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American Journal of Physics

Error Limits of Expectation Values in Quantum Mechanical Calculations

Joseph F. Mucci, Peter J. Haskins

Published Aug 1, 1973• 2 citations
Publisher

Abstract

We present a general treatment of error limits of expectation values in quantum-mechanical calculations. Important expressions and their derivations are included along with an indication of the state of research in this area.
DOI
10.1119/1.1987447

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Usually, expected values for various physical quantities, such as the number of electrons occupying certain states or the Coulomb interaction between different states of electrons, can be expressed in terms of integrals. In contrast, our method, based on differential forms, shows that expected values can be obtained by averaging over time. To confirm the validity of our method, we prepare the two cases: one is a very simple case with no many-body interaction, and the other is the case where the many-body term is included (the simplest Anderson Hamiltonian). Regarding the simple case without inclusion of many-body term, we prove analytically that the number of electrons occupying any state derived from our method is equivalent to the analytical one evaluated from the Greenʼs function method. When the many-body term is included, our results show good numerical agreement with the analytical ones derived from the Greenʼs function method. By the two cases, the calculation of expected values based on our method is considered valid.
Fractional calculus is becoming increasingly important nowadays in studying and understanding fundamental physical phenomena, both simple and complex, through the formulation of generalized models. In the present work, a treatment of non-integer order is developed for the study of the wave function that describes the accessible states of a one-dimensional particle confined in a finite well with finite and constant walls, using a conformable derivative in describing the temporal evolution of the states of the system and Caputo-Fabrizio fractional derivative in describing its spatial evolution. By plotting the complete fractional solution ψ(x,t) for given values ​​of the Kernel indices α, β, γ, we recover the flat-wave front oscillations predicted by the integer-order calculation, whose superposition represents the particle state with uniform probability in all space. However, for another range of values ​​of the indices of the integral transformation, damping effects are observed, which do not seem to have a quantum mechanical physical meaning in themselves, but they do have a special physical-mathematical interest since such effects are related to the nature of the Caputo-Fabrizio operator, which as a non-local operator saves memory, that is, non-local spatial effects entail long-range interactions in a physical region, so the generalized wave function obtained does not depend exclusively on a point in space but on a region of space. When calculating the probability density, the fractional wave function shows a definite parity when γ is unitary, just as expected for a symmetric potential well. The generalized model obtained by incorporating a greater number of free parameters can represent an appropriate alternative formulation in the description of the probabilistic behavior of simple and complex quantum mechanical systems. Palabras clave: Ecuación de Schrödinger fraccional; Derivada fraccionaria de Caputo, derivada fraccionaria de Caputo-Fabrizio, derivada general conformable de Khalil.