Quantum mechanics is one of the largest achievements in th 20th centurty,the theory has been developed. The main concept of the theory was 'quanta',which is the smallest scale of discrete things. This concept was arised in the 19th century when main stress was given to the study of heat properties of various objects. The model considered for the study was black body(an object that absorbs and re-emit all the radiations) . The energy of radiatin waves with frequency v is ,
Basic quantum mechanics is set up to describe how fixed numbers of particles behave - say in externally applied electromagnetic or other fields. But to describe things like fields one must allow particles to be created and destroyed. In the mid-1920s there was already discussion of how to set up a formalism for this, with an underlying idea again being to think in terms of virtual oscillators - but now one for each possible state of each possible one of any number of particles. At first this was just applied to a pure electromagnetic field of non-interacting photons, but by the end of the 1920s there was a version of quantum electrodynamics (QED) for interacting photons and electrons that is essentially the same as today. To find predictions from this theory a so-called perturbation expansion was made, with successive terms representing progressively more interactions, and each having a higher power of the so-called coupling constant α1/137. It was immediately noticed, however, that self-interactions of particles would give rise to infinities, much as in classical electromagnetism. At first attempts were made to avoid this by modifying the basic theory. But by the mid-1940s detailed calculations were being done in which infinite parts were just being dropped - and the results were being found to agree rather precisely with experiments. In the late 1940s this procedure was then essentially justified by the idea of renormalization: that since in all possible QED processes only three different infinities can ever appear, these can in effect systematically be factored out from all predictions of the theory. Then in 1949 Feynman diagrams were introduced to represent terms in the QED perturbation expansion - and the rules for these rapidly became what defined QED in essentially all practical applications. By the 1970s the dozen or so standard processes discussed in QED had been calculated to order α^2 - and by the mid-1980s the anomalous magnetic moment of the electron had been calculated to order α^4, and nearly one part in a trillion.[3]
OLD QUANTUM THEORY:-
The old quantum theory was a collection of results from the years 1900-1925 which predate modern quantum mechanics. The theory was never complete or self-consistent, but was a collection of heuristic prescriptions which are now understood to be the first quantum corrections to classical mechanics.[1] The Bohr model was the focus of study, and Arnold Sommerfeld[2] made a crucial contribution by quantizing the z-component of the angular momentum, which in the old quantum era was inappropriately called space quantization (Richtungsquantelung). This allowed the orbits of the electron to be ellipses instead of circles, and introduced the concept of quantum degeneracy. The theory would have correctly explained the Zeeman effect, except for the issue of electron spin.
The main tool was Bohr Sommerfeld quantization, a procedure for selecting out certain discrete set of states of a classical integrable motion as allowed states. These are like the allowed orbits of the Bohr model of the atom, the system can only be in one of these states, and not in any states in between. The theory did not extend to chaotic motions, because it required a full multiply periodic trajectory of the classical system for all time in order to pose the quantum conditions.
The old quantum theory lives on as an approximation technique in quantum mechanics, called the WKB method. Semi-classical approximations were a popular research subject in the 1970s and 1980s, after Gutzwiller discovered a semi-classical description for systems which are classically chaotic (see quantum chaos).
BASIC PRINCIPLES: The basic idea of the old quantum theory is that the motion in an atomic system is quantized, or discrete. The system obeys classical mechanics except that not every motion is allowed, only those motions which obey the old quantum condition:
where the pi are the momenta of the system and the qi are the corresponding coordinates. The quantum numbers ni are integers and the integral is taken over one period of the motion. The integral is an area in phase space, which is a quantity called the action, which is quantized in units of Planck's constant. For this reason, Planck's constant was often called the quantum of action.
In order for the old quantum condition to make sense, the classical motion must be separable, meaning that there are separate coordinates qi in terms of which the motion is periodic. The periods of the different motions do not have to be the same, they can even be incommensurate, but there must be a set of coordinates where the motion decomposes in a multi-periodic way.
The motivation for the old quantum condition was the correspondence principle, complemented by the physical observation that the quantities which are quantized must be adiabatic invariants. Given Planck's quantization rule for the harmonic oscillator, either condition determines the correct classical quantity to quantize in a general system up to an additive constant.
WHY DO WE NEED QUANTUM THEORY?:-
Classical mechanics works perfectly in explaining the world, and is enough for charting the trajectory of probes sent to Jupiter and beyond. So why are we not content with classical physics? A question, Where does the need for quantum theory arise? Quantum theory unveils a new level of reality, the world of intrinsic uncertainty, a world of possibilities, which is totally absent in classical physics. And world of quantum mechanics not only offers us the most compelling explanation of physical phenomena presently known, but is also one of the most important source of modern technologies, providing society with a cornucopia of devices and instruments.
PLANK'S QUANTUM HYPOTHYSIS:-
A black body which is maintained at a constant temperature T steadily loses energy from its surface in the form of electromagnetic radiation. Since the atoms composing the black body are in contact with a heat bath at temperature T, each atom has approximately kT amount of energy, where k = Boltzmann's constant. Since the atoms are jiggling around due to thermal motion, classical electromagnetic theory then predicts that all wavelength's of radiation, in particular upto infinitely short wavelengths, should be emitted by a black body. This classical prediction for the spectrum of radiation that is emitted by such a black-body is contradicted by experiment. Max Planck, a German physicist, correctly explained the experimentally measured black- body spectrum by making the epoch-making conjecture in 1900 that electromagnetic waves are the macroscopic manifestations of packets of wave-energy called photons. Planck further made the quantum hypothesis that the energy of photons is quantized in the sense that the energy of the photons only comes in discrete packets, the smallest packet called a quantum. Photons are massless quantum particles, and all phenomena involving electromagnetic radiation can be fully explained by the quantum theory of photons. The phenomenon of electromagnetic radiation is a classical approximation to the quantum theory of photons. As classical electro-magnetism is a valid approximation when the typical energy of the photon is less than the characteristic energy of the instrument with which the experiment is being performed. Photons can have wavelength from zero to infinity. For a wave of frequency f, or equivalently, of wavelength , the quanta of energy are given by ( is the velocity of light)
from the above that the shorter the wavelength of a photon, the greater is its quantum of energy. The constant h is an empirical constant of nature, required by dimensional analysis, and is called Planck's constant. Its numerical value is given by
h
Also,
The quantum postulate immediately solves the problem of the black-body spectrum; radiation with increasingly short (ultra-violet) wavelength is incorrectly predicted by classical physics to make an increasingly large contribution to the energy loss. Due to Planck's quantum postulate, to emit even a single quanta of ultra-violet radiation would require a minimum energy much larger than the typical thermal energy of about kT that is available for emission, and hence would not be present in the radiated spectrum of a black-body. The black-body spectrum obtained from Planck's postulate is confirmed by experiment, and was the first success of quantum theory.
BASIC POSTULATES OF QUANTUM THEORY:-
Consider a particle of mass m, 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 p=mv. Let us study what classical and quantum mechanics have to say about the particle confined to a box.
Figure 1: Particle in a Box
Classical Description: The classical 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 from to-v, and it continues to travel until it hits the other wall and bounces back and so on. We hence have
Figure 2: Spacetime diagram of a Classical Particle in a Box.
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 H-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. 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, but its position inside the well is a random variable. When it is not observed, the quantum particle exists in a random state, which in physics is called a virtual state; in particular, the position of the particle can be anywhere within the interval L. What happens if we perform a measurement to actually "see" what is the position occupied by the particle? The measurement will find the electron to be always at some definite point; the act of measurement causes the electron to make a quantum transition from its virtual state to an actual physical state. In summary, the quantum particle has two forms of existence: a virtual state when it is not being observed, and a physical state which is observed when a measurement is performed on the particle.
QUANTUM THEORY AND OTHER DISCIPLINES:-
It would not be an exaggeration to hold that quantum mechanics has revolutionized our understanding of nature. Our understanding of quantum mechanics is still far from complete and one can be sure there are a lot of surprises awaiting us in the future. On the more practical side, quantum mechanics has led to the creation of most of the modern 20th century technologies that has served society so well. This connection of quantum mechanics with engineering and technology in general has been covered in the Section on Physics and Technology. It can also be safely predicted that 21st century technology will depend even more on quantum mechanics, and those who will study quantum mechanics will be richly rewarded. Quantum theory has had, and continues to have, a far reaching impact on a number of related and not so related fields. About a hundred and fifty years ago, chemistry had almost no connection with physics and concepts of chemistry such as valency, activity, solubility and volatility had more of a qualitative character. The first application of physics to chemistry started in the 19th century with the theory of heat, and was led by the hope of understanding the laws of chemistry in terms of the mechanics of atoms. One of the most successful application of quantum mechanics is the explanation of all the atoms which form the periodic table, and which is the starting point of all chemistry. With the explanation of chemical processes and chemical laws in terms of the quantum mechanics of atoms and molecules, a complete understanding of the laws of chemistry can now be sought in the laws of quantum mechanics. The present relation of biology to physics and chemistry is similar to that of chemistry to physics a hundred years ago. Biological concepts such as life, organ, cell function, perception, adaptation, etc. presently have no explanation in terms of physical and chemical laws. However, with the discovery of the DNA molecule (which contains more than 100,000 atoms), molecular biology now seeks the explanation of terrestrial life in terms of the atoms and molecules which compose the DNA, proteins and other biological macro-molecules, and hence has taken a giant step towards basing itself on the principles of quantum mechanics. The laws of physics and chemistry together with the laws of history (as embodied in Darwin's theory of evolution), have been suggested as forming the conceptual basis for explaining life. Biological evolution has taken place for about 4 billion years, during which nature could try out an almost infinite variety of combinations of atoms and molecules to come up with quasi-stable self-replicating biological macro-molecules which form the basis of living organisms; hence the element of history will probably have to be added to the laws of quantum mechanics to achieve an explanation of the principles of biology. Quantum mechanics has had a profound impact on mathematics and vice versa. The concept of being the probability for occurrence of the different values of the position x has had a major influence on the formal theory of probability. Quantum mechanics has made major contributions to the theory of functional analysis since physically measurable quantities such as position, charge etc. are realized by operators acting on function space and which forms the subject matter of functional analysis. The theory of renormalization, which is an essential feature quantum field theory (the application of the principles of quantum mechanics to the quantization of a classical field), is even today beyond the scope of rigorous mathematics, and find its final justification in the experimental validation of its predictions. More recently, the rapid progress of string theory has opened up new connections between theoretical physics and mathematics, in particular with the more specialized branches of mathematics such as (algebraic and differential) geometry and topology of higher dimensional manifolds, number theory, singularity theorems, knot and link theory, infinite dimensional algebra's and groups and so on. The importance of quantum theory for disciplines more closely related to physics has been even more seminal. Astronomy and astrophysics is concerned with formation and distribution of galaxies, and with the stars which compose them. The processed taking place inside a star as well as its composition are largely determined by quantum theory; in particular, the synthesis via fusion of heavier elements inside a star, the evolution of a star, whether it will become a supernova or a neutron star or a white dwarf or a black hole are all the result of quantum mechanical processes going on inside a star. Cosmology studies the large scale structure of the universe, and one would have thought that quantum mechanics, which apparently is concerned with the micro-world, would not have any relevance to cosmology; in fact, nothing could be further from the truth. The current hot big-bang theory of cosmology relies solely on quantum mechanics to explain the events which occurred within the first 1000 seconds and which are the determinate events which shaped all later evolution of the universe. If one probes even closer to within a few trillionth's of a second after the big-bang, then the universe is seen to be completely dominated by the fundamental quanta's of nature and which find their explanation in the quantum field theory, string theory, gauge fields, quantum gravity etc.[4]
REFRENCES:
[1] http://www.quantiki.org/mediawiki/images/9/9d/Blackbody.jpg
[2] http://www.wolframscience.com/reference/notes/1056a
[3] Stephen Wolfram, A New Kind of Science Notes for Chapter 9: Fundamental Physics Section: Quantum Phenomena Page 1056.
[4] http://srikant.org/core/node12.html
[5] Quantum chemistry by John P. Lowe, Kirk A. Peterson.
