Part of the book: Theoretical Concepts of Quantum Mechanics
Part of the book: Advances in Quantum Mechanics
We explore the properties of quantum states and operators that are conjugate to the Hamiltonian eigenstates and operator when the Hamiltonian spectrum is continuous, i.e., we find time-like operators T^ such that [T^,H^]=iℏ. This is a property expected for a time operator. We explicitly unfold the momentum sign degeneracy of energy states. We consider the free-particle case, and we find, among other things, that the time states are also the solution of the quantized version of the classical motion of the particle.
Part of the book: Dynamical Systems
We introduce a finite difference derivative, on a non-uniform partition, with the characteristic that the derivative of the exponential function is the exponential function itself, times a constant, which is similar to what happens in the continuous variable case. Aside from its application to perform numerical computations, this is particularly useful in defining a quantum mechanical discrete momentum operator.
Part of the book: Numerical Simulations in Engineering and Science
We introduce and study a matrix which has the exponential function as one of its eigenvectors. We realize that this matrix represents a set of finite differences derivation of vectors on a partition. This matrix leads to new expressions for finite differences derivatives which are exact for the exponential function. We find some properties of this matrix, the induced derivatives and of its inverse. We provide an expression for the derivative of a product, of a ratio, of the inverse of vectors, and we also find the equivalent of the summation by parts theorem of continuous functions. This matrix could be of interest to discrete quantum mechanics theory.
Part of the book: Matrix Theory
We introduce three sets of solutions to the nonlinear Schrödinger equation for the free particle case. A well-known solution is written in terms of Jacobi elliptic functions, which are the nonlinear versions of the trigonometric functions sin, cos, tan, cot, sec, and csc. The nonlinear versions of the other related functions like the real and complex exponential functions and the linear combinations of them is the subject of this chapter. We also illustrate the use of these functions in Quantum Mechanics as well as in nonlinear optics.
Part of the book: Nonlinear Optics