Showing papers on "Ladder operator published in 1971"

Self-adjoint algebras of unbounded operators

Abstract: Unbounded *-representations of *-algebras are studied. Representations called self-adjoint representations are defined in analogy to the definition of a self-adjoint operator. It is shown that for self-adjoint representations certain pathologies associated with commutant and reducing subspaces are avoided. A class of well behaved self-adjoint representations, called standard representations, are defined for commutative *-algebras. It is shown that a strongly cyclic self-adjoint representation of a commutative *-algebra is standard if and only if the representation is strongly positive, i.e., the representations preserves a certain order relation. Similar results are obtained for *-representations of the canonical commutation relations for a finite number of degrees of freedom.

Exact Solution of a Time‐Dependent Quantal Harmonic Oscillator with a Singular Perturbation

Abstract: The quantal problem of a particle interacting in one dimension with an external time‐dependent quadratic potential and a constant inverse square potential is exactly solved. The solutions are found both in the Schrodinger representation, by using a generating function or a time‐dependent raising operator, and in the Heisenberg picture. They depend only on the solution of the classical harmonic oscillator. The generalizations to the n‐dimensional problem and to the problem of N particles in one dimension, interacting pairwise via a quadratic time‐dependent potential and a constant inverse square potential, are finally sketched.

Density operator for unpolarized radiation

Abstract: The form of density operator for unpolarized radiation is obtained by a simple method.

Quantization in generalized coordinates

Abstract: The operator form of the generalized canonical momenta in quantum mechanics is derived by a new, instructive method and the uniqueness of the operator form is proven. If one wishes to find the correct representation of the generalized momentum operator, he finds the Hermitian part of the operator —iħ ∂/∂q, whereq q is the generalized coordinate. There are interesting philosophical implications involved in this: It is like saying that a physical structure is composed of two parts, one which is real (the measurable quantity) and one which is pure imaginary. However, in order to understand the theoretical generation of the physical structure, one must look at the imaginary part as well as the real part since the sum of these two parts gives the simplified physical theory. That is why we can choose the total generalized momentum operator as simply —iħ ∂/∂q, but in order to arrive at the “measurable” momentum operator, we must choose the real (Hermitian) part, the other part being anti-Hermitian (corresponding to pure imaginary eigenvalues). We also discuss the operator form of the generalized Hamiltonian and show that the primary focus in developing fundamental concepts and prescriptions in quantum mechanics should be on the generalized momenta rather than on the Hamiltonian.

On the cosine-sine operator functional equations

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