" in German). In the everyday world, it is natural and intuitive to think of everything being in an eigenstate of every observable. Everything appears to have a definite position, a definite momentum, and a definite time of occurrence.

However, Quantum Mechanics does not pinpoint the exact values for the position or momentum of a certain particle in a given space in a finite time, but, rather, it only provides a range of probabilities of where that particle might be. Therefore, it became necessary to use different words for a) the state of something having an uncertainty relation and b) a state that has a definite value.

The latter is called the "eigenstate" of the property being measured.A concrete example will be useful here. Let us consider a free particle.

In quantum mechanics, there is wave-particle duality so the properties of the particle can be described as a wave. Therefore, its quantum state can be represented as a wave, of arbitrary shape and extending over all of space, called a wavefunction.

The position and momentum of the particle are observables. The Uncertainty Principle of quantum mechanics states that both the position and the momentum cannot simultaneously be known with infinite precision at the same time.

Normalization

The that born interpretation suggests that there should be another requirement for acceptable wavefunctions. If the probability for a particle having wavefunction were evaluted over the entire space in witch the particle exits, then the probability should be equal to 1, or 100%. In order for this to be the core, wavefunctions are expected to be normalized. In mathematical terms, a wave function is normalized if and only if it is equal to 1,

According to the born interpretation, normalization also guarantees that the probability of a particle existing in all space is 100%.

Quantization of energy

It is interesting to consider what the hydrogen atom quantum numbers signify in terms of the classical model of the atom . this model, corresponds exactly to planetary motion in the solar system except that the inverse - square force holding the electron to the nucleus is electrical rather than gravitatiopnal. Two quantities are conserved that is, maintain a constant value at all time -in planetary motion : the scaler total energy and vector angular momentum of each planet.

Classically the total energy can have any value whatever, but it must, of course, be negative if the planet is to be trapped permanently in the solar system. In the quantum theory of hydrogen atom the electron energy is also constant, what while it may have any positive value, the only negative value the electron can have are specified by the formula. the quantization of electron energy in the hydrogen atom is therefore described by the principal quantum number n.

The theory of planetary motion can also be worked out from schrodingers equation, and it yields a similar energy restriction. However, the total quantum number n for any of the planet turns outs to be so immense that the separation of permitted levels is far too small to be observable. For this reason classical physics provides an adequate description of planetary motion but fails within the atom.

A CLASSICAL QUANTUM THEORY

The Quantum Theory was developed to explain the structure of atomic spectra and the random nature of radioactive decay both of which seemed to contradict the principles of classical physics. The historical context is that towards the end of the nineteenth century Classical Physics seemed well developed.

Everything was governed by exact physical laws and nature went about her business with the precision of a clock. Maxwell's laws explained almost everything and had enabled the technological revolution of electrical engineering. Then Thompson's discovery of the electron opened the way to Rutherford's revelation that atoms

were not solid balls, but empty space inhabited by a positive nucleus and orbiting electrons. The discovery ofradioactivity showed nature acting randomly breaking the neat laws of Newtonian determinism. The attempts of Lorentz to show that even mass was electromagnetic in nature had come to nothing. In 1905, Einstein reinterpreted the work of Lorentz and Poincaré in his theory of Special Relativity which gained wide

acceptance some 11 years later with the publication of his theory of General Relativity.

In my earlier works, I have shown that the historical ordering of discoveries is a determining factor in the development of theories. By imagining a different history in which we take selected modern discoveries and tools back in time, we are able to take alternative routes of reasoning and develop alternative theories

ENERGY IN QUANTUM MECHANICS

We return to our two-state system. The two-state system in general changes with time, and at any instance of time , the wavefunction of the system is given by

For the case when the two state refers to a spin system, we can think of as describing the precession of the spin about the z-axis. diagram of spin precessing The change of the wavefunction, for a time interval from to , is given by

Energy in quantum mechanics is the physical quantity that determines how the system will evolve in time, and becomes an operator; for this reason energy is given the special name of the Hamiltonian operator which acts on the wavefunction, causing it to change. In other words, for a small interval , the Hamiltonian is defined by

BASIC POSTRULAT OF QUANTUM THEOUR

The modern formulation of quantum theory rests primarily on the ideas of Erwin Schr dinger, Werner Heisenberg and P.A.M. Dirac. In the period from 1926-29 they laid the mathematical foundations for quantum mechanics, and this theory has successfully stood the test of innumerable experiments over the last seventy years. At present, there is not a single experimental result which cannot be explained by the principles of quantum theory. Unlike Einstein's theory of relativity which reinterpret's the meaning of classical concepts such as time, position, velocity, mass and so on, quantum theory introduces brand new and radical ideas which have no pre-existing counterpart in classical physics. To understand the counter-intuitive and paradoxical ideas that are essential for the understanding of quantum mechanics, we develop it in contrast to what one would expect from classical physics, and from intuition based on our perceptions of the macroscopic world. Recall that a classical system is fully described by Newton's laws. In particular, if we specify the position and velocity of a particle at some instant, its future evolution is fully determined by Newton's second law. In quantum mechanics, the behaviour of a quantum particle is radically different from a classical particle. The essence of the difference lies in the concept of measurement, which results in an observation of the state of the system. A classical particle, whether it is observed or unobserved, is in the same state. By contrast, a quantum particle has two completely different modes of existence, something like Dr Jekyll and Mr Hyde. When a quantum particle is observed it appears to be a classical particle having say a definite position or momentum, and is said to be in a physical state. However, when it is not observed, it exists in a counter-intuitive state, called a virtual, or a probabilistic, state. To illustrate the difference between a classical and quantum particle, let us study the behaviour of a classical and quantum particle confined inside a potential well of infinite depth. Consider a particle of mass , confined to a one-dimensional box, with perfectly reflecting sides due to the infinite potential, of length . Suppose the particle has a velocity , and hence momentum . Let us study what classical and quantum physics have to say about the particle confined to a box.

Classical Description The classical (Newtonian) description of a particle is that the particle travels along a well-defined path, with a velocity . Since the box has perfectly reflecting boundaries, every time the particle hits the wall, its velocity is reversed fromto , and it continues to travel until it hits the other wall and bounces back and so on. We hence have

The point to note is that the position and velocity of the classical particle are determined at every instant, regardless of whether it is being observed or not. Quantum Description A particle inside a potential well is similar to an electron inside an atom, and hence is be described by a resonant wave. The reason we choose the example is because it has all the features of the -atom, but is much simpler. The specific features of an electron inside an atom discussed earlier reflect the general principle of quantum theory which states that, if the momentum of the particle is fully known, we then have correspondingly no knowledge of its position. The precise relation between the uncertainty in position and momentum is given by the Heisenberg Uncertainty Principle discussed in Section 11.76. Note we can interchange the role of momentum with position, and a similar analysis follows. Hence, similar to the case of the Bohr atom, the electron in the potential well is in a bound state with a definite energy, but at the same time it no longer has a definite position. In summary, the particle inside the potential well has a definite momentum (and hence has definite energy), but its position inside the well is a random variable.

APPLICATIONS OF QUANTUM THEORY

Quantum mechanics has had enormous success in explaining many of the features of our world. The individual behavior of the subatomic particles that make up all forms of matter - electrons, protons, neutrons, and so forth - can often only be satisfactorily described using quantum mechanics.Quantum mechanics has strongly influenced string theory, a candidate for a theory of everything (see Reductionism. It is also related to statistical mechanics.

Quantum mechanics is important for understanding how individual atoms combine covalently to form chemicals or molecules. The application of quantum mechanics to chemistry is known as quantum chemistry (relativistic) quantum mechanics can in principle mathematically describe most of chemistry.

Quantum mechanics can provide quantitative insight into ionic and covalent bonding processes by explicitly showing which molecules are energetically favorable to which others, and by approximately how much. Most of the calculations performed in computational chemistry rely on quantum mechanics.

Much of modern technology operates at a scale where quantum effects are significant. Examples include the laser, the transistor, the electron microscope, and magnetic resonance imaging. The study of semiconductors led to the invention of the diode and the transistor, which are indispensable for modern electronics.

Researchers are currently seeking robust methods of directly manipulating quantum states. Efforts are being made to develop quantum cryptography, which will allow guaranteed secure transmission of information.

A more distant goal is the development of quantum computers, which are expected to perform certain computational tasks exponentially faster than classical computers. Another active research topic is quantum teleportation, which deals with techniques to transmit quantum states over arbitrary distances.

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