Heisenberg mechanics

The miderstanding of the quantum mechanics of atoms was pioneered by Bohr, in his theory of the hydrogen atom. This combined the classical ideas on planetary motion—applicable to the atom because of the fomial similarity of tlie gravitational potential to tlie Coulomb potential between an electron and nucleus—with the quantum ideas that had recently been introduced by Planck and Einstein. This led eventually to the fomial theory of quaiitum mechanics, first discovered by Heisenberg, and most conveniently expressed by Schrodinger in the wave equation that bears his name. 

A classical Hamiltonian is obtained from the spectroscopic fitting Hamiltonian by a method that has come to be known as the Heisenberg correspondence [46], because it is closely related to the teclmiques used by Heisenberg in fabricating the fomi of quantum mechanics known as matrix mechanics. 

Heisenberg W 1925 Z. Phyz. 33 879 (Engl.Transl. van der Waerden B L (ed) 1967 Sources of Quantum Mechanics (New York Dover)... 

How to extract from E(qj,t) knowledge about momenta is treated below in Sec. III. A, where the structure of quantum mechanics, the use of operators and wavefunctions to make predictions and interpretations about experimental measurements, and the origin of uncertainty relations such as the well known Heisenberg uncertainty condition dealing with measurements of coordinates and momenta are also treated. 

The electromagnetic spectrum is a quantum effect and the width of a spectral feature is traceable to the Heisenberg uncertainty principle. The mechanical spectrum is a classical resonance effect and the width of a feature indicates a range of closely related r values for the model elements. 

In 1913 Niels Bohr proposed a system of rules that defined a specific set of discrete orbits for the electrons of an atom with a given atomic number. These rules required the electrons to exist only in these orbits, so that they did not radiate energy continuously as in classical electromagnetism. This model was extended first by Sommerfeld and then by Goudsmit and Uhlenbeck. In 1925 Heisenberg, and in 1926 Schrn dinger, proposed a matrix or wave mechanics theory that has developed into quantum mechanics, in which all of these properties are included. In this theory the state of the electron is described by a wave function from which the electron s properties can be deduced. 

Ortho- and para-hydrogen discovered spectroscopically by R. Mecke and interpreted quantum-mechanically by W. Heisenberg, 1927. 

W. Heisenberg (Leipzig) the creation of quantum mechanics, the application of which has, inter alia, led to the discovery of the allotropic forms of hydrogen. 

The year 1926 was an exciting one. Schrddinger, Heisenberg and Dirac, all working independently, solved the hydrogen atom problem. Schrddinger s treatment, which we refer to as wave mechanics, is the version that you will be fanuliar with. The only cloud on the horizon was summarized by Dirac, in his famous statement ... 

The notion of electrons in orbitals consists essentially of ascribing four distinct quantum numbers to each electron in a many-electron atom. It can be shown that this notion is strictly inconsistent with quantum mechanics (7). Definite quantum numbers for individual electrons do not have any meaning in the framework of quantum mechanics. The erroneous view stems from the original formulation of the Pauli principle in 1925, which stated that no two electrons could share the same four quantum numbers (8), This version of the principle was superseded by a new formulation that avoids any reference to individual quantum numbers for separate electrons. The new version due to the independent work of Heisenberg and Dirac in 1926 states that the wave function of a many-electron atom must be antisymmetrical with respect to the interchange of any two particles (9,10). 

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 new quantum mechanics contradicts this independent electron model as it is often called. In Heisenberg s formulation of quantum mechanics the fundamental equation is,... 

Pauli s original version of the exclusion principle was found lacking precisely because it ascribes stationary states to individual electrons. According to the new quantum mechanics, only the atomic system as a whole possesses stationary states. The original version of the exclusion principle was replaced by the statement that the wavefunction for a system of fermions must be antisymmetrical with respect to the interchange of any two particles (Heisenberg [1925], Dirac [1928]). 

It was a characteristic feature of Bohr s classical quantum mechanics that it could never be generalized to give good quantitative results for systems containing more than one electron. The extension from N = 1 to N = 2, 3,. . . came first with modern wave mechanics and Heisenberg s discovery in 1926 of the exchange phenomenon in the He-atom, which, with the identity... 

Both quantum mechanical and classical theories of Raman scattering have been developed. The quantum mechanical treatment of Kramers and Heisenberg 5) preceded the classical theory of Cabannes and Rochard 6). 

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. 

The physical interpretation of the quantum mechanics and its generalization to include aperiodic phenomena have been the subject of papers by Dirac, Jordan, Heisenberg, and other authors. For our purpose, the calculation of the properties of molecules in stationary states and particularly in the normal state, the consideration of the Schrodinger wave equation alone suffices, and it will not be necessary to discuss the extended theory. 

The application of the quantum mechanics to the interaction of more complicated atoms, and to the non-polar chemical bond in general, is now being made (45). A discussion of this work can not be given here it is, however, worthy of mention that qualitative conclusions have been drawn which are completely equivalent to G. N. Lewis s theory of the shared electron pair. The further results which have so far been obtained are promising and we may look forward with some confidence to the future explanation of chemical valence in general in terms of the Pauli exclusion principle and the Heisenberg-Dirac resonance phenomenon. 

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. 

In Schrodinger s wave mechanics (which has been shown4 to be mathematically identical with Heisenberg s quantum mechanics), a conservative Newtonian dynamical system is represented by a wave function or amplitude function [/, which satisfies the partial differential equation... 

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. 

One of the first to show up was Werner Heisenberg, who later won a Nobel Prize. Soon afterward came George Gamow, the fun-loving Russian physicist who sorted out the nuclear reactions that power the stars. Erwin Schrodinger, who also won a Nobel Prize in physics, stopped by to lecture on his new wave theory. Wolfgang Pauli, who would also win a Nobel Prize for his contributions to quantum mechanics, was there, too. 

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