Quantum Mechanical Theorem Research Articles

Degenerate Systems and Mass Singularities

For a system with degenerate energies, the power series expansions of the $S$-matrix elements may become singular. An elementary theorem in quantum mechanics is proved which shows that under certain general conditions such singularities do not appear in the power series expansions of the transition probabilities, provided these are averaged over an appropriate ensemble of degenerate states. Application of this theorem leads to the cancellations of mass singularities and infrared divergences in quantum electrodynamics. The question of whether a charged particle can have zero mass is studied.

An improved version of an ergodic theorem in quantum mechanics

An improved version of an ergodic theorem in quantum mechanics

Ergodic theorem in quantum mechanics. Evaluation of the probability of an exceptional initial condition

In this paper an evaluation is made of the probability of finding an exceptional initial condition in the sense of the Bocchieri-Loinger formulation of the quantum ergodic theorem. The result is compared with that obtained byPauli andFierz who made a corresponding evaluation of the probability of finding a non-thermodynamical macro-observer in the original von Neumann formulation.

Ergodic Theorem in Quantum Mechanics

Landsberg and Farquhar have shown that the von Neumann-Fierz approach to the quantum ergodic problem does not give any criterion of ergodicity.In this paper it is shown that von Neumann's procedure of averaging over the macro-observers is sufficient by itself to "justify" at every time the use of the quantum microcanonical ensemble. It is then proved that the time behavior of the system is completely irrelevant to the demonstration of the von Neumann and von Neumann-Fierz ergodic theorems. It is concluded that the quantum ergodic problem must be attacked on entirely new lines.

The theorem on incoming waves

The general form of the following theorem is stated and proved : if a particle is scattered by two potentials, then the resulting amplitude for a real transition may be written as the sum of two parts. The first is the amplitude for scattering by either of the potentials acting alone. The second is that for scattering by the other potential, where the incident wave is the wave function belonging to the first potential in which the asymptotic form of the « scattered » wave moves inward. Some remarks are added concerning the applications of this theorem in quantum mechanics and the theory of fields.

XLIV. The virial theorem in quantum mechanics.

Summary A derivation is given of a form of the virial theorem in quantum mechanics suitable for the discussion of a system of particles in a box.

On the Adiabatic Theorem of Quantum Mechanics

On the Adiabatic Theorem of Quantum Mechanics

Larmor's Theorem in Quantum Mechanics

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On the probability of artificial nuclear transformations and its connection with the vector model of the nucleus

1. The theorem of quantum mechanics that “the spin part of the angular momentum is approximately a constant of motion provided that the forces depending on the direction of the spins are small compared with the total interaction forces” is introduced into the discussion of artificial nuclear transformations. A definition is given for the terms “probable” and “not probable” nuclear reaction.2. Attention is drawn to the experimental result that the reaction Li6 + H1 → He4 + He3 is more probable than the reaction Li7 + H1 → 2He4, and an explanation is suggested.3. The nuclear spin of Lie is derived from nuclear transformations and found to be 1.4. Similar derivations for the spins of H3, He3 and B11 are summarised in a table.

An elementary theory of electronic semi-conductors, and some of their possible properties

The formal theory of electronic semi-conductors was started by A. H. Wilson. It has recently been carried further by Bronstein, and has, of course, been applied (for example to the theory of rectifying contacts) and discussed in a number of other places. There is, however, I think, still room for a more general but quite elementary discussion of a number of possible models, all of which represent possible varieties of semi-conductors; it is to be understood throughout that only electronic conduction is in question. Granted certain general quantum mechanical theorems, the elementary theory can be made both simple and exact. After presenting such a version of the theory here, it is my purpose to show how the proposed model semi-conductors can account for the positive or negative or zero Hall coefficients which are observed, and also for the large positive or negative thermoelectric power of a semi-conductor-metal thermocouple and for the sign relationships of the two effects. The possibility of the explanation of an abnormal sign for the Hall coefficient is, of course, no new thing; it was first contemplated in the work of Peierls on metals and at his suggestion taken over (but not elaborated) by Bronstein ( loc. cit .) for semi-conductors. But a satisfactory (even elementary) theory requires us to consider the conduction in a solid as a mixed phenomenon due to two almost independent families of electrons. This has not been undertaken by Bronstein or by Peierls and such a theory is given here. It may prove of importance in further study of semi-conductors, beyond the phenomena on which attention is concentrated here. The scope of this paper then is as follows. We give an elementary theory of the effective number of “conduction” electrons in model semi-conductors of various types. We take account both of the few ordinary electrons in otherwise empty levels, and of the few holes or vacancies in otherwise completely full electron levels which function as positive electrons. We work out for semi conductors, conducting partly by electrons and partly by holes, the isothermal Hall coefficient and the thermo-electric power of the thermocouple formed by the semi-conductor and an ideal metal. We thus show in detail how abnormal signs of the Hall effect and the thermo-electric power can be fitted into the theory.