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Communications in Theoretical Physics

Quantum Mechanical Version of z -Transform Related to Eigenkets of Boson Creation Operator

Fan Hong-Yi, Liang Fu, Alfred Wünsche

Published Nov 15, 2004• 4 citations
Publisher

Abstract

Using the completeness relation composed of the coherent state and of the eigenket of bosonic creation operator, we establish a one-to-one correspondence between the z -transform and the quantum-mechanical transform from the representation by number states to the representation by coherent states (Bargmann representation). In this way, the quantum-mechanical version of the various properties of z -transform are obtained and the operators for embodying these properties in the Fock space are derived, which may find applications in quantum states engineering.

DOI
10.1088/0253-6102/42/5/675

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Using the completeness relation composed of the coherent state and of the eigenket of bosonic creation operator, we establish a one-to-one correspondence between the z -transform and the quantum-mechanical transform from the representation by number states to the representation by coherent states (Bargmann representation). In this way, the quantum-mechanical version of the various properties of z -transform are obtained and the operators for embodying these properties in the Fock space are derived, which may find applications in quantum states engineering.
Abstract This chapter defines and determines the properties of the important (though non-physical) boson and fermion coherent states, which are specified in terms of c-number (bosons) or Grassmann (fermions) variables and their complex conjugates, and defined via unitary displacement operators acting on the vacuum state. Important non-orthogonality, overcompleteness and coherence properties of coherent states are obtained. Coherent states are shown to be eigenstates of mode annihilation operators, with eigenvalues given by the c-number or Grassmann variable specifying the state. Related (unnormalised) Bargmann states are defined, depending only on the variable and not its complex conjugate, and are used to obtain useful representations of physical states and canonical forms for operators in terms of Bargmann state projectors, as well as obtaining important results such as for traces of operator products in terms of phase space integrals. Mode creation operators acting on Bargmann states are shown as equivalent to differentiation.
A q-analogue of Bargmann space is defined, using the properties of coherent states associated with a pair of q-deformed bosons. The space consists of a class of entire functions of a complex variable z, and has a reproducing kernel. On this space, the q-boson creation and annihilation operators are represented as multiplication by z and q-differentiation with respect to z, respectively. A q-integral analogue of Bargmann's scalar product is defined, involving the q-exponential as a weight function. Associated with this is a completeness relation for the q-coherent states.
In the Bargmann representation of quantum mechanics, physical states are mapped into entire functions of a complex variable z *, whereas the creation and annihilation operators and play the role of multiplication and differentiation with respect to z *, respectively. In this paper we propose an alternative representation of quantum states, conjugate to the Bargmann representation, where the roles of and are reversed, much like the roles of the position and momentum operators in their respective representations. We derive expressions for the inner product that maintain the usual notion of distance between states in the Hilbert space. Applications to simple systems and to the calculation of semiclassical propagators are presented.