scholarly journals PT symmetric, Hermitian and P-self-adjoint operators related to potentials in PT quantum mechanics

2012 ◽  
Vol 53 (1) ◽  
pp. 012109 ◽  
Author(s):  
Tomas Ya. Azizov ◽  
Carsten Trunk
Keyword(s):  
Quantum Mechanics ◽  
Adjoint Operators ◽  
Pt Quantum Mechanics

scholarly journals Coordinate representation for non-Hermitian position and momentum operators

2017 ◽  
Vol 473 (2205) ◽  
pp. 20170434 ◽  
Author(s):  
F. Bagarello ◽  
F. Gargano ◽  
S. Spagnolo ◽  
S. Triolo
Keyword(s):  
Quantum Mechanics ◽  
The Self ◽  
The Other ◽  
Alternative Strategy ◽  
Coordinate Representation ◽  
Suitable Sense ◽  
Adjoint Operators ◽  
Ordinary Quantum

In this paper, we undertake an analysis of the eigenstates of two non-self-adjoint operators q ^ and p ^ similar, in a suitable sense, to the self-adjoint position and momentum operators q ^ 0 and p ^ 0 usually adopted in ordinary quantum mechanics. In particular, we discuss conditions for these eigenstates to be biorthogonal distributions , and we discuss a few of their properties. We illustrate our results with two examples, one in which the similarity map between the self-adjoint and the non-self-adjoint is bounded, with bounded inverse, and the other in which this is not true. We also briefly propose an alternative strategy to deal with q ^ and p ^ , based on the so-called quasi *-algebras .

q-Equivalent Hamiltonian Operators

1972 ◽  
Vol 50 (17) ◽  
pp. 2037-2047 ◽  
Author(s):  
M. Razavy
Keyword(s):  
Quantum Mechanics ◽  
Classical Limit ◽  
First Integral ◽  
Canonical Commutation Relation ◽  
Equation Of Motion ◽  
First Integrals ◽  
Position Operator ◽  
Integrals Of Motion ◽  
Adjoint Operators ◽  
Hamiltonian Operators

From the equation of motion and the canonical commutation relation for the position of a particle and its conjugate momentum, different first integrals of motion can be constructed. In addition to the proper Hamiltonian, there are other operators that can be considered as the generators of motion for the position operator (q-equivalent Hamiltonians). All of these operators have the same classical limit for the probability density of the coordinate of the particle, and many of them are symmetric and self-adjoint operators or have self-adjoint extensions. However, they do not satisfy the Heisenberg rule of quantization, and lead to incorrect commutation relations for velocity and position operators. Therefore, it is concluded that the energy first integral and the potential, rather than the equation of motion and the force law, are the physically significant operators in quantum mechanics.

scholarly journals Vector Models in $\mathcal{PT}$ Quantum Mechanics

2013 ◽  
Vol 52 (7) ◽  
pp. 2187-2195 ◽  
Author(s):  
Katherine Jones-Smith ◽  
Rudolph Kalveks
Keyword(s):  
Quantum Mechanics ◽  
Pt Quantum Mechanics

scholarly journals Transfer principle in quantum set theory

2007 ◽  
Vol 72 (2) ◽  
pp. 625-648 ◽  
Author(s):  
Masanao Ozawa
Keyword(s):  
Quantum Mechanics ◽  
Quantum Logic ◽  
Set Theory ◽  
Von Neumann Algebra ◽  
Transfer Principle ◽  
Real Numbers ◽  
Von Neumann ◽  
Equality Relation ◽  
Adjoint Operators ◽  
Transfer Theorems

AbstractIn 1981, Takeuti introduced quantum set theory as the quantum counterpart of Boolean valued models of set theory by constructing a model of set theory based on quantum logic represented by the lattice of closed subspaces in a Hilbert space and showed that appropriate quantum counterparts of ZFC axioms hold in the model. Here, Takeuti's formulation is extended to construct a model of set theory based on the logic represented by the lattice of projections in an arbitrary von Neumann algebra. A transfer principle is established that enables us to transfer theorems of ZFC to their quantum counterparts holding in the model. The set of real numbers in the model is shown to be in one-to-one correspondence with the set of self-adjoint operators affiliated with the von Neumann algebra generated by the logic. Despite the difficulty pointed out by Takeuti that equality axioms do not generally hold in quantum set theory, it is shown that equality axioms hold for any real numbers in the model. It is also shown that any observational proposition in quantum mechanics can be represented by a corresponding statement for real numbers in the model with the truth value consistent with the standard formulation of quantum mechanics, and that the equality relation between two real numbers in the model is equivalent with the notion of perfect correlation between corresponding observables (self-adjoint operators) in quantum mechanics. The paper is concluded with some remarks on the relevance to quantum set theory of the choice of the implication connective in quantum logic.

scholarly journals ON NON-SELF-ADJOINT OPERATORS FOR OBSERVABLES IN QUANTUM MECHANICS AND QUANTUM FIELD THEORY

2010 ◽  
Vol 25 (09) ◽  
pp. 1785-1818 ◽  
Author(s):  
ERASMO RECAMI ◽  
VLADISLAV S. OLKHOVSKY ◽  
SERGEI P. MAYDANYUK
Keyword(s):  
Quantum Mechanics ◽  
Quantum Electrodynamics ◽  
Nuclear Physics ◽  
Relativistic Quantum ◽  
Relativistic Quantum Mechanics ◽  
Quantum Dissipation ◽  
Interesting Possibility ◽  
Nonrelativistic Quantum Mechanics ◽  
Adjoint Operators ◽  
Unstable States

The aim of this paper is to show the possible significance, and usefulness, of various non-self-adjoint operators for suitable Observables in nonrelativistic and relativistic quantum mechanics, and in quantum electrodynamics. More specifically, this work deals with: (i) the maximal Hermitian (but not self-adjoint) time operator in nonrelativistic quantum mechanics and in quantum electrodynamics; (ii) the problem of the four-position and four-momentum operators, each one with its Hermitian and anti-Hermitian parts, for relativistic spin-zero particles. Afterwards, other physically important applications of non-self-adjoint (and even non-Hermitian) operators are discussed: in particular, (iii) we reanalyze in detail the interesting possibility of associating quasi-Hermitian Hamiltonians with (decaying) unstable states in nuclear physics. Finally, we briefly mention the cases of quantum dissipation, as well as of the nuclear optical potential.

scholarly journals A sharp version of Ehrenfest’s theorem for general self-adjoint operators

2010 ◽  
Vol 466 (2119) ◽  
pp. 2137-2143 ◽  
Author(s):  
Gero Friesecke ◽  
Bernd Schmidt
Keyword(s):  
Quantum Mechanics ◽  
Ehrenfest Theorem ◽  
Adjoint Operators ◽  
Sharp Version

We prove the Ehrenfest theorem of quantum mechanics under sharp assumptions on the operators involved.

scholarly journals From Boolean Valued Analysis to Quantum Set Theory: Mathematical Worldview of Gaisi Takeuti

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 397
Author(s):  
Masanao Ozawa
Keyword(s):  
Quantum Mechanics ◽  
Quantum Logic ◽  
Set Theory ◽  
World View ◽  
Open Problems ◽  
Boolean Valued Analysis ◽  
Mathematical World ◽  
Distributive Law ◽  
And Algebra ◽  
Adjoint Operators

Gaisi Takeuti introduced Boolean valued analysis around 1974 to provide systematic applications of the Boolean valued models of set theory to analysis. Later, his methods were further developed by his followers, leading to solving several open problems in analysis and algebra. Using the methods of Boolean valued analysis, he further stepped forward to construct set theory that is based on quantum logic, as the first step to construct "quantum mathematics", a mathematics based on quantum logic. While it is known that the distributive law does not apply to quantum logic, and the equality axiom turns out not to hold in quantum set theory, he showed that the real numbers in quantum set theory are in one-to-one correspondence with the self-adjoint operators on a Hilbert space, or equivalently the physical quantities of the corresponding quantum system. As quantum logic is intrinsic and empirical, the results of the quantum set theory can be experimentally verified by quantum mechanics. In this paper, we analyze Takeuti’s mathematical world view underlying his program from two perspectives: set theoretical foundations of modern mathematics and extending the notion of sets to multi-valued logic. We outlook the present status of his program, and envisage the further development of the program, by which we would be able to take a huge step forward toward unraveling the mysteries of quantum mechanics that have persisted for many years.

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