Bicomplex hamiltonian systems in quantum mechanics

Bicomplex Quantum Mechanics: II. The Hilbert Space

Using the bicomplex numbers T which is a commutative ring with zero divisors defined by T = {w0+w1i1+w2i2+ w3j | w0, w1, w2, w3 ∈ R} where i21 = −1, i22 = −1, j = 1, i1i2 = j = i2i1, we construct hyperbolic and bicomplex Hilbert spaces. Linear functionals and dual spaces are considered on these spaces and properties of linear operators are obtain; in particular it is established that the eigenv...

Hamiltonian Formalism in Quantum Mechanics

To Vladimir Igorevich Arnol'd with admiration, on occasion of his 60 th birthday. Abstract Heisenberg motion equations in Quantum mechanics can be put into the Hamilton form. The difference between the commutator and its principal part, the Poisson bracket, can be accounted for exactly. Canonical transformations in Quantum mechanics are not, or at least not what they appear to be; their propert...

Bicomplex quantum mechanics: I. The generalized Schrödinger equation

We introduce the set of bicomplex numbers T which is a commutative ring with zero divisors defined by T = {w0 + w1i1 + w2i2 + w3j| w0, w1, w2, w3 ∈ R} where i21 = −1, i22 = −1, j = 1, i1i2 = j = i2i1. We present the conjugates and the moduli associated with the bicomplex numbers. Then we study the bicomplex Schrödinger equation and found the continuity equations. The discrete symmetries of the ...

Minimal length in Quantum Mechanics and non-Hermitian Hamiltonian systems

Non-Hamiltonian commutators in quantum mechanics.

The symplectic structure of quantum commutators is first unveiled and then exploited to describe generalized non-Hamiltonian brackets in quantum mechanics. It is easily recognized that quantum-classical systems are described by a particular realization of such a bracket. In light of previous work, this paper explains a unified approach to classical and quantum-classical non-Hamiltonian dynamics...