arXiv subjects
Publications and source records attributed to Rafael Sala Mayato.
We give a modified Hamiltonian for a particle in a box with infinite potential walls that takes into account wall effects. The Hamiltonian is expressed in both the position and momentum representation. In the momentum representation the eigenvalue problem for energy is an integral equation.
Probability densities that are not uniquely determined by their moments are said to be "moment-indeterminate", or "M-indeterminate". Determining whether or not a density is M-indeterminate, or how to generate an M-indeterminate density, is a challenging problem with a long history. Quantum mechanics is inherently probabilistic, yet the way in which probability densities are obtained is dramatically different in comparison to standard probability theory, involving complex wave functions and operators, among other aspects. Nevertheless, the end results are standard probabilistic quantities, such as expectation values, moments and probability density functions. We show that the quantum mechanics procedure to obtain densities leads to a simple method to generate an infinite number of M-indeterminate densities. Different self-adjoint operators can lead to new classes of M-indeterminate densities. Depending on the operator, the method can produce densities that are of the Stieltjes class or new formulations that are not of the Stieltjes class. As such, the method complements and extends existing approaches and opens up new avenues for further development. The method applies to continuous and discrete probability densities. A number of examples are given.
We consider the non-overlapping wave function paradox of Aharanov \textit{et al.}, wherein the relative phase between two wave functions cannot be measured by the moments of position or momentum. We show that there is an unlimited number of other expectation values that depend on the phase. We further show that the Wigner distribution is M-indeterminate, that is, a distribution whose moments do not uniquely determine the distribution. We generalize to more than two non-overlapping functions. We consider arbitrary representations and show there is an unlimited number of M-indeterminate distributions. The dual case of non-overlapping momentum functions is also considered.
We demonstrate that it is possible to discern the way that has been followed to measure a quantum observable that can be expressed in terms of different products of observables, whereas no such discrimination is possible by assigning predetermined values. Specifically we show how to distinguish different routes (contexts) to measure C=AB=A'B', when C,A,B and C,A',B' commute with each other, but A and B do not commute with A' and B'.