Diagonalization of Hamiltonians, Uncertainty Matrices and Robertson Inequality

International Journal of Applied Mathematics and Theoretical Physics, 2015

We present a study on linear canonical transformation in the framework of a phase space representation of quantum mechanics that we have introduced in our previous work . We begin with a brief recall about the so called phase space representation. We give the definition of linear canonical transformation with the transformation law of coordinate and momentum operators. We establish successively the transformation laws of mean values, dispersions, basis state and wave functions. Then we introduce the concept of isodispersion linear canonical transformation.

2020

Linear Canonical Transformations (LCTs) are known in signal processing and optics as the generalization of certain useful integral transforms such as Fourier and fractional Fourier ones. In quantum theory, they are the linear transformations which keep invariant the canonical commutation relations between coordinates and momenta operators. There is also a similarity between them and Bogolioubov transformation. In this work, the possibility of considering LCTs as the elements of a symmetry group for relativistic quantum physics is studied using the principle of covariance. It is established that Lorentz transformations and Fourier transforms are particular cases of LCTs and some of the main symmetry groups currently considered in relativistic theories can be obtained from the contractions of LCTs groups. It is also shown that a link can be established between the spinorial representation of LCTs and some properties of elementary fermions. A classification which suggests the existence...

Physics Letters A, 1996

A comoving reference frame in phase space is introduced in order to analyze the behaviour of time-dependent multi-dimensional quadratic Hamiltonians. It is found that an explicit (time-dependent) representation exists in which the wave function always remains in a factorized form and independent of time. In this way the previous finding that the Wigner functions of such systems can be cast into factorized time-independent form by the introduction of appropriate time-dependent coordinates related to its integrals of motion is explained.

Journal of Physics A: Mathematical and Theoretical, 2012

A set of generalized squeezed-coherent states for the finite u(2) oscillator is obtained. These states are given as linear combinations of the mode eigenstates with amplitudes determined by matrix elements of exponentials in the su(2) generators. These matrix elements are given in the (N + 1)-dimensional basis of the finite oscillator eigenstates and are seen to involve 3 × 3 matrix multi-orthogonal polynomials Q n (k) in a discrete variable k which have the Krawtchouk and vector-orthogonal polynomials as their building blocks. The algebraic setting allows for the characterization of these polynomials and the computation of mean values in the squeezedcoherent states. In the limit where N goes to infinity and the discrete oscillator approaches the standard harmonic oscillator, the polynomials tend to 2 × 2 matrix orthogonal polynomials and the squeezed-coherent states tend to those of the standard oscillator.

Physical Review D, 1982

We discuss how the Hamiltonian changes in quantum canonical transformations. To the operator M(p, q) one can associate (in a given ordering rule) a c-number function A (p, q). It is this function that appears in the action of the phase-space path integral. A quantum canonical transformation A-+A ' can now be expressed as an integral transformation A (p, q) = dp dqW(p, q;p, q)Pi (p, q). The kernel W is constructed explicitly for point transformations and for the p =q, q =p reflection by studying changes of variables in the path integral. The ordering dependence of W is displayed. The invariance of commutation rules is also discussed.

Entropy

The problem of finding covariance matrices that remain constant in time for arbitrary multi-dimensional quadratic Hamiltonians (including those with time-dependent coefficients) is considered. General solutions are obtained.

Journal of Physics Communications

The main purpose of this work is to identify invariant quadratic operators associated with Linear Canonical Transformations (LCTs) which could play important roles in physics. In quantum physics, LCTs are the linear transformations which keep invariant the Canonical Commutation Relations (CCRs). In this work, LCTs corresponding to a general pseudo-Euclidian space are considered and related to a phase space representation of quantum theory. Explicit calculations are firstly performed for the monodimensional case to identify the corresponding LCT-invariant quadratic operators then multidimensional generalizations of the obtained results are deduced. The eigenstates of these operators are also identified. A first kind of LCT-invariant operator is a second order polynomial of the coordinates and momenta operators. The coefficients of this polynomial depend on the mean values and the statistical variances-covariances of the coordinates and momenta operators themselves. It is shown that t...

Canadian Journal of Physics, 1999

Canonical transformations using the idea of quantum generating functions are applied to construct a quantum Hamilton-Jacobi theory, based on the analogy with the classical case. An operator and a c-number form of the time-dependent quantum Hamilton-Jacobi equation are derived and used to find dynamical solutions of quantum problems. The phase-space picture of quantum mechanics is discussed in connection with the present theory.PACS Nos.: 03.65-w, 03.65Ca, 03.65Ge

Chinese Physics C, 2012

Recently a f-deformed Fock space which is spanned by |n λ has been introduced. These bases are indeed the eigen-states of a deformed non-Hermitian Hamiltonian. In this contribution, we will use a rather new non-orthogonal basis vectors for the construction of coherent and squeezed states, which in special case lead to the earlier known states. For this purpose, we first generalize the previously introduced Fock space spanned by |n λ bases, to a new one, spanned by an extended two-parameters bases |n λ 1 ,λ 2. These bases are now the eigen-states of a non-Hermitian Hamiltonian H λ 1 ,λ 2 = a † λ 1 ,λ 2 a + 1 2 , where a † λ 1 ,λ 2 = a † + λ 1 a + λ 2 and a are respectively, the deformed creation and ordinary bosonic annihilation operators. The bases |n λ 1 ,λ 2 are non-orthogonal (squeezed states), but normalizable. Then, we deduce the new representations of coherent and squeezed states, in our two-parameters Fock space. Finally, we discuss the quantum statistical properties, as well as the non-classical properties of the obtained states, numerically.

Physics Letters A, 2004

A general expression of the Uncertainty Principle is presented. Applying a density matrix formalism, we show that it is an invariant of motion for time-dependent Hamiltonian systems whose dynamics can be contained into an antisymmetric matrix G . The procedure to find the G -matrix is outlined and some examples are briefly shown.

Physical Review A, 2002

We introduce a linear, canonical transformation of the fundamental single-mode field operators a and a † that generalizes the linear Bogoliubov transformation familiar in the construction of the harmonic oscillator squeezed states. This generalization is obtained by adding to the linear transformation a nonlinear function of any of the fundamental quadrature operators X1 and X2, making the original Bogoliubov transformation quadrature-dependent. Remarkably, the conditions of canonicity do not impose any constraint on the form of the nonlinear function, and lead to a set of nontrivial algebraic relations between the c-number coefficients of the transformation. We examine in detail the structure and the properties of the new quantum states defined as eigenvectors of the transformed annihilation operator b. These eigenvectors define a class of multiphoton squeezed states. The structure of the uncertainty products and of the quasiprobability distributions in phase space shows that besides coherence properties, these states exhibit a squeezing and a deformation (cooling) of the phase-space trajectories, both of which strongly depend on the form of the nonlinear function. The presence of the extra nonlinear term in the phase of the wave functions has also relevant consequences on photon statistics and correlation properties. The non quadratic structure of the associated Hamiltonians suggests that these states be generated in connection with multiphoton processes in media with higher nonlinearities.

After introducing Wigner Quantum Systems, we give a short review of the one-dimensional Wigner Quantum Oscillator. Then we define the threedimensional N-particle Wigner Quantum Oscillator, and its relation to the Lie superalgebra sl(1|3N). In this framework (and first for N = 1), energy, coordinates, momentum and the angular momentum of the particles are investigated.

ArXiv, 2021

This work considers uncertainty relations on time frequency distributions from a signal processing viewpoint. An uncertainty relation on the marginalizable time frequency distributions is given. A result from quantum mechanics is used on Wigner distributions and marginalizable time frequency distributions to investigate the change in variance of time and frequency variables from a signal processing perspective. Moreover, operations on signals which leave uncertainty relations unchanged are studied.

Physical Review A, 2010

We address the multiplicity of solutions to the time-energy canonical commutation relation for a given Hamiltonian. Specifically, we consider a particle spatially confined in a potential free interval, where it is known that two distinct self-adjoint and compact time operators conjugate to the system Hamiltonian exist. The dynamics of the eigenvectors of these operators indicate that different time operators posses distinguishing properties that can unambiguously associate them to specific aspects of the quantum time problem.

Annals of Physics, 1980

The ambiguities in the classical canonical transformation leading to action and angle variables of the attractive Coulomb Problem (suitably generalized for positive energies) and the phase space structure they entail are analyzed. The ambiguity group is found to be different for positive and negative energies. Nevertheless it continues to be the essential concept for the construction of the quantum mechanical representation of the classical transformation, which we explicitly obtain. * Dedicated to E. Ruth on his 60th birthday.