Starting from a discrete Heisenberg algebra we solve several representation problems for a discretized quantum oscillator in a weighted sequence space. The Schrödinger operator for a discrete harmonic oscillator is derived. The representation problem for a q‐oscillator algebra is studied in detail. The main result of the article is the fact that the energy representation for the discretized momentum operator can be interpreted as follows: It allows to calculate quantum properties of a large number of non‐interacting harmonic oscillators at the same time. The results can be directly related to current research on squeezed laser states in quantum optics. They reveal and confirm the observation that discrete versions of continuum Schrodinger operators allow more structural freedom than their continuum analogs do.
Starting from a discrete Heisenberg algebra we solve several representation problems for a discretized quantum oscillator in a weighted sequence space. The Schrödinger operator for a discrete harmonic oscillator is derived. The representation problem for a q‐oscillator algebra is studied in detail. The main result of the article is the fact that the energy representation for the discretized momentum operator can be interpreted as follows: It allows to calculate quantum properties of a large number of non‐interacting harmonic oscillators at the same time. The results can be directly related to current research on squeezed laser states in quantum optics. They reveal and confirm the observation that discrete versions of continuum Schrodinger operators allow more structural freedom than their continuum analogs do.
The aim of this paper is to study the position and momentum operators in q-deformed oscillator algebras. The natural form of the position operator is Xp=qpN(a++a)qpN, where p is a real number. This operator is an operator representable by a Jacobi matrix. Using the theory of Jacobi matrices, the theory of classical moment problem and the theory of basic hypergeometric functions, it is shown that, depending on values of q and p, Xp can be unbounded symmetric operator [which has the deficiency indices (1,1) and, hence, is not self-adjoint, but has self-adjoint extensions], bounded self-adjoint operator with continuous simple spectrum or self-adjoint operator of trace class (therefore, with discrete spectrum with zero as the point of accumulation of eigenvalues). The connection of the q-deformed Heisenberg relation PX−qXP=1 for the position and momentum operators with a q-deformation of the quantum harmonic oscillator is also considered.
As a very important example for dynamical symmetries in the context of q‐generalized quantum mechanics the algebra aa† − q−2a†a = 1 is investigated. It represents the oscillator symmetry SUq(1, 1) and is regarded as a commutation phenomenon of the q‐Heisenberg algebra which provides a discrete spectrum of momentum and space, i.e., a discrete Hilbert space structure. Generalized q‐Hermite functions and systems of creation and annihilation operators are derived. The classical limit q → 1 is investigated. Finally the SUq(1, 1) algebra is represented by the dynamical variables of the q‐Heisenberg algebra.
The finite q -oscillator is a model that obeys the dynamics of the harmonic oscillator, with the operators of position, momentum and Hamiltonian being functions of elements of the q -algebra su q (2). The spectrum of position in this discrete system, in a fixed representation j , consists of 2 j + 1 'sensor'-points , and similarly for the momentum observable. The spectrum of energies is finite and equally spaced, so the system supports coherent states. The wavefunctions involve dual q -Kravchuk polynomials, which are solutions to a finite-difference Schrödinger equation. Time evolution (times a phase) defines the fractional Fourier– q -Kravchuk transform. In the classical limit as q → 1 we recover the finite oscillator Lie algebra, the N = 2 j → ∞ limit returns the Macfarlane–Biedenharn q -oscillator and both limits contract the generators to the standard quantum-mechanical harmonic oscillator.
This chapter examines the linear harmonic oscillator in the energy (Fock) representation, emphasizing its algebraic structure and spectral properties. It is shown that the energy spectrum is discrete and bounded from below. The energy raising and lowering (ladder) operators are introduced, and the energy eigenvalues are derived. The eigenstates are constructed algebraically from the ground state by repeated application of the raising operator, forming the Fock basis. These energy eigenstates are then transformed into the coordinate representation, where the corresponding wavefunctions are expressed in terms of Hermite polynomials. This establishes a connection between the algebraic and coordinate formulations of the quantum harmonic oscillator. Metadata/Keywords: linear harmonic oscillator, Fock representation, ladder operators, energy spectrum, energy eigenstates, ground state, Hermite polynomials, raising and lowering operators
The harmonic oscillator is a useful model for the interpretation of many physical phenomena. It is used as a first approximation to the vibration of molecules and crystal lattices and for the quantization of electromagnetic fields, among other applications. One can solve the time-independent Schrodinger equation for the harmonic oscillator in many different ways; operator methods give rise to recurrence relations and generating functions for the stationary states. The coordinate representation renders the coordinate operator diagonal. In this particular representation the eigenvectors of the harmonic oscillator are functions of the coordinate that are commonly expressed in terms of Hermite polynomials. The chapter shows that calculation with the eigenfunctions of the harmonic oscillator is remarkably simple in the realm of the Lie algebras. The application of the bosonic algebra in theoretical physics and chemistry is so vast that it is impossible even to give a reasonable summary. An important application of the bosonic algebra is the modeling of simple quantum-mechanical systems.
Until this point, we have been solving problems that involve just two states or are reducible to two-state problems. While we could examine a plethora of complex quantum phenomena, it is not sufficient to describe the behavior of electrons in atoms or how quantum particles interact with each other. That all changes now as we work to set-up the quantum mechanics of the atom. There is much for us to do before we get to the hydrogen atom. We start, in this chapter, with the simple harmonic oscillator in one dimension. We show how to determine the canonical commutation relation between position and momentum from the fact that atomic spectra have sharp lines and then move into describing free particles and free particles on a circle. We introduce two of our operator identities, the Hadamard lemma and the Baker-Campbell-Hausdorff formula and we revisit exponential disentangling, now for the symplectic group Sp(2). Then we show how to solve the harmonic oscillator energy eigenstates using operator algebra, discuss Heisenberg uncertainty, determine harmonic oscillator wavefunctions, and examine coherent and squeezed states. We end with a discussion of applications of the simple harmonic oscillator.