Difference ladder operators for a harmonic Schrödinger oscillator using unitary linear lattices

Relativistic Ladder Operators for the Three-Dimensional Harmonic Oscillator

The quantum constraint equations for a relativistic three-dimensional harmonic oscillator are shown to find concise expression in terms of Lorentz covariant ladder operators. These ladder operators consist of two conjugate 4-vectors that are each constrained to generate three linearly independent combinations of ladder operator components for raising and lowering the eigenstates of the oscillat...

Trajectory of the harmonic oscillator in the Schrödinger wave

A trajectory of a harmonic oscillator obeying the Schrödinger wave equation is exactly derived and illustrated. The trajectory resembles well the classical orbit between the turning points, and also runs through the tunneling region. The dynamics of the ‘particle’ motion and the wave function associated with the motion are proposed. The period of a round trip on the trajectory is exactly equal ...

Eigenvalue Problem for Schrödinger Operators and Time-Dependent Harmonic Oscillator

It is shown that the eigenvalue problem for the Hamiltonians of the standard form, H = p/(2m) + V (x), is equivalent to the classical dynamical equation for certain harmonic oscillators with time-dependent frequency. This is another indication of the central role played by time-dependent harmonic oscillators in quantum mechanics. The utility of the known results for eigenvalue problem in the so...

Darboux Transformations for Linear Operators on Two Dimensional Regular Lattices

Darboux transformations for linear operators on regular two dimensional lattices are reviewed. The six point scheme is considered as the master linear problem, whose various specifications, reductions, and their sublattice combinations lead to other linear operators together with the corresponding Darboux transformations. The second part of the review deals with multidimensional aspects of (bas...

Harmonic Oscillator