Abstract The main objective of this research paper is to develop an approach within phase-space quantum mechanics for the pseudo-harmonic oscillator potential, enabling the derivation of analytical expressions for the Wigner distribution function and the formulation of a general expression for the moments associated with higher-order position and momentum operators. This approach also allows for the identification of a key dimensionless parameter α in the phase-space transformations. This parameter contributes to improving the general analytical expression for the energy levels. These energy levels are evaluated numerically and show strong agreement with existing numerical results reported in the literature for various diatomic molecular systems. Furthermore, it has been verified that our results are accurate and satisfy both the Heisenberg-Born-Jordan-Dirac commutation relation and the Heisenberg uncertainty principle.
Abstract The main objective of this research paper is to develop an approach within phase-space quantum mechanics for the pseudo-harmonic oscillator potential, enabling the derivation of analytical expressions for the Wigner distribution function and the formulation of a general expression for the moments associated with higher-order position and momentum operators. This approach also allows for the identification of a key dimensionless parameter α in the phase-space transformations. This parameter contributes to improving the general analytical expression for the energy levels. These energy levels are evaluated numerically and show strong agreement with existing numerical results reported in the literature for various diatomic molecular systems. Furthermore, it has been verified that our results are accurate and satisfy both the Heisenberg-Born-Jordan-Dirac commutation relation and the Heisenberg uncertainty principle.
Abstract In this work, we develop a phase space formulation of quantum mechanics to investigate the inversely quadratic Hellmann-Kratzer (IQHK) potential, with the aim of simultaneously constructing the Wigner distribution and the corresponding characteristic functions. By employing Weyl transformations, we establish two independent computational approaches, each yielding explicit and generalized analytical expressions for higher-order moments and momentum. In addition, we derive a generalized analytical formulation of the momentum operator and establish a corresponding generalized expression of Heisenberg’s uncertainty principle, explicitly dependent on the quantum numbers n and L. This constitutes a novel and complementary contribution to the study of the IQHK potential. Moreover, our approach enables the identification of a dimensionless principal parameter β in phase-space transformations, which further refines the generalized analytical expressions for the energy levels En,L. These energy levels are numerically evaluated and exhibit excellent agreement with previously published results for all special cases of the IQHK potential across various diatomic molecular systems. Furthermore, we verify that our results are consistent with the canonical Heisenberg-Born-Jordan-Dirac commutation relations and Heisenberg’s uncertainty principle, thereby confirming the robustness and reliability of the proposed methodology within this framework.
The objective of this thesis is to describe the fundamental concepts relating to the reformulation of quantum mechanics in phase space. It is assumed that the interested reader is familiar with the principles and techniques of ordinary quantum mechanics. In classical statistical mechanics the expectation values of physical quantities are calculated as averages over phase space distribution functions. It is possible to obtain a similar procedure in quantum mechanics. However, phase space is a classical concept and thus has no quantum mechanical equivalent, owing to the non-commutability of the operators p and q. A sensible basic requirement for the formulation of quantum mechanics in phase space is a linear one-one mapping between quantum operators and classical functions. This was achieved by the pioneering work of Weyl, Wigner and Moyal. The phase space that they devised we shall call "pseudo phase space". It is completely quantum. The pseudo phase space variables are represented by commuting c-numbers of momentum, p, and position, q; they are the result of applying the Weyl correspondence rule to the operators p and q, respectively. In some ways the pseudo phase space formulation of quantum mechanics appears to return us to an "almost" classical arena, since we dispense with the calculus of non-commuting operators for purely algebraic methods. The pseudo phase space formulation of quantum mechanics is self-contained. That is, there is no need, in principle, to switch to the Schroedinger or Heisenberg pictures when solving physical problems, although it may often be convenient to do so. The classical appearance of the pseudo phase space scheme is especially useful when considering the semi-classical limit of quantum mechanics. In fact, for potentials up to and including the harmonic oscillator the equations represent classical motion. The structure of the thesis is such that it takes the reader steadily through the major concepts of pseudo phase space theory. The mathematical techniques that shall be needed are developed early in Chapter 1. These are used throughout the thesis. This presents the advantage that the reader is not faced with learning new mathematical methods as the thesis proceeds. The pseudo phase space formulation of quantum mechanics as found applications in all areas of mathematical physics. Hence, there is a considerable amount of specialist literature available on the subject. Itis hoped that this thesis serves has an introduction.
The Weyl transform is applied in quantum dynamics to derive and extend Moyal's statistical theory of phase-space distributions for noncommuting coordinate and momentum operators. The distinction is made between Weyl transforms in Schrödinger and Heisenberg pictures; the general case of time-dependent Hamiltonians is considered. The Wigner function for the probability distribution in a phase space of Cartesian coordinates Q and momenta K propagates according to a conditional probability P(t, Q, K | t0, Q0, K0), which is exhibited as a Feynman path integral in phase space. Properties of P(t, Q, K | t0, Q0, K0) are developed; it is expressed in terms of the quantum generalization of the classical Liouville operator. The Weyl transform of a Heisenberg operator propagates according to P(t, Q0, K0 | t0, Q, K) which is also given as a Feynman path integral. An equation for the time evolution of Weyl transforms of Heisenberg operators is obtained, according to which the transform of Heisenberg coordinate and momentum operators obey a quantum form of Hamilton's equations of motion. If the initial density operator of the system commutes with the coordinate operator, then the state of the system is a mixture of pure coordinate states; the spectrum of the density operator in this case is continuous. For a Heisenberg operator AH(t) with Weyl transform AH(t, Q, K), the function A(t,Q)=∫dKAH(t,Q,K) is the expectation at time t of the dynamical property for a quantum system initially in a pure state of coordinate Q; it is the quantum-mechanical generalization of the dynamical property of the system along the classical trajectory in configuration space at time t. The probability amplitude for the time dependence of A(t,Q) can be expressed as a Feynman path integral with a Heisenberg Lagrangian. The amplitude of the conditional probability P(t, Q | t0, Q0) considered by Feynman is expressed as a path integral with a Schrödinger Lagrangian. The velocity in the Heisenberg Lagrangian is the negative of that in the Schrödinger Lagrangian of Feynman, but it agrees with the velocity appearing in the Hamiltonian equations. It is the Heisenberg Lagrangian that is the Lagrangian of classical dynamics. For a particle whose potential energy is a function of position, a quantum form of Newton's second law is obtained. An extension of the formalism to non-Cartesian coordinate systems is given.
We present a phase space formulation of quantum mechanics in the Schrödinger representation and derive the associated Weyl pseudo-differential calculus. We prove that the resulting theory is unitarily equivalent to the standard "configuration space" formulation and show that it allows for a uniform treatment of both pure and mixed quantum states. In the second part of the paper we determine the unitary transformation (and its infinitesimal generator) that maps the phase space Schrödinger representation into another (called Moyal) representation, where the wave function is the cross-Wigner function familiar from deformation quantization. Some features of this representation are studied, namely the associated pseudo-differential calculus and the main spectral and dynamical results. Finally, the relation with deformation quantization is discussed.
Abstract Phase spaces as given by the Wigner distribution function provide a natural description of infinite-dimensional quantum systems. They are an important tool in quantum optics and have been widely applied in the context of time–frequency analysis and pseudo-differential operators. Phase-space distribution functions are usually specified via integral transformations or convolutions which can be averted and subsumed by (displaced) parity operators proposed in this work. Building on earlier work for Wigner distribution functions (Grossmann in Commun Math Phys 48(3):191–194, 1976. https://doi.org/10.1007/BF01617867 ), parity operators give rise to a general class of distribution functions in the form of quantum-mechanical expectation values. This enables us to precisely characterize the mathematical existence of general phase-space distribution functions. We then relate these distribution functions to the so-called Cohen class (Cohen in J Math Phys 7(5):781–786, 1966. https://doi.org/10.1063/1.1931206 ) and recover various quantization schemes and distribution functions from the literature. The parity operator approach is also applied to the Born–Jordan distribution which originates from the Born–Jordan quantization (Born and Jordan in Z Phys 34(1):858–888, 1925. https://doi.org/10.1007/BF01328531 ). The corresponding parity operator is written as a weighted average of both displacements and squeezing operators, and we determine its generalized spectral decomposition. This leads to an efficient computation of the Born–Jordan parity operator in the number-state basis, and example quantum states reveal unique features of the Born–Jordan distribution.
The objective of this work is to present some aspects of the phase space representation of quantum mechanics on the Euclidean Motion Group of rank three. In particular, the main contribution of this paper is the statement and proof of fundamental formula relating the Weyl transform for the group and the Wigner functions on the group, given by 〈F1|WkF2〉=∫R3∫so(3)k(c,C)WE(3)(F1,F2)dc dC. Here, Wk is the Weyl transform of the function k on E(3) and WE(3) is the Wigner transform. Thus, the various phase space objects such as the Wigner transform and the Weyl transform are also explicitly formulated on the Euclidean Motion Group.