Quantenmechanische Phasenoperatoren im Zusammenhang mit orthogonalen Polynomsystemen

The approach of Susskind and Glogower to the quantum phase problem defines Hermitian operators, which may be interpreted as cosine and sine operators. Their eigenstates in the Fock representation are the Chebyshev polynomials of the second kind. On the basis of this approach more general cosine and sine operators are introduced, whose eigenstates in the Fock representation are given by arbitrary orthogonal polynomial sets on the interval [−1,+1] with respect to a weight function. To every polynomial set there corresponds a pair of cosine and sine operators. Depending on the symmetry of the weight function one distinguishes between generalized and extended cosine and sine operators. Corresponding arccosine and arcsine operators of the generalized and extended type are introduced. The eigenstates of the trigonometric and inverse trigonometric operators are studied and used to define corresponding representations of an arbitrary quantum state as well as corresponding probability distributions. Explicit examples are given for the classical orthogonal polynomials. Further, exponential operators generalizing the Susskind-Glogower exponential phase operators are introduced in terms of the cosine and sine operators. The eigenstates of the lowering exponential operators are defined on the unit disk and yield generalized coherent states if they admit a resolution of unity. In this case they can be used to define two-dimensional probability distributions (Q-functions) on the unit disk and corresponding phase distributions as marginal distributions. In the case of the classical orthogonal polynomials the eigenstates of the generalized (extended) operators are normalized to the hypergeometric functions 2 F 1 ( 4 F 3 ); the resolution of unity needs a special treatment as a limiting case. Finally, extending the class of states encounted above, generalized hypergeometric states normalized to the generalized hypergeometric functions p F q are introduced. Depending on the radius of convergence of p F q , one distinguishes between generalized hypergeometric states on the (whole) plane, on the unit disk and on the unit circle. The states yielding a resolution of unity define the generalized hypergeometric coherent states. They can be used to define representations of an arbitrary state in the appropriate Bargmann and Hardy spaces, respectively, as well as corresponding generalized hypergeometric Husimi distributions and (marginal) phase distributions.