Canonical transforms. III. Configuration and phase descriptions of quantum systems possessing an sl (2,R) dynamical algebra (original) (raw)
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Journal of Physics A: Mathematical and General, 1999
The algebra of generalized linear quantum canonical transformations is examined in the perspective of Schwinger's unitary-canonical operator basis. Formulation of the quantum phase problem within the theory of quantum canonical transformations and in particular with the generalized quantum action-angle phase space formalism is established and it is shown that the conceptual foundation of the quantum phase problem lies within the algebraic properties of the canonical transformations in the quantum phase space. The representations of the Wigner function in the generalized action-angle unitary operator pair for certain Hamiltonian systems with dynamical symmetry is examined. This generalized canonical formalism is applied to the quantum harmonic oscillator to examine the properties of the unitary quantum phase operator as well as the action-angle Wigner function.
Journal of Physics A: Mathematical and General, 2002
Following the discussion -in state space language -presented in a preceding paper, we work on the passage from the phase space description of a degree of freedom described by a finite number of states (without classical counterpart) to one described by an infinite (and continuously labeled) number of states. With that it is possible to relate an original Schwinger idea to the Pegg and Barnett approach to the phase problem. In phase space language, this discussion shows that one can obtain the Weyl-Wigner formalism, for both Cartesian and angular coordinates, as limiting elements of the discrete phase space formalism. PACS: 03.65.-w, 03.65.Bz, 03.65.Ca
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Fizika B, 1996
Starting with a real abstract algebra which encapsulates the algebraic structures of both classical and quantum mechanics, this paper presents a self-contained realization of the latter in phase space. Having both mechanics formulated in the same space opens two new windows into the comparative study of foundations. This stems from the fact that the same physical problem, as defined by a given Hamiltonian, can be solved in several independent ways. The exact solutions can then be compared. Thus, comparison of the classical and quantum solutions in phase space offers new epistemological insights into Bohr's correspondence principle, while comparison of the quantum solutions in the different formalisms of Hilbert space and phase space yields new physical insights. These general ideas are then tested on the harmonic oscillator. The analytic ground work is presented in Part I, the exact solutions will be derived in Part II.
On generalized phase operators for the quantum harmonic oscillator
Il Nuovo Cimento B Series 10, 1970
Following the demonstration by Lerner that the phase (cosine and sine) operators for a harmonic oscillator can be much more general than the specific set considered earlier in the literature, we make a systematic study of the properties of such generalized phase operators. We determine their eigenstates and show that the coefficients of transformation from the number eigenstates to these are, quite generally, orthogonal polynomials in the eigenvalue parameter. Using these generalized C and S operators the number-phase minimum uncertainty state is given. Finally we point out that though the commutator [C, S] cannot be made to vanish, nevertheless one can construct C and S in such a way that their commutator has a vanishing expectation value with respect to all coherent states having a given mean occupation number exceeding unity. 1.-Introduction.
Who is afraid of nonhermitian operators? A quantum description of angle and phase
Annals of Physics, 1976
The physical characteristics of a quantum property consist essentially of its eigenvalues and eigenstates. As a consequence, hermitian operators are shown to define an unduly restricted framework for the theoretical description of quantum properties; nonhermitian operators, for instance unitary, but also nonnormal ones, may be acceptable as well if the projectors onto their eigenstates allow for a resolution of the identity operator, so as to preserve the probabilistic interpretation of the Hilbert space formalism. This extension of the conventional rules permits a simple and natural solution of the long-standing difficulties in achieving a consistent quantum description of the angular coordinate for the plane rotator and the phase variable for the harmonic oscillator. A unitary operator for the angle, a one-sided unitary (nonnormal) operator for the phase, exist which yield all the necessary physical information. Valid commutation relations hold with the angular momentum in the first case, with the number operator in the second one. They lead to new Heisenberg inequalities with a straightforward interpretation, in agreement with intuitive expectations. The relevance of the proposed description for the analysis of phase measurements performed on complex physical systems, and the possibility of experimental tests, are discussed.
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arXiv (Cornell University), 2002
A consistent realization of the quantum operators corresponding to the canonically conjugate phase and number variables is proposed, resorting to the κ = 1 2 positive discrete series of the irreducible unitary representation of the Lie algebra su(1, 1) of the double covering group of SO ↑ (1, 2).
A finite-dimensional representation of the quantum angular momentum operator
Arxiv preprint quant-ph/0008120, 2000
A useful finite-dimensional matrix representation of the derivative of periodic functions is obtained by using some elementary facts of trigonometric interpolation. This N × N matrix becomes a projection of the angular derivative into polynomial subspaces of finite dimension and it can be interpreted as a generator of discrete rotations associated to the z-component of the projection of the angular momentum operator in such subspaces, inheriting thus some properties of the continuum operator. The group associated to these discrete rotations is the cyclic group of order N. Since the square of the quantum angular momentum L 2 is associated to a partial differential boundary value problem in the angular variables θ and ϕ whose solution is given in terms of the spherical harmonics, we can project such a differential equation to obtain an eigenvalue matrix problem of finite dimension by extending to several variables a projection technique for solving numerically two point boundary value problems and using the matrix representation of the angular derivative found before. The eigenvalues of the matrix representing L 2 are found to have the exact form n(n + 1), counting the degeneracy, and the eigenvectors are found to coincide exactly with the corresponding spherical harmonics evaluated at a certain set of points.