Showing papers on "Ladder operator published in 2023"

Generalized $ \left\{ h (1) \oplus h(1) \right\} \uplus u(2) $ commensurate anisotropic Hamiltoninan and ladder operators; energy spectrum, eigenstates and associated coherent and squeezed states

Abstract: In this article a study was made of the conditions under which a Hamiltonian which is an element of the complex $ \left\{ h (1) \oplus h(1) \right\} \uplus u(2) $ Lie algebra admits ladder operators which are also elements of this algebra. The algebra eigenstates of the lowering operator constructed in this way are computed and from them both the energy spectrum and the energy eigenstates of this Hamiltonian are generated in the usual way with the help of the corresponding raising operator. Thus, several families of generalized Hamiltonian systems are found, which, under a suitable similarity transformation, reduce to a basic set of systems, among which we find the 1:1, 2:1, 1:2, $su(2)$ and some other non-commensurate and commensurate anisotropic 2D quantum oscillator systems. Explicit expressions for the normalized eigenstates of the Hamiltonian and its associated lowering operator are given, which show the classical structure of two-mode separable and non-separable generalized coherent and squeezed states. Finally, based on all the above results, a proposal for new ladder operators for the $p:q$ coprime commensurate anisotropic quantum oscillator is made, which leads us to a class of Chen $SU(2)$ coherent states.

A Lanczos approach to the Adiabatic Gauge Potential

Abstract: The Adiabatic Gauge Potential (AGP) is the generator of adiabatic deformations between quantum eigenstates. There are many ways to construct the AGP operator and evaluate the AGP norm. Recently, it was proposed that a Gram-Schmidt-type algorithm can be used to explicitly evaluate the expression of the AGP. We employ a version of this approach by using the Lanczos algorithm to evaluate the AGP operator in terms of Krylov vectors and the AGP norm in terms of the Lanczos coefficients. It has the advantage of minimizing redundancies in evaluating the nested commutators in the analytic expression for the AGP operator. The algorithm is used to explicitly construct the AGP operator for some simple systems. We derive an integral transform relation between the AGP norm and the autocorrelation function of the deformation operator. We present a modification of the Variational approach to derive the regulated AGP norm. Using this, we approximate the AGP to varying degrees of success. Finally, we compare and contrast the quantum-chaos-probing capacities of the AGP and K-complexity in view of the Operator Growth Hypothesis.

The B<sub>2</sub> Harmonic Oscillator with Reflections and Superintegrability

Abstract: The two-dimensional quantum harmonic oscillator is modified with reflection terms associated with the action of the Coxeter group B₂, which is the symmetry group of the square. The angular momentum operator is also modified with reflections. The wavefunctions are known to be built up from Jacobi and Laguerre polynomials. This paper introduces a fourth-order differential-difference operator commuting with the Hamiltonian but not with the angular momentum operator; a specific instance of superintegrability. The action of the operator on the usual orthogonal basis of wavefunctions is explicitly described. The wavefunctions are classified according to the representations of the group: four of degree one and one of degree two. The identity representation encompasses the wavefunctions invariant under the group. The paper begins with a short discussion of the modified Hamiltonians associated to finite reflection groups, and related raising and lowering operators. In particular, the Hamiltonian for the symmetric groups describes the Calogero-Sutherland model of identical particles on the line with harmonic confinement.

Supersymmetric Quantum Mechanics, multiphoton algebras and coherent states

Abstract: The multiphoton algebras for one-dimensional Hamiltonians with infinite discrete spectrum, and for their associated kth-order SUSY partners are studied. In both cases, such an algebra is generated by the multiphoton annihilation and creation operators, as well as by Hamiltonians which are functions of an appropriate number operator. The algebras obtained turn out to be polynomial deformations of the corresponding single-photon algebra previously studied. The Barut-Girardello coherent states, which are eigenstates of the annihilation operator, are obtained and their uncertainty relations are explored by means of the associated quadratures.

Operator techniques and the algebraic solutions of problems

Abstract: Abstract Modern applications of quantum mechanics are illustrated, and the simplifications offered by operator and algebraic techniques are highlighted. The various "pictures" (Schrödinger, Heisenberg, and Interaction or Dirac) are discussed and their origins are motivated. It is shown that operator techniques simplify the harmonic oscillator and angular momentum through the use of ladder operators. The WKB approximation boundary conditions are developed through the use of complex number algebra. Other topics that benefit from these techniques are discussed, including the squeezing of optical fields and various alternative solutions to the hydrogen atom.