In this work we analyze some aspects related to the introduction, in the non-relativistic quantum mechanics,of a translation operator whose action is a function of the point of application. From the introduction of this operator, we derive a modified canonical commutation relation and a modified representation for the momentum operator and we show that such changes require a restructuring of the Hilbert space. We also see that the “new” Hilbert space can be associated with the case of a particle with position dependent mass and also that of a particle moving in a curved space. Still, we compare our formalism with the formalism of von Roos, we analyze the classical limit of this formalism,we solve the problem of a particle with position-dependent mass in two dimensions and we end up analyzing some quantities of information theory for the case of a one-dimensional harmonic particle.
This thesis addresses aspects of quantum phase and sets out to draw attention to a collection of new and extend some not so new techniques utilizing Weyl's correspondence rule and its inverse. The phase formulation of quantum mechanics is as old as quantum mechanics itself and evolved over a period of two decades due to the contributions of H. Weyl \cite{Weyl1950}, E. P. Wigner \cite{Wigner1932}, H. Groenewold \cite{Groenewold1946} and J. E. Moyal \cite{Moyal1949}. Over the ensuing years the phase picture has matured into a practical analytical tool, and has generated a multitude of technical papers, reports, reviews, and expositions from many authors and commentators. One reason for its success is that it offers an intuitive picture of quantum mechanics owing to its \it{classical look and feel}, and it has made an impression, in one way or another, in many fields such as mathematical physics, quantum optics, and physical chemistry.The thesis opens with a brief history of classical phase space, tracing the possible origin of the term ``phase space itself, and chronicles the events leading to the conceptualization (and generalization) of quantum phase space. In Chapter 2 the Weyl correspondence is recast in terms of von Neumann's integral kernel transform and a heuristic analysis is given from this perspective. A characteristic of Weyl's correspondence rule and its inverse is that there are two inherent representations of the same operator. These have been referred to as the Weyl representation and Wigner representation and are related by a symplectic Fourier transform. For an arbitrary outer dyadic operator it is shown that the Weyl representation leads to the \it{modified} Gabor transform (known to Signal Processing), while the Wigner representation specializes to the Bargmann transform and a representation in terms of analytic functions. Calculating Weyl-Wigner (WW) transforms of products of
n operators, from first principles, has proved to be a technically difficult problem. However, with the machinery developed in Chapter 2 the problem is reduced to a straightforward algebraic exercise. Finally, forming an
s-parametrized generalization of Weyl's correspondence rule and focusing on the Weyl representation it is shown to lead to an intimate relationship with the relative state formalism \cite{Ban1998}. Weyl's correspondence rule is only applicable to operators and consequently state vectors
∣ψ⟩ are not represented in the usual formulation of quantum phase space. This problem is circumvented in Chapter 3 by introducing the notion of a normalized fixed state (or window) vector
∣φ0⟩ to form an outer dyadic product $\wh{\Psi} = |\psi\rangle\langle\varphi_0|$. The WW transform of this product under the action of the Weyl's rule is a complex valued function of real variables called a phase amplitude and denoted by the symbol
Ψ. The phase amplitudes have many (and sometimes surprising) properties. For example they: \begin{itemize} \item form the building blocks of Wigner functions
W by admitting a generalized Born rule
W=Ψ⋆Ψˉ, and as such can be considered to be a more fundamental objects than Wigner functions; \item superpose and interfere like the state vectors they represent; \item specialize to the Bargmann and Gabor transforms when a Gaussian window is chosen; \end{itemize} \ni In Chapter 4 further aspects of phase amplitudes are explored and discussed. For example it is shown that they: \begin{itemize} \item define phase state vectors which form an over complete set and are elements of a reproducing kernel Hilbert space; \item allow unitary ray representations of the group symmetries of translation, rotation and time reversal; \item evolve in phase under the influence of a propagator
R which is intimately related to the Wigner propagator
P. \end{itemize} Chapter 5 considers an extension of phase amplitudes to include systems with finitely many degrees of freedom. These are called spin amplitudes and are obtained by first forming the outer dyadic product of a spin state vector with an arbitrary fixed spin state vector, and then applying a finite-dimensional analogue of the WW transform. The spin amplitude operator is represented as an
n×n matrix, whose elements are products of the spin components of the spin and fixed state vectors. Two kinds of spin amplitudes are considered, arising on the sphere
S2 or on a
(2j+1) dimensional lattice. In the special case of spin-$\half$ systems it is shown that the spin amplitudes transform as spinors under rotations on both the sphere and lattice. As a physical application the spin amplitudes for a spin-$\half$ magnetic dipole are found for a Nuclear Magnetic Resonance (NMR) environment. Chapter 6 considers Schwinger's abstract approach to quantum mechanics which takes into account the inability to indefinitely compensate for fluctuations in the measurement process. The resulting measurement algebra provides a symbolic representation of the physical laws as opposed to the mathematical abstraction (of von Neumann and Dirac) that was justified \it{\'a posteriori} by being in agreement with experiment. Despite Schwinger's prolific work on the subject of measurement algebra, its interpretation in quantum phase remained an open issue. It is shown that the key to formulating a quantum phase in terms of the measurement operators and transformation functions is the determination of a suitable set of basis operators for an arbitrary measurement operator. Many new results follow as a consequence of expanding an arbitrary measurement operator in this basis. In Chapter 7 the quantum-classical interface is explored by considering Hamilton's principal function or action
S(q,v,t). Starting from a purely heuristic classical argument, a phase function
U(p,q,t) is derived in terms of the classical action and is shown to be proportional to the WW transform of the quantum evolution operator. This association leads to the transform \be U(p,q,t) = A(t) \int e^{ipv/\hbar} e^{iS(q,v,t)/\hbar} \; \rm{d}v, \ee which enables one, in principle, to deduce the Wigner functions of a given system when only the classical action is known. Simple physical systems are used to provide examples illustrating this approach.
The physical content of a theory is not intrinsically tied to any single mathematical formalism. Both classical and quantum mechanics admit equivalent representations, notably in phase space and in Hilbert space, related by the Wigner-Weyl correspondence. While this correspondence has long been studied in mathematical physics, its foundational and operational implications are often left implicit. Here we give a systematic account of what changes, and what does not, when classical and quantum theories are expressed in each other's native language. This representational viewpoint separates artifacts (such as the appearance of non-positivity or negativity under certain maps) from robust structural distinctions that persist across representations, in particular noncommutativity and its
ℏ-dependent
⋆-deformation of the classical algebra. We develop the comparison at the level of states, kinematics, and dynamics, and extend it to measurement by formulating both outcome statistics and state-update rules within the same framework.
In the context of quantum mechanics reformulated in a modified Hilbert space, we can formulate the Feynman’s path integral approach for the quantum systems with position-dependent mass particle using the formulation of position-dependent infinitesimal translation operator. Which is similar a deformed quantum mechanics based on modified commutation relations. An illustration of calculation is given in the case of the harmonic oscillator, the infinite square well and the inverse square plus Coulomb potentials.
We present an algebra of deformed Heisenberg-type [^ x; p ^] = i~f(^ x) arising from a rede nition of the translation operator. We point out that the Hilbert space should be used for the representation of the algebra in question and show the explicit form of the position and momentum operators in this space, in addition, we also write the equations that governs the time evolution of quantum systems (the Schrödinger equation) and use it to solve a traditional problem in the usual quantum mechanics: the harmonic oscillator immersed in a uniform electric eld . It should also point out that we used an unusual method to the solution of the harmonic oscillator, the method of the Laplace transform.
Abstract We develop a deformation quantization approach to quantum mechanics that exhibits a nonzero minimal uncertainty in position through the modification of commutation relations for the operators of position and momentum. Deformation quantization is a method of quantizing a classical Hamiltonian system by suitably deforming the Poisson algebra of the system. The developed theory of deformation quantization is non-formal. An appropriate integral formula for the star-product is introduced, along with a suitable space of functions on which the star-product is well defined. A C*-algebra of observables and a space of states are constructed. Moreover, an operator representation in momentum space is presented. Finally, an example of states of maximal localization is given in terms of generalized Wigner functions.