On the completeness problem for the eigenfunction formulae of relativistic quantum mechanics

1961 ◽  
Vol 262 (1311) ◽  
pp. 489-502 ◽  
Keyword(s):  
Wave Function ◽  
Quantum Mechanics ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Relativistic Quantum ◽  
Relativistic Quantum Mechanics ◽  
Partial Differential ◽  
Completeness Problem ◽  
Complete Set

Dirac’s theory of relativistic quantum mechanics leads to the problem of solving a set of four partial differential equations for the four components of the wave function. Solutions of these equations in the case where the potential is a function of the radial co-ordinate only were obtained by Darwin. It is proved that these solutions form a complete set in the sense that we can simultaneously expand four arbitrary functions in terms of them.

scholarly journals Deformed Shape Invariant Superpotentials in Quantum Mechanics and Expansions in Powers of ℏ

Symmetry ◽  
2020 ◽  
Vol 12 (11) ◽  
pp. 1853
Author(s):  
Christiane Quesne
Keyword(s):  
Differential Equation ◽  
Quantum Mechanics ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Supersymmetric Quantum Mechanics ◽  
Invariance Condition ◽  
Partial Differential ◽  
Shape Invariance ◽  
Shape Invariant ◽  
Infinite Set

We show that the method developed by Gangopadhyaya, Mallow, and their coworkers to deal with (translational) shape invariant potentials in supersymmetric quantum mechanics and consisting in replacing the shape invariance condition, which is a difference-differential equation, which, by an infinite set of partial differential equations, can be generalized to deformed shape invariant potentials in deformed supersymmetric quantum mechanics. The extended method is illustrated by several examples, corresponding both to ℏ-independent superpotentials and to a superpotential explicitly depending on ℏ.

Probability relations between separated systems

1936 ◽  
Vol 32 (3) ◽  
pp. 446-452 ◽  
Author(s):  
E. Schrödinger
Keyword(s):  
Wave Function ◽  
Quantum Mechanics ◽  
Pure State ◽  
Relativistic Quantum ◽  
Present Theory ◽  
Relativistic Quantum Mechanics ◽  
Quantum Mechanical ◽  
Alternative Possibility ◽  
Linearly Independent ◽  
Statistical Operator

The paper first scrutinizes thoroughly the variety of compositions which lead to the same quantum-mechanical mixture (as opposed to state or pure state). With respect to a given mixture every state has a definite probability (or mixing fraction) between 0 and 1 (including the limits), which is calculated from the mixtures Statistical Operator and the wave function of the state in question.A well-known example of mixtures occurs when a system consists of two separated parts. If the wave function of the whole system is known, either part is in the situation of a mixture, which is decomposed into definite constituents by a definite measuring programme to be carried out on the other part. All the conceivable decompositions (into linearly independent constituents) of the first system are just realized by all the possible measuring programmes that can be carried out on the second one. In general every state of the first system can be given a finite chance by a suitable choice of the programme.It is suggested that these conclusions, unavoidable within the present theory but repugnant to some physicists including the author, are caused by applying non-relativistic quantum mechanics beyond its legitimate range. An alternative possibility is indicated.

scholarly journals NEUTRINO OSCILLATIONS AND ENERGY–MOMENTUM CONSERVATION

2010 ◽  
Vol 25 (07) ◽  
pp. 479-487 ◽  
Author(s):  
T. GOLDMAN
Keyword(s):  
Wave Function ◽  
Quantum Mechanics ◽  
Fourth Order ◽  
Relativistic Quantum ◽  
Momentum Conservation ◽  
Relativistic Quantum Mechanics ◽  
Entangled State ◽  
Neutrino Masses ◽  
Mass Eigenstates ◽  
Energy Momentum Conservation

A description of neutrino oscillation phenomena is presented which is based on relativistic quantum mechanics with four-momentum conservation. This is different from both conventional approaches which arbitrarily use either equal energies or equal momenta for the different neutrino mass eigenstates. Both entangled state and source dependence aspects are also included. The time dependence of the wave function is found to be crucial to recovering the conventional result to second order in the neutrino masses. An ambiguity appears at fourth order which generally leads to source dependence, but the standard formula can be promoted to this order by a plausible convention.

OCTONIC FIRST-ORDER EQUATIONS OF RELATIVISTIC QUANTUM MECHANICS

2009 ◽  
Vol 24 (22) ◽  
pp. 4157-4167 ◽  
Author(s):  
VICTOR L. MIRONOV ◽  
SERGEY V. MIRONOV
Keyword(s):  
Wave Function ◽  
Quantum Mechanics ◽  
Maxwell Equations ◽  
Relativistic Quantum ◽  
Order Equation ◽  
Second Order ◽  
Relativistic Quantum Mechanics ◽  
Quantum Fields ◽  
First Order ◽  
Spatial Operators

We demonstrate a generalization of relativistic quantum mechanics using eight-component octonic wave function and octonic spatial operators. It is shown that the second-order equation for octonic wave function describing particles with spin 1/2 can be reformulated in the form of a system of first-order equations for quantum fields, which is analogous to the system of Maxwell equations for the electromagnetic field. It is established that for the special types of wave functions the second-order equation can be reduced to the single first-order equation analogous to the Dirac equation. At the same time it is shown that this first-order equation describes particles, which do not have quantum fields.

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