Heisenberg quantum theory

The original Placzek theory of Raman scattering [30] was in terms of the linear, or first order microscopic polarizability, a (a second rank tensor), not the third order h3q)erpolarizability, y (a fourth rank tensor). The Dirac and Kramers-Heisenberg quantum theory for linear dispersion did account for Raman scattering. It turns out that this link of properties at third order to those at first order works well for the electronically nonresonant Raman processes, but it cannot hold rigorously for the fully (triply) resonant Raman spectroscopies. However, provided one discards the important line shaping phenomenon called pure dephasing , one can show how the third order susceptibility does reduce to the treatment based on the (linear) polarizability tensor [6, 27]. 

I restrict my attention to non-relativistic pioneer quantum mechanics of 1925-6, and even further to the time independent formulation. Numerous other developments have taken place in quantum theory, such as Dirac s relativistic treatment of the hydrogen atom (Dirac [1928]) and various modern quantum field theories have been constructed (Redhead [1986]). Also, much work has been done in the philosophy of quantum theory such as the question of E.P.R. correlations (Bell [1966]). However, it seems fair to say that no fundamental change has occurred in quantum mechanics since the pioneer version was established. The version of quantum mechanics used on a day-to-day basis by most chemists and physicists remains as the 1925-6 version (Heisenberg [1925], Schrodinger [1926]). 

The first consistent attempt to unify quantum theory and relativity came after Schrddinger s and Heisenberg s work in 1925 and 1926 produced the rules for the quantum mechanical description of nonrelativistic systems of point particles. Mention should be made of the fact that in these developments de Broglie s hypothesis attributing wave-corpuscular properties to all matter played an important role. Central to this hypothesis are the relations between particle and wave properties E — hv and p = Ilk, which de Broglie advanced on the basis of relativistic dynamics. 

Now in quantum theory the description of a physical system in the Heisenberg picture for a given observer O is by means of operators Q, which satisfy certain equations of motion and commutation rules with respect to O s frame of reference (coordinate system x). The above notion of an invariance principle can be stated alternatively as follows If, when we change this coordinate frame of reference (i.e., for observer O ) we are able to find a new set of operators that obeys the same equations of motion and the same commutation rules with respect to the new frame of reference (coordinate system x ) we then say that these observers are equivalent and the theory invariant under the transformation x - x. The observable consequences of theory in the new frame (for observer O ) will then clearly be the same as those in the old frame. 

Heisenberg operators Q+ t), 600 Heisenberg picture in quantum theory, 665... 

In recent years the old quantum theory, associated principally with the names of Bohr and Sommerfeld, encountered a large number of difficulties, all of which vanished before the new quantum mechanics of Heisenberg. Because of its abstruse and difficultly interpretable mathematical foundation, Heisenberg s quantum mechanics cannot be easily applied to the relatively complicated problems of the structures and properties of many-electron atoms and of molecules in particular is this true for chemical problems, which usually do not permit simple dynamical formulation in terms of nuclei and electrons, but instead require to be treated with the aid of atomic and molecular models. Accordingly, it is especially gratifying that Schrodinger s interpretation of his wave mechanics3 provides a simple and satisfactory atomic model, more closely related to the chemist s atom than to that of the old quantum theory. 

This equation, including succeeding terms, was obtained originally by Sommerfeld from relativistic considerations with the old quantum theory the first term, except for the screening constant sQ> has now been derived by Heisenberg and Jordan] with the use of the quantum mechanics and the idea of the spinning electron. The value of the screening constant is known for a number of doublets, and it is found empirically not to vary with Z. 

W. Heisenberg, The Physical Principles of the Quantum Theory, Translated by C. Ekart and F. C. Hoyt, p. 14211, Dover Publications, New York, USA (1930). 

W. Z. Heisenberg, The translation of kinematical and mechanical relations into terms of the quantum theory, Phys. 33 879 (1927). 

The fact that quantum observables are represented by matrices immediately suggests problems of non-commutation. For instance, the observables can be measured at the same time only if they have a complete orthogonal set of eigenvectors in common. This happens only when they commute, i.e. XY = YX, or the commutator [X, Y] = XY — YX = 0. This is a central feature of the matrix formulation of quantum theory discovered by Heisenberg, Born and Jordan while trying to explain the observed spectral transitions of the hydrogen atom in a more fundamental way than the quantization... 

Heisenberg s formulation of quantum theory consisted of associating with each observable a square Hermitian matrix, with the view of determining its measurable values. Applied to the quantity H, which, classically is a function of the cartesian coordinates ql and the momenta piy... 

The fundamental equivalence between Schrodinger s wave-mechanical and Heisenberg s matrix-mechanical representation of quantum theory implies that H (or Hm>) may be viewed as a differential operator or a matrix. The latter viewpoint is usually more convenient in the... 

To explain this behaviour, physicists appeal to the very foundations of quantum theory. Because of their much reduced freedom to move in space, the particles can be considered to be more and more localised. Then, by Heisenberg s uncertainty principle, the spread in their velocities has to grow. In other words, some particles may have much higher velocities than those allowed by the temperature. A quantum pressure arises at high densities, when the mean distance between electrons becomes comparable with their associated wavelength... 

Discussing physics with Bohr was sometimes quite exhausting. In 1925, while working at Bohr s institute, Heisenberg discovered quantum mechanics, the theory that superseded the old quantum theory that had been developed by Bohr and his colleagues. Then, in 1926, a theory that looked much different from Heisenberg s, but which turned out to be mathematically equivalent, was propounded by the Austrian physicist Erwin Schrodinger. 

Here q and p are Heisenberg operators, y is the usual damping coefficient, and (t) is a random force, which is also an operator. Not only does one have to characterize the stochastic behavior of g(t), but also its commutation relations, in such a way that the canonical commutation relation [q(t), p(t)] = i is preserved at all times and the fluctuation-dissipation theorem is obeyed. ) Moreover it appears impossible to maintain the delta correlation in time in view of the fact that quantum theory necessarily cuts off the high frequencies. ) We conclude that no quantum Langevin equation can be obtained without invoking explicitly the equation of motion of the bath that causes the fluctuations.1 That is the reason why this type of equation has so much less practical use than its classical counterpart. 

A new quantum theory called wave mechanics (as formulated by Schrodinger) or quantum mechanics (as formulated by Heisenberg, Born and Dirac) was developed in 1926. This was immediately successful m accounting for a wide variety of experimental observations, and there is little doubt that, in principle, the theory is capable of describing any physical system. A strange feature of the new mechanics, however, is that nowhere does the path or velocity of the electron enter the description. In fact it is often impossible to visualize any classical motion that could be consistent with the quantum mechanical picture of the atom,... 

In the Quantum picture of the world, each individual event cannot be determined exactly, but has to be described by a wave of probability. There is a kind of polarity between the position and energy of any particle in which they cannot be simultaneously determined. This was not a failing of experimental method but a property of the kinds of mathematical structures that physicists have to use to describe this realm of the world. The famous equation of Quantum theory embodying Heisenberg s Uncertainty Principle is ... 

An important experiment carried out as recently as summer 1982 by the French physicist, Aspect, has unequivocally demonstrated the fact that physicists cannot get round the Uncertainty Principle and simultaneously determine the quantum states of particles, and confirmed that physicists cannot divorce the consciousness of the observer from the events observed. This experiment (in disproving the separabilty of quantum measurements) has confirmed what Einstein, Bohr and Heisenberg were only able to philosophically debate over - that with quantum theory we have to leave behind our naive picture of reality as an intricate clockwork. We are challenged by quantum theory to build new ways in which to picture reality, a physics, moreover, in which consciousness plays a central role, in which the observer is inextricably interwoven in the fabric of reality. 

The recognition of electronic delocalization had to wait until the 1930s. The notion has gained a proper perspective with the development of quantum theory and with the recognition that electrons are also waves which are intrinsically delocalized. Heisenberg s theory of resonance,54 which was applied by Heitler and London to the H2 molecule,55 demon-... 

Following the development of quantum theory by Heisenberg [1] and Schrodinger [2] and a few further discoveries, the basic principles of the structure of atoms and molecules were described around 1930. Unfortunately, the complexity of the Schrodinger equation increases dramatically with the number of electrons involved in a system, and thus for a long time the hydrogen and helium atoms and simple molecules as H2 were the only species whose properties could really be calculated from these first principles. In 1929, Dirac [3] wrote ... 

On a more philosophical or meta-physical level, one may suspect that free will and consciousness may have some quantum mechanical origin rooted in the Heisenberg Uncertainty Principle. Perhaps at some neurological level an electron at a synapse exists in a superposition of two or more states that ultimately results in someone making some sort of decision. Should I run for President, or not Should I get married, or not . Perhaps there are two states with eigenvalues yes or no that asymptotically lead to very different actions. Does quantum theory enter into our decision making process Perhaps the brain itself acts as some sort of quantum computer taking... 

The orthodox or Copenhagen interpretation of quantum theory originated with three seminal papers published in 1925-26 by Heisenberg, Born and Jordan and an independent paper by Dirac (1926) all of these are available in English (translation) in a single volume [13]. A detailed summary was published by Heisenberg [9]. The primary aim of these studies was to formulate a mathematical system for the mechanics of atomic and electronic motion, based entirely on relations between experimentally observable quantities. An immediate consequence of this stipulation was that the motion of electrons could no longer be described in terms of the familiar concepts of space and time, but rather in terms of state functions constructed from matrix elements that relate to the Fourier sums over observed spectroscopic frequencies. The procedure became known as matrix mechanics. 

Because of the obvious validity of matrix mechanics it was natural to assume that the initial premises on which Heisenberg first approached the quantum problem also had to be valid beyond dispute. The same courtesy is not extended to Schrodinger, who after all, produced the most useful version of the theory. The damage can clearly not be undone, but the relevance of the Copenhagen orthodoxy to wave mechanics can certainly be re-examined. Whereas the contribution of orthodox quantum theory to the understanding of chemistry has been minimal, there is the realistic hope that unfettered use of wave mechanics can only be an improvement on the current situation. 

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