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

Abstract

Classical and Quantum Mechanics make use of different objects and they also were written on different background spaces. As a consequence, they have followed different paths regarding the mathematical objects that they use. Classical Mechanics makes use of points and functions on a real cotangent space T * Q of a mechanical system, and makes use of differential geometry as the basic language with which the theory is developed. This is due to the existence of trajectories of single points. On the other side, Quantum Mechanics makes use of state vectors in a complex Hilbert space, with operators, commutators, and eigenvectors, and makes use of some postulates that look weird from a classical point of view. This is a point of view that was induced by the lack of trajectories and by the use of probabilistic interpretations of state vectors. There have been efforts to define similar classical and quantum functions that can be compared with each other. Quantum densities were written as functions on phase-space by means of integral transformations. Two of these transformations are the

A Method for Choosing an Initial Time Eigenstate in Classical and Quantum Systems

“…With this choice, we ensure that the initial F eigensurface intersects all of the constant G shells, and this eigensurface is a simple surface that can be generated easily. This method is particularly convenient in quantum mechanics because we now have an easy way to define a zero-time eigenstate and justifies the use of coordinate eigenstates as the initial time eigenstates for other potential functions besides the free particle case [2,3,[5][6][7]. The additional condition to be considered is the use of a coordinate eigenstate at one of the extremal points of the G function so that the time eigenstates have components with all of the energy eigenvalues.…”

“…As we observed in Equation (1), X F · ∇ and G, as well as X G · ∇ and F , are conjugate pairs. Therefore, X F · ∇ and X G · ∇ can be used to translate functions in phase space along the G or F directions as follows: [2][3][4] …”

“…where k, a and ℓ are the parameters of the model [3,8]. The curves generated with the gradient of the Hamiltonian,…”

A Quantum Time Coordinate

“…In that calculation, we take the line q = 0, E ∈ [E n , E x ] as the initial time curve and propagate it in time. The generated set of curves are the time curves, curves for which the time variable is a constant and contain all energy values of interest [34]. We can compare these results with the ones obtained by means of the quantization of the classical time of a free particle.…”