Correspondence, Time, Energy, Uncertainty, Tunnelling, and Collapse of Probability Densities

Part of the book: Theoretical Concepts of Quantum Mechanics

Classical and Quantum Conjugate Dynamics – The Interplay Between Conjugate Variables

Part of the book: Advances in Quantum Mechanics

Emergence of Classical Distributions from Quantum Distributions: The Continuous Energy Spectra Case

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

Exact Finite Differences for Quantum Mechanics

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

Matrices Which are Discrete Versions of Linear Operations

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

Three Solutions to the Nonlinear Schrödinger Equation for a Constant Potential

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

The Inverse of the Discrete Momentum Operator

In the search of a quantum momentum operator with discrete spectrum, we obtain some properties of the discrete momentum operator for nonequally spaced spectrum. We find the inverse operator. We use the matrix representation of these operators, and we find that there is one more eigenvalue and eigenfunction than the dimension of the matrix. We apply the results to obtain the discrete adjoint of the momentum operator. We conclude that we can have discrete operators which can be self-adjoint and that it is possible to define a self-adjoint extension of the corresponding Hilbert space. These results help us understand the quantum time operator.

Part of the book: Schrödinger Equation

Matrices with a Diagonal Commutator

View all chaptersIt is well known that there are no two matrices with a diagonal commutator. However, the commutator can behave as if it is diagonal when acting on a particular vector. We discuss pairs of matrices that give rise to a diagonal commutator when applied to a given arbitrary vector. Some properties of these matrices are discussed. These matrices have additional, continuous eigenvalues and eigenvectors than the dimension of the matrix, and their inverse also has this property. Some of these matrices are discrete approximations of the derivative and integration of a function and are exact for the exponential function. We also determine the adjoint of the obtained discrete derivative.

Part of the book: Nonlinear Systems and Matrix Analysis - Recent Advances in theory and Applications [Working title]