Heisenberg’s matrix mechanics

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... 

During the Sturm-und-Drang quantum period of the 1920s, Heisenberg s matrix mechanics and Schrodinger s equation have been established as very mathematized cornerstones of the rigorous theory of elementary chemical processes, which changed the field of chemistry dramatically. 

The stability of the hydrogen molecule within the newly developed quantum mechanics was first successfully explained by Walter Heitler and Fritz London in their paper of 1927 (Gavroglu and Simdes 1994 Gavroglu 1995 Karachalios 2000). In April of that year, Heitler and London, both recipients of a Rockefeller Fellowship, decided to go to the University of Zurich where Erwin Schrodinger was—they both felt more at ease with his more intuitive approach than with Werner Heisenberg s matrix mechanics. Schrodinger agreed to their stay, but there was not much collaboration with him. 

One should not overlook the fact that both of these models represent just different starting points in quantum chemical calculations. It has been known for some time that if both methods were pushed to the limit by including the higher order contributions, they would both yield the same results. The situation is analogous to the relationship between Heisenberg s matrix mechanics and Schrodinger s wave mechanics , which are mathematically equivalent. For an account of The History of Quantum Theory, see the book by Hund (of Hund s rules) with that title. 

With his quantum mechanics, Schrodinger created a powerful formalism for treating atoms and molecules. Schrodinger s version of quantum mechanics was preferred by physicists through the 1930s. As the years passed, however, physicists mastered the matrix mathematics that was the basis of Heisenberg s quantum mechanics and with that mastery they discovered that some problems are treated more naturally with the matrix approach. Today, both approaches are used equally. 

It is commonly accepted that the old quantum theory era spans from the birth of Planck s quantum hypothesis to the formulation of Schrodinger s equation. This section describes the old quantum theory in three parts the failure of classical mechanics, the birth of the quantum theory, and the completion of wave mechanics.5 8) This century obviously began with the birth of quantum theory. Many researchers appeared on the scene of quantum theory at the time, but we remember mostly the contributions of four researchers Max Planck (1901), Albert Einstein (1905), Niels Bohr (1913), and de Broglie (1923). Then Schrodinger proposed the new wave equation to conclude the age of the old quantum theory. Heisenberg established matrix mechanics and formulated the uncertainty principle. 

The only assumption, in addition to Bohr s conjecture, is that the electron appears as a continuous fluid that carries an indivisible charge. As already shown, Bohr s conjecture, in this case, amounts to the representation of angular momentum by an operator L —> ihd/dp, shown to be equivalent to the fundamental quantum operator of wave mechanics, p —> —ihd/dq, or the difference equation (pq — qp) = —ih(I), the assumption by which the quantum condition enters into matrix mechanics. In view of this parallel, Heisenberg s claim [13] (page 262), quoted below, appears rather extravagent ... 

Schrodinger and Bohm both accepted that quantum motion follows a wave pattern. To account for wave-particle dualism the interpretation of matrix mechanics, developed by Heisenberg and others, was extended on the assumption of probability densities. Schrodinger developed the notion of wave structures to simulate particle behaviour, but this model has been rejected almost universally and apparently irretrievably, in favour of proba-bities, arguably prematurely and for questionable reasons. Bohm s attempt to revive the wave interpretation advocated a literary interpretation of wave-particle dualism in the form of a classical particle accompanied and piloted by a quantum wave. 

Letter, Heisenberg to Pauli, 8 June 1926. Pauli s letter appears in full in B. L. van der Waerden, From Matrix Mechanics and Wave Me-... 

These papers present a comprehensive survey of the striking and important recent advances in atomic mechanics. M. Brillouin contributes a detailed and lucid elementary exposition of the Matrix Mechanics of Heisenberg and Dirac. The papers by M. de Broglie include his classical contributions to the new Wave Mechanics, as well as others dealing with Schrbdinger s development of- the idea. 

HEISENBERG S FORMULATION was based on matrix algebra, but was shown to be equivalent to the wave mechanics formulation. 

A paper by 24-year-old Werner Heisenberg turned out to be a breakthrough in quantum theory. He wrote in a letter My whole effort is to destroy without a trace the idea of orbits. Max Born recognized matrix algebra in Heisenberg s formulation (who, himself, had not yet realized it), and in the same year, a more solid formulation of the new mechanics ( matrix mechanics ) was proposed by Werner Heisenberg, Max Born, and Pascual Jordan. ... 

An attemative fwmulation of quantum mechanics, based on matrices, was developed independently by Heisenberg at about the same time. It was later shown that this matrix mechanics is equivalent to Schr6dinga s theory. 

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