Harmonic oscillator

Abstract: A formalism has been developed, using Feynman's space-time formulation of nonrelativistic quantum mechanics whereby the behavior of a system of interest, which is coupled to other external quantum systems, may be calculated in terms of its own variables only. It is shown that the effect of the external systems in such a formalism can always be included in a general class of functionals (influence functionals) of the coordinates of the system only. The properties of influence functionals for general systems are examined. Then, specific forms of influence functionals representing the effect of definite and random classical forces, linear dissipative systems at finite temperatures, and combinations of these are analyzed in detail. The linear system analysis is first done for perfectly linear systems composed of combinations of harmonic oscillators, loss being introduced by continuous distributions of oscillators. Then approximately linear systems and restrictions necessary for the linear behavior are considered. Influence functionals for all linear systems are shown to have the same form in terms of their classical response functions. In addition, a fluctuation-dissipation theorem is derived relating temperature and dissipation of the linear system to a fluctuating classical potential acting on the system of interest which reduces to the Nyquist–Johnson relation for noise in the case of electric circuits. Sample calculations of transition probabilities for the spontaneous emission of an atom in free space and in a cavity are made. Finally, a theorem is proved showing that within the requirements of linearity all sources of noise or quantum fluctuation introduced by maser-type amplification devices are accounted for by a classical calculation of the characteristics of the maser.

Abstract: The theory of explicitly time‐dependent invariants is developed for quantum systems whose Hamiltonians are explicitly time dependent. The central feature of the discussion is the derivation of a simple relation between eigenstates of such an invariant and solutions of the Schrodinger equation. As a specific well‐posed application of the general theory, the case of a general Hamiltonian which settles into constant operators in the sufficiently remote past and future is treated and, in particular, the transition amplitude connecting any initial state in the remote past to any final state in the remote future is calculated in terms of eigenstates of the invariant. Two special physical systems are treated in detail: an arbitrarily time‐dependent harmonic oscillator and a charged particle moving in the classical, axially symmetric electromagnetic field consisting of an arbitrarily time‐dependent, uniform magnetic field, the associated induced electric field, and the electric field due to an arbitrarily time‐dependent uniform charge distribution. A class of explicitly time‐dependent invariants is derived for both of these systems, and the eigenvalues and eigenstates of the invariants are calculated explicitly by operator methods. The explicit connection between these eigenstates and solutions of the Schrodinger equation is also calculated. The results for the oscillator are used to obtain explicit formulas for the transition amplitude. The usual sudden and adiabatic approximations are deduced as limiting cases of the exact formulas.

Abstract: A new realisation of the quantum group SUq(2) is constructed by means of a q-analogue to the Jordan-Schwinger mapping, determining thereby both the complete representation structure and q-analogues to the Wigner and Racah operators. To achieve this realisation, a new elementary object is defined, a q-analogue to the harmonic oscillator. The uncertainty relation for position and momentum in a q-harmonic oscillator is quite unusual.

Abstract: Synopsis It is shown that the non-Gaussian-Markoff process for Brownian motion derived on a statistical mechanical basis by Prigogine and Balescu, and Prigogine and Philippot, is related through a transformation of variables to the Gaussian-Markoff process of the conventional phenomenological theory of Brownian motion. First the mathematical equivalence of the two types of processes is established by expressing the well-known formulae and equations for the random process {Vx(t), Vy(t)} which describe the motion of a charged Brownian particle in two-dimensional space under the influence of a magnetic field (Vx and Vy are the components of the velocity), in terms of the new variables ∈ = 1 2 m v x 2 + 1 2 m v y 2 and α = arccos { v x ( 2 ∈ / m ) − 1 2 } . The transformed process is called the e(t), a(t) process. It is then shown that the phenomenological theory of the Brownian motion of a strongly underdamped linear harmonic oscillator, if expressed in action-angle variables, leads under well specified conditions to the same e(t), a(t) process, i.e. to the process obtained by Prigogine e.a. in their statistical theory of irreversible processes (in which action-angle variables are used) for a system of weakly coupled harmonic oscillators.

Abstract: This paper proposes to broaden the canonical formulation of quantum mechanics. Ordinarily, one imposes the condition H†=H on the Hamiltonian, where † represents the mathematical operation of complex conjugation and matrix transposition. This conventional Hermiticity condition is sufficient to ensure that the Hamiltonian H has a real spectrum. However, replacing this mathematical condition by the weaker and more physical requirement H‡=H, where ‡ represents combined parity reflection and time reversal PT, one obtains new classes of complex Hamiltonians whose spectra are still real and positive. This generalization of Hermiticity is investigated using a complex deformation H=p2+x2(ix)e of the harmonic oscillator Hamiltonian, where e is a real parameter. The system exhibits two phases: When e⩾0, the energy spectrum of H is real and positive as a consequence of PT symmetry. However, when −1