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Nature of excitation measured The type of spectroscopy depends on the physical quantity measured. Normally, the quantity that is measured is an intensity, either of energy absorbed or produced. Auger spectroscopy involves inducing the Auger effect with an electron beam.

In this case the measurement typically involves the kinetic energy of the electron as variable. The term "mass spectroscopy" is deprecated, for the technique is primarily a form of measurement, though it does produce a spectrum for observation. This spectrum has the mass m as variable, but the measurement is essentially one of the kinetic energy of the particle. Measurement process Most spectroscopic methods are differentiated as either atomic or molecular based on whether or not they apply to atoms or molecules.

The substance first must absorb energy. This energy can be from a variety of sources, which determines the name of the subsequent emission, like luminescence. Molecular luminescence techniques include spectrofluorimetry. Spectroscopy 4. Absorption Absorption spectroscopy is a technique in which the power of a beam of light measured before and after interaction with a sample is compared.

When performed with tunable diode laser, it is often referred to as Tunable diode laser absorption spectroscopy TDLAS. It is also often combined with a modulation technique, most often wavelength modulation spectrometry WMS and occasionally frequency modulation spectrometry FMS in order to reduce the noise in the system. Fluorescence Fluorescence spectroscopy uses higher energy photons to excite a sample, which will then emit lower energy photons. This technique has become popular for its biochemical and medical applications, and can be used for confocal microscopy, fluorescence resonance energy transfer, and fluorescence lifetime imaging.

X-ray Spectrum of light from a fluorescent lamp showing prominent mercury peaks When X-rays of sufficient frequency energy interact with a substance, inner shell electrons in the atom are excited to outer empty orbitals, or they may be removed completely, ionizing the atom. The inner shell "hole" will then be filled by electrons from outer orbitals. The energy available in this de-excitation process is emitted as radiation fluorescence or will remove other less-bound electrons from the atom Auger effect.

The absorption or emission frequencies energies are characteristic of the specific atom. In addition, for a specific atom small frequency energy variations occur which are characteristic of the chemical bonding. With a suitable apparatus, these characteristic X-ray frequencies or Auger electron energies can be measured.

X-ray absorption and emission spectroscopy is used in chemistry and material sciences to determine elemental composition and chemical bonding. X-ray crystallography is a scattering process; crystalline materials scatter X-rays at well-defined angles. If the wavelength of the incident X-rays is known, this allows calculation of the distances between planes of atoms within the crystal. The intensities of the scattered X-rays give information about the atomic positions and allow the arrangement of the atoms within the crystal structure to be calculated.

Spectroscopy 5. The use of a flame during analysis requires fuel and oxidant, typically in the form of gases. Common fuel gases used are acetylene ethyne or hydrogen. Common oxidant gases used are oxygen, air, or nitrous oxide. These methods are often capable of analyzing metallic element analytes in the part per million, billion, or possibly lower concentration ranges. Light detectors are needed to detect light with the analysis information coming from the flame.

This method commonly uses a total consumption burner with a round burning outlet. A higher temperature flame than atomic absorption spectroscopy AA is typically used to produce excitation of analyte atoms. Since analyte atoms are excited by the heat of the flame, no special elemental lamps to shine into the flame are needed. A high resolution polychromator can be used to produce an emission intensity vs.

Alternatively, a monochromator can be set at one wavelength to concentrate on analysis of a single element at a certain emission line. Plasma emission spectroscopy is a more modern version of this method. See Flame emission spectroscopy for more details.

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The temperature of the flame is low enough that the flame itself does not excite sample atoms from their ground state. The nebulizer and flame are used to desolvate and atomize the sample, but the excitation of the analyte atoms is done by the use of lamps shining through the flame at various wavelengths for each type of analyte. In AA, the amount of light absorbed after going through the flame determines the amount of analyte in the sample. A graphite furnace for heating the sample to desolvate and atomize is commonly used for greater sensitivity.

The graphite furnace method can also analyze some solid or slurry samples. Because of its good sensitivity and selectivity, it is still a commonly used method of analysis for certain trace elements in aqueous and other liquid samples. The flame is used to solvate and atomize the sample, but a lamp shines light at a specific wavelength into the flame to excite the analyte atoms in the flame.

The atoms of certain elements can then fluoresce emitting light in a different direction. The intensity of this fluorescing light is used for quantifying the amount of analyte element in the sample. A graphite furnace can also be used for atomic fluorescence spectroscopy. This method is not as commonly used as atomic absorption or plasma emission spectroscopy. Plasma Emission Spectroscopy In some ways similar to flame atomic emission spectroscopy, it has largely replaced it.

A plasma support gas is necessary, and Ar is common. Samples can be deposited on one of the electrodes, or if conducting can make up one electrode. Spectroscopy 6. For non-conductive materials, a sample is ground with graphite powder to make it conductive. In traditional arc spectroscopy methods, a sample of the solid was commonly ground up and destroyed during analysis. An electric arc or spark is passed through the sample, heating the sample to a high temperature to excite the atoms in it.

The excited analyte atoms glow emitting light at various wavelengths which could be detected by common spectroscopic methods. Since the conditions producing the arc emission typically are not controlled quantitatively, the analysis for the elements is qualitative. Nowadays, the spark sources with controlled discharges under an argon atmosphere allow that this method can be considered eminently quantitative, and its use is widely expanded worldwide through production control laboratories of foundries and steel mills.

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Visible Many atoms emit or absorb visible light. In order to obtain a fine line spectrum, the atoms must be in a gas phase. This means that the substance has to be vaporised. The spectrum is studied in absorption or emission. Although this form may be uncommon as the human eye is a similar indicator, it still proves useful when distinguishing colours.

Ultraviolet All atoms absorb in the Ultraviolet UV region because these photons are energetic enough to excite outer electrons. If the frequency is high enough, photoionization takes place. UV spectroscopy is also used in quantifying protein and DNA concentration as well as the ratio of protein to DNA concentration in a solution. Reasonable estimates of protein or DNA concentration can also be made this way using Beer's law. Infrared Infrared spectroscopy offers the possibility to measure different types of inter atomic bond vibrations at different frequencies.

Especially in organic chemistry the analysis of IR absorption spectra shows what type of bonds are present in the sample. It is also an important method for analysing polymers and constituents like fillers, pigments and plasticizers. Near Infrared NIR The near infrared NIR range, immediately beyond the visible wavelength range, is especially important for practical applications because of the much greater penetration depth of NIR radiation into the sample than in the case of mid IR spectroscopy range.

Raman Raman spectroscopy uses the inelastic scattering of light to analyse vibrational and rotational modes of molecules. The resulting 'fingerprints' are an aid to analysis. Nuclear magnetic resonance Nuclear magnetic resonance spectroscopy analyzes the magnetic properties of certain atomic nuclei to determine different electronic local environments of hydrogen, carbon, or other atoms in an organic compound or other compound. This is used to help determine the structure of the compound. It is often used in connection with electron microscopy.

An example is acoustic spectroscopy, involving sound waves. The techniques are widely used by organic chemists, mineralogists, and planetary scientists. Background subtraction Background subtraction is a term typically used in spectroscopy when one explains the process of acquiring a background radiation level or ambient radiation level and then makes an algorithmic adjustment to the data to obtain qualitative information about any deviations from the background, even when they are an order of magnitude less decipherable than the background itself.

Background subtraction can effect a number of statistical calculations Continuum, Compton, Bremsstrahlung leading to improved overall system performance. Spectroscopy 9. Dubois, G. Sando, E. Laboratory Journal Europe, No. Lewis, E. Lee and L. Microscopy Today, Volume 12, No. Luthria, Editor pp.

Baianu, D. Costescu, N. Hofmann and S. Evans and X. Quantum mechanics Quantum mechanics is a set of principles underlying the most fundamental known description of all physical systems at the submicroscopic scale at the atomic level. Notable among these principles are both a dual wave-like and particle-like behavior of matter and radiation, and prediction of probabilities in situations where classical physics predicts certainties.

Classical physics can be derived as a good approximation to quantum physics, typically in circumstances with large numbers of particles. Thus quantum phenomena are particularly relevant in systems whose dimensions are close to the atomic scale, such as molecules, atoms, electrons, protons Fig. Brighter which exhibit quantum mechanical areas correspond to higher probability density in a position effects on macroscopic scale; measurement.

Wavefunctions like these are directly superfluidity is one well-known comparable to Chladni's figures of acoustic modes of vibration in classical physics and are indeed modes of example. Quantum theory provides oscillation as well: they possess a sharp energy and thus a accurate descriptions for many keen frequency. The angular momentum and energy are previously unexplained phenomena quantized, and only take on discrete values like those such as black body radiation and stable shown as is the case for resonant frequencies in acoustics.

It has also given insight into the workings of many different biological systems, including smell receptors and protein structures. Overview The word quantum is Latin for "how great" or "how much. The discovery that waves have discrete energy packets called quanta that behave in a manner similar to particles led to the branch of physics that deals with atomic and subatomic systems which we today call quantum mechanics. For example, if classical mechanics governed the workings of an atom, electrons would rapidly travel towards and collide with the nucleus, making stable atoms impossible.

However, in the natural world the electrons normally remain in an uncertain, non-deterministic "smeared" wave-particle wave function orbital path around or "thru" the nucleus, defying classical electromagnetism. The quantum theory of the atom was developed as an explanation for the electron's staying in its orbital, which could not be explained by Newton's laws of motion and by Maxwell's laws of classical electromagnetism. For example, it allows one to compute the probability of finding an electron in a particular region around the nucleus at a particular time.

Contrary to classical mechanics, one can never make simultaneous predictions of conjugate variables, such as position and momentum, with arbitrary accuracy. For instance, electrons may be considered to be located somewhere within a region of space, but with their exact positions being unknown. Heisenberg's uncertainty principle quantifies the inability to precisely locate the particle given its conjugate. When it was found in by Max Planck that the energy of waves could be described as consisting of small packets or quanta, Albert Einstein exploited this idea to show that an electromagnetic wave such as light could be described by a particle called the photon with a discrete energy dependent on its frequency.

This led to a theory of unity between subatomic particles and electromagnetic waves called wave—particle duality in which particles and waves were neither one nor the other, but had certain properties of both. Each of these phenomena is described in detail in subsequent sections. Planck insisted[12] that this was simply an aspect of the processes of absorption and emission of radiation and had nothing to do with the physical reality of the radiation itself. However, this did not explain the photoelectric effect , i. These later came to be called photons From Einstein's simple postulation was born a flurry of debating, theorizing and testing, and thus, the entire field of quantum physics Quantum mechanics and classical physics Predictions of quantum mechanics have been verified experimentally to a very high degree of accuracy.

Thus, the current logic of correspondence principle between classical and quantum mechanics is that all objects obey laws of quantum mechanics, and classical mechanics is just a quantum mechanics of large systems or a statistical quantum mechanics of a large collection of particles. Laws of classical mechanics thus follow from laws of quantum mechanics at the limit of large systems or large quantum numbers.

Essentially the difference boils down to the statement that quantum mechanics is coherent addition of amplitudes , whereas classical theories are incoherent addition of intensities. Thus, such quantities as coherence lengths and coherence times come into play. For microscopic bodies the extension of the system is certainly much smaller than the coherence length; for macroscopic bodies one expects that it should be the other way round.

For example, stability of bulk matter which consists of atoms and molecules which would quickly collapse under electric forces alone , rigidity of this matter, mechanical, thermal, chemical, optical and magnetic properties of this matter—they are all results of interaction of electric charges under the rules of quantum mechanics. Quantum mechanics Theory There are numerous mathematically equivalent formulations of quantum mechanics. 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. 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. Oftentimes these results are skewed by many causes, such as dense probability clouds or quantum state nuclear attraction.

Much of the time, these small anomalies are attributed to different causes such as quantum dislocation. Naturally, these probabilities will depend on the quantum state at the instant of the measurement. When the probability amplitudes of four or more quantum nodes are similar, it is called a quantum parallelism.

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" can be roughly translated from German as inherent or as a characteristic. 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; 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.

For example, 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 wave function. 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. However, one can measure just the position alone of a moving free particle creating an eigenstate of position with a wavefunction that is very large at a particular position x, and almost zero everywhere else. In other words, the position of the free particle will almost be known.

This is called an eigenstate of position mathematically more precise: a generalized eigenstate eigendistribution. If the particle is in an eigenstate of position then its momentum is completely unknown. An eigenstate of momentum, on the other hand, has the form of a plane wave. Usually, a system will not be in an eigenstate of whatever observable we are interested in.

However, if one measures the observable, the wavefunction will instantaneously be an eigenstate or generalized eigenstate of that observable. This process is known as wavefunction collapse. It involves expanding the system under study to include the measurement device, so that a detailed quantum calculation would no longer be feasible and a classical description must be used. If one knows the corresponding wave function at the instant before the measurement, one will be able to compute the probability of collapsing into each of the possible eigenstates.

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For example, the free particle in the previous example will usually have a wavefunction that is a wave packet centered around some mean position x0, neither an eigenstate of position nor of momentum. When one measures the position of the particle, it is impossible to predict with certainty the result that we will obtain. It is probable, but not certain, that it will be near x0, where the amplitude of the wave function is large. After the measurement is performed, having obtained some result x, the wave function collapses into a position eigenstate centered at x.

Wave functions can change as time progresses. However, the wave packet will also spread out as time progresses, which means that the position becomes more uncertain. This also has the effect of turning position eigenstates which can be thought of as infinitely sharp wave packets into broadened wave packets that are no longer position eigenstates.

Some wave functions produce probability distributions that are constant in time. Many systems that are treated dynamically in classical mechanics are described by such "static" wave functions. For example, a single electron in an unexcited atom is pictured classically as a particle moving in a circular trajectory around the atomic nucleus, whereas in quantum mechanics it is described by a static, spherically symmetric wavefunction surrounding the nucleus Fig. Note that only the lowest angular momentum states, labeled s, are spherically symmetric.

The time evolution of wave functions is deterministic in the sense that, given a wavefunction at an initial time, it makes a definite prediction of what the wavefunction will be at any later time. During a measurement, the change of the wavefunction into another one is not deterministic, but rather unpredictable, i. The probabilistic nature of quantum mechanics thus stems from the act of measurement.

This is one of the most difficult aspects of quantum systems to understand. It was the central topic in the famous Bohr-Einstein debates, in which the two scientists attempted to clarify these fundamental principles by way of thought experiments. In the decades after the formulation of quantum mechanics, the question of what constitutes a "measurement" has been extensively studied. Interpretations of quantum mechanics have been formulated to do away with the concept of "wavefunction collapse"; see, for example, the relative state interpretation. The basic idea is that when a quantum system interacts with a measuring apparatus, their respective wavefunctions become entangled, so that the original quantum system ceases to exist as an independent entity.

For details, see the article on measurement in quantum mechanics. Mathematical formulation In the mathematically rigorous formulation of quantum mechanics, developed by Paul Dirac[16] and John von Neumann[17] , the possible states of a quantum mechanical system are represented by unit vectors called "state vectors" residing in a complex separable Hilbert space variously called the "state space" or the "associated Hilbert space" of the system well defined up to a complex number of norm 1 the phase factor.

In other words, the possible states are points in the projectivization of a Hilbert space, usually called the complex projective space. The exact nature of this Hilbert space is dependent on the system; for example, the state space for position and momentum states is the space of square-integrable functions, while the state space for the spin of a single proton is just the product of two complex planes.

Each observable is represented by a maximally-Hermitian precisely: by a self-adjoint linear operator acting on the state space. Each eigenstate of an observable corresponds to an eigenvector of the operator, and the associated eigenvalue corresponds to the value of the observable in that eigenstate. If the operator's spectrum is discrete, the observable can only attain those discrete eigenvalues. The inner product between two state vectors is a complex number known as a probability amplitude. During a measurement, the probability that a system collapses from a given initial state to a particular eigenstate is given by the square of the absolute value of the probability amplitudes between the initial and final states.

The possible results of a measurement are the eigenvalues of the operator - which explains the choice of Hermitian operators, for which all the eigenvalues are real. We can find the probability distribution of an observable in a given state by computing the spectral decomposition of the corresponding operator. Heisenberg's uncertainty principle is represented by the statement that the operators corresponding to certain observables do not commute.

Whereas the absolute value of the probability amplitude encodes information about probabilities, its phase encodes information about the interference between quantum states. This gives rise to the wave-like behavior of quantum states. Even the helium atom, which contains just one more electron than hydrogen, defies all attempts at a fully analytic treatment. There exist several techniques for generating approximate solutions. For instance, in the method known as perturbation theory one uses the analytic results for a simple quantum mechanical model to generate results for a more complicated model related to the simple model by, for example, the addition of a weak potential energy.

Another method is the "semi-classical equation of motion" approach, which applies to systems for which quantum mechanics produces weak deviations from classical behavior. The deviations can be calculated based on the classical motion. This approach is important for the field of quantum chaos. Interactions with other scientific theories The fundamental rules of quantum mechanics are very deep. They assert that the state space of a system is a Hilbert space and the observables are Hermitian operators acting on that space, but do not tell us which Hilbert space or which operators, or if it even exists.

These must be chosen appropriately in order to obtain a quantitative description of a quantum system. An important guide for making these choices is the correspondence principle, which states that the predictions of quantum mechanics reduce to those of classical physics when a system moves to higher energies or equivalently, larger quantum numbers.

In other words, classic mechanics is simply a quantum mechanics of large systems. This "high energy" limit is known as the classical or correspondence limit. One can therefore start from an established classical model of a particular system, and attempt to guess the underlying quantum model that gives rise to the classical model in the correspondence limit. Unsolved problems in physics: In the correspondence limit of quantum mechanics: Is there a preferred interpretation of quantum mechanics?

How does the quantum description of reality, which includes elements such as the "superposition of states" and "wavefunction collapse", give rise to the reality we perceive? When quantum mechanics was originally formulated, it was applied to models whose correspondence limit was non-relativistic classical mechanics. For instance, the well-known model of the quantum harmonic oscillator uses an explicitly non-relativistic expression for the kinetic energy of the oscillator, and is thus a quantum version of the classical harmonic oscillator.

While these theories were successful in explaining many experimental results, they had certain unsatisfactory qualities stemming from their neglect of the relativistic creation and annihilation of particles. The first complete quantum field theory, quantum electrodynamics, provides a fully quantum description of the electromagnetic interaction. The full apparatus of quantum field theory is often unnecessary for describing electrodynamic systems. A simpler approach, one employed since the inception of quantum mechanics, is to treat charged particles as quantum mechanical objects being acted on by a classical electromagnetic field.

For example, the elementary quantum model of the hydrogen atom describes the electric field of the hydrogen atom using a classical Coulomb potential. This "semi-classical" approach fails if quantum fluctuations in the electromagnetic field play an important role, such as in the emission of photons by charged particles. Quantum field theories for the strong nuclear force and the weak nuclear force have been developed. The weak nuclear force and the electromagnetic force were unified, in their quantized forms, into a single quantum field theory known as electroweak theory, by the physicists Abdus Salam, Sheldon Glashow and Steven Weinberg.

It has proven difficult to construct quantum models of gravity, the remaining fundamental force. Semi-classical approximations are workable, and have led to predictions such as Hawking radiation. However, the formulation of a complete theory of quantum gravity is hindered by apparent incompatibilities between general relativity, the most accurate theory of gravity currently known, and some of the fundamental assumptions of quantum theory. The resolution of these incompatibilities is an area of active research, and theories such as string theory are among the possible candidates for a future theory of quantum gravity.

Example The particle in a 1-dimensional potential energy box is the most simple example where restraints lead to the quantization of energy levels. The box is defined as zero potential energy inside a certain interval and infinite everywhere outside that interval. The presence of the walls of the box restricts the acceptable solutions of the wavefunction.

At each wall:. To satisfy the cos term has to be removed. Relativity and quantum mechanics The modern world of physics is founded on the two tested and demonstrably sound theories of general relativity and quantum mechanics —theories which appear to contradict one another. The defining postulates of both Einstein's theory of relativity and quantum theory are indisputably supported by rigorous and repeated empirical evidence.

However, while they do not directly contradict each other theoretically at least with regard to primary claims , they are resistant to being incorporated within one cohesive model. Einstein himself is well known for rejecting some of the claims of quantum mechanics. While clearly inventive in this field, he did not accept the more philosophical consequences and interpretations of quantum mechanics, such as the lack of deterministic causality and the assertion that a single subatomic particle can occupy numerous areas of space at one time.

He also was the first to notice some of the apparently exotic consequences of entanglement and used them to formulate the Einstein-Podolsky-Rosen paradox, in the hope of showing that quantum mechanics had unacceptable implications. This was , but in it was shown by John Bell see Bell inequality that Einstein's assumption was correct, but had to be completed by hidden variables and thus based on wrong philosophical assumptions. According to the paper of J. The Einstein-Podolsky-Rosen paradox shows in any case that there exist experiments by which one can measure the state of one particle and instantaneously change the state of its entangled partner, although the two particles can be an arbitrary distance apart; however, this effect does not violate causality, since no transfer of information happens.

These experiments are the basis of some of the most topical applications of the theory, quantum cryptography, which works well, although at small distances of typically km, being on the market since There do exist quantum theories which incorporate special relativity—for example, quantum electrodynamics QED , which is currently the most accurately tested physical theory [19] —and these lie at the very heart of modern particle physics.

Gravity is negligible in many areas of particle physics, so that unification between general relativity and quantum mechanics is not an urgent issue in those applications. However, the lack of a correct theory of quantum gravity is an important issue in cosmology. Attempts at a unified theory Inconsistencies arise when one tries to join the quantum laws with general relativity, a more elaborate description of spacetime which incorporates gravitation. Resolving these inconsistencies has been a major goal of twentieth- and twenty-first-century physics.

Many prominent physicists, including Stephen Hawking, have labored in the attempt to discover a "Grand Unification Theory" that combines not only different models of subatomic physics, but also derives the universe's four forces—the strong force, electromagnetism, weak force, and gravity— from a single force or phenomenon. Leading the charge in this field is Edward Witten, a physicist that formulated the groundbreaking M-theory which is an attempt at describing the supersymmetrical based string theory.

Applications Quantum mechanics has had enormous success in explaining many of the features of our world. The individual behaviour of the subatomic particles that make up all forms of matter—electrons, protons, neutrons, photons and others—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 and the multiverse hypothesis. 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.

Another active research topic is quantum teleportation, which deals with techniques to transmit quantum states over arbitrary distances. In many devices, even the simple light switch, quantum tunneling is vital, as otherwise the electrons in the electric current could not penetrate the potential barrier made up, in the case of the light switch, of a layer of oxide. Flash memory chips found in USB drives also use quantum tunneling to erase their memory cells.

Philosophical consequences Since its inception, the many counter-intuitive results of quantum mechanics have provoked strong philosophical debate and many interpretations. Even fundamental issues such as Max Born's basic rules concerning probability amplitudes and probability distributions took decades to be appreciated. The Copenhagen interpretation, due largely to the Danish theoretical physicist Niels Bohr, is the interpretation of quantum mechanics most widely accepted amongst physicists.

According to it, the probabilistic nature of quantum mechanics predictions cannot be explained in terms of some other deterministic theory, and does not simply reflect our limited knowledge. Quantum mechanics provides probabilistic results because the physical universe is itself probabilistic rather than deterministic. Albert Einstein, himself one of the founders of quantum theory, disliked this loss of determinism in measurement this dislike is the source of his famous quote, "God does not play dice with the universe.

Einstein held that there should be a local hidden variable theory underlying quantum mechanics and that, consequently, the present theory was incomplete. John Bell showed that the EPR paradox led to experimentally testable differences between quantum mechanics and local realistic theories. Experiments have been performed confirming the accuracy of quantum mechanics, thus demonstrating that the physical world cannot be described by local realistic theories.

The Bohr-Einstein debates provide a vibrant critique of the Copenhagen Interpretation from an epistemological point of view. The Everett many-worlds interpretation, formulated in , holds that all the possibilities described by quantum theory simultaneously occur in a "multiverse" composed of mostly independent parallel universes. This is not accomplished by introducing some new axiom to quantum mechanics, but on the contrary by removing the axiom of the collapse of the wave packet: All the possible consistent states of the measured system and the measuring apparatus including the observer are present in a real physical not just formally mathematical, as in other interpretations quantum superposition.

Such a superposition of consistent state combinations of different systems is called an entangled state. While the multiverse is deterministic, we perceive non-deterministic behavior governed by probabilities, because we can observe only the universe, i. Everett's interpretation is perfectly consistent with John Bell's experiments and makes them intuitively understandable. However, according to the theory of quantum decoherence, the parallel universes will never be accessible to us.

This inaccessibility can be understood as follows: once a measurement is done, the measured system becomes entangled with both the physicist who measured it and a huge number of other particles, some of which are photons flying away towards the other end of the universe; in order to prove that the wave function did not collapse one would have to bring all these particles back and measure them again, together with the system that was measured originally. This is completely impractical, but even if one could theoretically do this, it would destroy any evidence that the original measurement took place including the physicist's memory.

Mehra and H. Rechenberg, The historical development of quantum theory, Springer-Verlag, Kuhn, Black-body theory and the quantum discontinuity , Clarendon Press, Oxford, A biography of Born details his role as the creator of the matrix formulation of quantum mechanics. This was recognized in a paper by Heisenberg, in , honoring Max Planck.

Princeton University Press. Buffalo NY: Prometheus Books. Includes cosmological and philosophical considerations.

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Primer of Quantum Mechanics. John Wiley. Neill Graham, eds. The Principles of Quantum Mechanics. The beginning chapters make up a very clear and comprehensible introduction. The Feynman Lectures on Physics. Introduction to Quantum Mechanics 2nd ed. Prentice Hall. ISBN A standard undergraduate text. The Conceptual Development of Quantum Mechanics. McGraw Hill. Singapore: World Scientific. Draft of 4th edition.

Wave Mechanics. London: Pergamon Press. The mathematical foundations of quantum mechanics. Dover Publications. Quantum Mechanics Vol. I , English translation from French by G. IV, section III. Understanding Quantum Mechanics. Oxford University Press. Considers the extent to which chemistry and the periodic system have been reduced to quantum mechanics. What is Quantum Mechanics? A Physics Adventure. Language Research Foundation, Boston. Mathematical Foundations of Quantum Mechanics. O'Connor and E. See 5. See 8.

The New York Times. December 27, It is widely used in particle physics and condensed matter physics.

Most theories in modern particle physics, including the Standard Model of elementary particles and their interactions, are formulated as relativistic quantum field theories. In condensed matter physics, quantum field theories are used in many circumstances, especially those where the number of particles is allowed to fluctuate—for example, in the BCS theory of superconductivity. In quantum field theory QFT the forces between particles are mediated by other particles. The electromagnetic force between two electrons is caused by an exchange of photons.

Intermediate vector bosons mediate the weak force and gluons mediate the strong force. There is currently no complete quantum theory of the remaining fundamental force, gravity, but many of the proposed theories postulate the existence of a graviton particle which mediates it. These force-carrying particles are virtual particles and, by definition, cannot be detected while carrying the force, because such detection will imply that the force is not being carried.

In QFT photons are not thought of as 'little billiard balls', they are considered to be field quanta - necessarily chunked ripples in a field that 'look like' particles. Fermions, like the electron, can also be described as ripples in a field, where each kind of fermion has its own field. In summary, the classical visualisation of "everything is particles and fields", in quantum field theory, resolves into "everything is particles", which then resolves into "everything is fields". In the end, particles are regarded as excited states of a field field quanta.

In , Max Born, Pascual Jordan, and Werner Heisenberg constructed such a theory by expressing the field's internal degrees of freedom as an infinite set of harmonic oscillators and by employing the usual procedure for quantizing those oscillators canonical quantization. This theory assumed that no electric charges or currents were present and today would be called a free field theory.

This quantum field theory could be used to model important processes such as the emission of a photon by an electron dropping into a quantum state of lower energy, a process in which the number of particles changes — one atom in the initial state becomes an atom plus a photon in the final state. It is now understood that the ability to describe such processes is one of the most important features of quantum field theory.

It was evident from the beginning that a proper quantum treatment of the electromagnetic field had to somehow incorporate Einstein's relativity theory, which had after all grown out of the study of classical electromagnetism. This need to put together relativity and quantum mechanics was the second major motivation in the development of quantum field theory.

Pascual Jordan and Wolfgang Pauli showed in that quantum fields could be made to behave in the way predicted by special relativity during coordinate transformations specifically, they showed that the field commutators were Lorentz invariant , and in Niels Bohr and Leon Rosenfeld showed that this result could be interpreted as a limitation on the ability to measure fields at space-like separations, exactly as required by relativity.

A further boost for quantum field theory came with the discovery of the Dirac equation, a single-particle equation obeying both relativity and quantum mechanics, when it was shown that several of its undesirable properties such as negative-energy states could be eliminated by reformulating the Dirac equation as a quantum field theory. The third thread in the development of quantum field theory was the need to handle the statistics of many-particle systems consistently and with ease.

In , Jordan tried to extend the canonical quantization of fields to the many-body wavefunctions of identical particles, a procedure that is sometimes called second quantization. In , Jordan and Eugene Wigner found that the quantum field describing electrons, or other fermions, had to be expanded using anti-commuting creation and annihilation operators due to the Pauli exclusion principle. This thread of development was incorporated into many-body theory, and strongly influenced condensed matter physics and nuclear physics. Despite its early successes, quantum field theory was plagued by several serious theoretical difficulties.

Many seemingly-innocuous physical quantities, such as the energy shift of electron states due to the presence of the electromagnetic field, gave infinity — a nonsensical result — when computed using quantum field theory. This "divergence problem" was solved during the s by Bethe, Tomonaga, Schwinger, Feynman, and Dyson, through the procedure known as renormalization.

This phase of development culminated with the construction of the modern theory of quantum electrodynamics QED. Beginning in the s with the work of Yang and Mills, QED was generalized to a class of quantum field theories known as gauge theories. The s and s saw the formulation of a gauge theory now known as the Standard Model of particle physics, which describes all known elementary particles and the interactions between them. The weak interaction part of the standard model was formulated by Sheldon Glashow, with the Higgs mechanism added by Steven Weinberg and Abdus Salam.

The theory was shown to be renormalizable and hence consistent by Gerardus 't Hooft and Martinus Veltman.

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Also during the s, parallel developments in the study of phase transitions in condensed matter physics led Leo Kadanoff, Michael Fisher and Kenneth Wilson extending work of Ernst Stueckelberg, Andre Peterman, Murray Gell-Mann and Francis Low to a set of ideas and methods known as the renormalization group. The study of quantum field theory is alive and flourishing, as are applications of this method to many physical problems.

It remains one of the most vital areas of theoretical physics today, providing a common language to many branches of physics. Furthermore, each observable corresponds, in a technical sense, to the classical idea of a degree of freedom. For instance, the fundamental observables associated with the motion of a single quantum mechanical particle are the position and momentum operators and.

Ordinary quantum mechanics deals with systems such as this, which possess a small set of degrees of freedom. It is important to note, at this point, that this article does not use the word "particle" in the context of wave—particle duality. In quantum field theory, "particle" is a generic term for any discrete quantum mechanical entity, such as an electron, which can behave like classical particles or classical waves under different experimental conditions.

A quantum field is a quantum mechanical system containing a large, and possibly infinite, number of degrees of freedom. This is not as exotic a situation as one might think. A classical field contains a set of degrees of freedom at each point of space; for instance, the classical electromagnetic field defines two vectors — the electric field and the magnetic field — that can in principle take on distinct values for each position. When the field as a whole is considered as a quantum mechanical system, its observables form an infinite in fact uncountable set, because is continuous.

Furthermore, the degrees of freedom in a quantum field are arranged in "repeated" sets. For example, the degrees of freedom in an electromagnetic field can be grouped according to the position , with exactly two vectors for each. Note that is an ordinary number that "indexes" the observables; it is not to be confused with the position operator encountered in ordinary quantum mechanics, which is an observable.

Thus, ordinary quantum mechanics is sometimes referred to as "zero-dimensional quantum field theory", because it contains only a single set of observables. It is also important to note that there is nothing special about because, as it turns out, there is generally more than one way of indexing the degrees of freedom in the field. In the following sections, we will show how these ideas can be used to construct a quantum mechanical theory with the desired properties.

We will begin by discussing single-particle quantum mechanics and the associated theory of many-particle quantum mechanics. Then, by finding a way to index the degrees of freedom in the many-particle problem, we will construct a quantum field and study its implications. Quantum field theory We wish to consider how this problem generalizes to particles. There are two motivations for studying the many-particle problem.

The first is a straightforward need in condensed matter physics, where typically the number of particles is on the order of Avogadro's number 6. The second motivation for the many-particle problem arises from particle physics and the desire to incorporate the effects of special relativity. If one attempts to include the relativistic rest energy into the above equation, the result is either the Klein-Gordon equation or the Dirac equation.

It turns out that such inconsistencies arise from neglecting the possibility of dynamically creating or destroying particles, which is a crucial aspect of relativity. Einstein's famous mass-energy relation predicts that sufficiently massive particles can decay into several lighter particles, and sufficiently energetic particles can combine to form massive particles.

For example, an electron and a positron can annihilate each other to create photons. Thus, a consistent relativistic quantum theory must be formulated as a many-particle theory. Furthermore, we will assume that the particles are indistinguishable. As described in the article on identical particles, this implies that the state of the entire system must be either symmetric bosons or antisymmetric fermions when the coordinates of its constituent particles are exchanged. These multi-particle states are rather complicated to write.

For example, the general quantum state of a system of bosons is written as. In general, this is a sum of factorial distinct terms, which quickly becomes unmanageable as increases. The way to simplify this problem is to turn it into a quantum field theory. Second quantization In this section, we will describe a method for constructing a quantum field theory called second quantization.

This basically involves choosing a way to index the quantum mechanical degrees of freedom in the space of multiple identical-particle states. It is based on the Hamiltonian formulation of quantum mechanics; several other approaches exist, such as the Feynman path integral[2] , which uses a Lagrangian formulation. For an overview, see the article on quantization. Second quantization of bosons For simplicity, we will first discuss second quantization for bosons, which form perfectly symmetric quantum states.

Let us denote the mutually orthogonal single-particle states by and so on. For example, the 3-particle state with one particle in state and two in state is. The first step in second quantization is to express such quantum states in terms of occupation numbers, by listing the number of particles occupying each of the single-particle states etc. This is simply another way of labelling the states. For instance, the above 3-particle state is denoted as. The next step is to expand the -particle state space to include the state spaces for all possible values of.

This extended state space, known as a Fock space, is composed of the state space of a system with no particles the so-called vacuum state , plus the state space of a 1-particle system, plus the state space of a 2-particle system, and so forth. It is easy to see that there is a one-to-one correspondence between the occupation number representation and valid boson states in the Fock space.

At this point, the quantum mechanical system has become a quantum field in the sense we described above. The field's elementary degrees of freedom are the occupation numbers, and each occupation number is indexed by a number , indicating which of the single-particle states it refers to. The properties of this quantum field can be explored by defining creation and annihilation operators, which add and subtract particles.

They are analogous to "ladder operators" in the quantum harmonic oscillator problem, which added and subtracted energy quanta. However, these operators literally create and annihilate particles of a given quantum state. The bosonic annihilation operator and creation operator have the following effects:. It can be shown that these are operators in the usual quantum mechanical sense, i. Furthermore, they are indeed Hermitian conjugates, which justifies the way we have written them. They can be shown to obey the commutation relation.

These are precisely the relations obeyed by the ladder operators for an infinite set of independent quantum harmonic oscillators, one for each single-particle state. Adding or removing bosons from each state is therefore analogous to exciting or de-exciting a quantum of energy in a harmonic oscillator. For instance, the Hamiltonian of a field of free non-interacting bosons is. Second quantization of fermions It turns out that a different definition of creation and annihilation must be used for describing fermions.

According to the Pauli exclusion principle, fermions cannot share quantum states, so their occupation numbers can only take on the value 0 or 1.