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.

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.

EFFECT OF MAXIMUM MOMENTUM ON QUANTUM MECHANICAL SCATTERING AND BOUND STATES

2013 ◽  
Vol 28 (15) ◽  
pp. 1350061 ◽  
Author(s):  
CHEE-LEONG CHING ◽  
RAJESH R. PARWANI
Keyword(s):  
Quantum Mechanics ◽  
Bound States ◽  
Relativistic Quantum ◽  
Relativistic Quantum Mechanics ◽  
Quantum Mechanical ◽  
Position Uncertainty ◽  
Exact Position ◽  
Usual Quantum ◽  
Quantum Mechanical Scattering ◽  
Mechanical Scattering

We construct the exact position representation for a deformed (non-relativistic) quantum mechanics which exhibits an intrinsic maximum momentum and use it to study problems such as a particle in a box and an asymmetric well. In particular, we show that unlike usual quantum mechanics, the present deformed case delays the formation of bound states in a finite potential well, a distinguishing feature that might be relevant for empirical investigations. We also contrast our results with the string-motivated type of deformed quantum mechanics which incorporates a minimum position uncertainty rather than a maximum momentum.

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.

scholarly journals On the Relativistic Quantum Mechanics of a Particle in Space with Minimal Length

2012 ◽  
Vol 57 (9) ◽  
pp. 942
Author(s):  
Ch.M. Scherbakov
Keyword(s):  
Quantum Mechanics ◽  
Commutation Relation ◽  
Relativistic Quantum ◽  
Heisenberg Algebra ◽  
Minimal Length ◽  
Group Symmetry ◽  
Relativistic Quantum Mechanics ◽  
Noncommutative Space ◽  
Quantum Mechanical

A noncommutative space and the deformed Heisenberg algebra [X,P] = iħ{1 – βP2}1/2 are investigated. The quantum mechanical structures underlying this commutation relation are studied. The rotational group symmetry is discussed in detail.

scholarly journals GRAVITOMAGNETIC MOMENTS AND DYNAMICS OF DIRAC (SPIN ½) FERMIONS IN FLAT SPACE–TIME MAXWELLIAN GRAVITY

2004 ◽  
Vol 19 (25) ◽  
pp. 4207-4229 ◽  
Author(s):  
HARIHAR BEHERA ◽  
P. C. NAIK
Keyword(s):  
Quantum Mechanics ◽  
Equivalence Principle ◽  
Relativistic Quantum ◽  
Flat Space ◽  
Space Time ◽  
Relativistic Quantum Mechanics ◽  
Quantum Mechanical ◽  
Gravitational Effects ◽  
Spin Particle ◽  
Dirac Spin

The gravitational effects in the relativistic quantum mechanics are investigated in a relativistically derived version of Heaviside's speculative gravity (in flat space–time) named here as "Maxwellian gravity." The standard Dirac's approach to the intrinsic spin in the fields of Maxwellian gravity yields the gravitomagnetic moment of a Dirac (spin ½) particle exactly equal to its intrinsic spin. Violation of the Equivalence Principle (both at classical and quantum-mechanical level) in the relativistic domain has also been reported in this work.

Quantum-mechanical, gravitating Kerr–Newton particles

1992 ◽  
Vol 439 (1906) ◽  
pp. 271-278 ◽  
Keyword(s):  
General Relativity ◽  
Quantum Mechanics ◽  
Relativistic Quantum ◽  
Relativistic Quantum Mechanics ◽  
Quantum Mechanical ◽  
Twistor Theory ◽  
Theory Of Gravitation

Newton’s theory of gravitation is extended to allow for sources with spin. Ideas from twistor theory are used while the laplacian relation between source and potential is retained. The resulting gravitational potentials display singularity structures analogous to those found in Kerr’s solutions in general relativity. However, incorporating the effects of non-relativistic quantum mechanics removes the singularity for sources with spins 1/2, 3/2, 5/2...

scholarly journals Big picture of relativistic molecular quantum mechanics

2015 ◽  
Vol 3 (2) ◽  
pp. 204-221 ◽  
Author(s):  
Wenjian Liu
Keyword(s):  
Wave Function ◽  
Quantum Mechanics ◽  
Quantum Electrodynamics ◽  
Relativistic Quantum ◽  
Coupling Constants ◽  
Quantum Mechanical ◽  
Big Picture ◽  
The Right ◽  
Molecular Quantum Mechanics ◽  
Energy Property

Abstract Any quantum mechanical calculation on electronic structure ought to choose first an appropriate Hamiltonian H and then an Ansatz for parameterizing the wave function Ψ, from which the desired energy/property E(λ) can finally be calculated. Therefore, the very first question is: what is the most accurate many-electron Hamiltonian H? It is shown that such a Hamiltonian i.e. effective quantum electrodynamics (eQED) Hamiltonian, can be obtained naturally by incorporating properly the charge conjugation symmetry when normal ordering the second quantized fermion operators. Taking this eQED Hamiltonian as the basis, various approximate relativistic many-electron Hamiltonians can be obtained based entirely on physical arguments. All these Hamiltonians together form a complete and continuous ‘Hamiltonian ladder’, from which one can pick up the right one according to the target physics and accuracy. As for the many-electron wave function Ψ, the most intriguing questions are as follows. (i) How to do relativistic explicit correlation? (ii) How to handle strong correlation? Both general principles and practical strategies are outlined here to handle these issues. Among the electronic properties E(λ) that sample the electronic wave function nearby the nuclear region, nuclear magnetic resonance (NMR) shielding and nuclear spin-rotation (NSR) coupling constant are especially challenging: they require body-fixed molecular Hamiltonians that treat both the electrons and nuclei as relativistic quantum particles. Nevertheless, they have been formulated rigorously. In particular, a very robust ‘relativistic mapping’ between the two properties has been established, which can translate experimentally measured NSR coupling constants to very accurate absolute NMR shielding scales that otherwise cannot be obtained experimentally. Since the most general and fundamental issues pertinent to all the three components of the quantum mechanical equation HΨ = EΨ (i.e. Hamiltonian H, wave function Ψ, and energy/property E(λ)) have fully been understood, the big picture of relativistic molecular quantum mechanics can now be regarded as established.

Sign in / Sign up

Export Citation Format

Share Document