The expectation value definitions over an extended space from the considered Hilbert space of the system under consideration is given in another paper of the second author in this symposium. There, in that paper, the conceptuality rather than specification is emphasized on. This work uses that conceptuality to investigate the time evolutions of the position related operators' expectation values not in its standard meaning but rather in a new version of the definition over not the original Hilbert space but in the space obtained by extensions via introducing the images of the given initial wave packet under the positive integer powers of the system Hamiltonian. These images may not be residing in the same space of the initial wave packet when certain singularities appear in the structure of the system Hamiltonian. This may break down the existence of the integrals in the definitions of the expectation values. The cure is the use of basis functions in the abovementioned extended space and the sandwiching of the target operator whose expectation value is under questioning by an appropriately chosen operator guaranteeing the existence of the relevant integrals. Work specifically focuses on the hydrogen-like quantum systems whose Hamiltonians contain a polar singularity at the origin.
The expectation value definitions over an extended space from the considered Hilbert space of the system under consideration is given in another paper of the second author in this symposium. There, in that paper, the conceptuality rather than specification is emphasized on. This work uses that conceptuality to investigate the time evolutions of the position related operators' expectation values not in its standard meaning but rather in a new version of the definition over not the original Hilbert space but in the space obtained by extensions via introducing the images of the given initial wave packet under the positive integer powers of the system Hamiltonian. These images may not be residing in the same space of the initial wave packet when certain singularities appear in the structure of the system Hamiltonian. This may break down the existence of the integrals in the definitions of the expectation values. The cure is the use of basis functions in the abovementioned extended space and the sandwiching of the target operator whose expectation value is under questioning by an appropriately chosen operator guaranteeing the existence of the relevant integrals. Work specifically focuses on the hydrogen-like quantum systems whose Hamiltonians contain a polar singularity at the origin.
The time variant power series expansion for the expectation value of a given quantum dynamical operator is well-known and well-investigated issue in quantum dynamics. However, depending on the operator and Hamiltonian singularities this expansion either may not exist or may not converge for all time instances except the beginning of the evolution. This work focuses on this issue and seeks certain cures for the negativities. We work in the extended space obtained by adding all images of the initial wave function under the system Hamiltonian’s positive integer powers. This requires the introduction of certain appropriately defined weight operators. The resulting better convergence in the temporal power series urges us to call the new defined entities “extended space expectation values” even though they are constructed over certain weight oparetors and are somehow pseudo expectation values.
Recent studies on quantum evolutionary problems in Demiralp’s group have arrived at a stage where the construction of an expectation value formula for a given algebraic function operator depending on only position operator becomes possible. It has also been shown that this formula turns into an algebraic recursion amongst some finite number of consecutive elements in a set of expectation values of an appropriately chosen basis set over the natural number powers of the position operator as long as the function under consideration and the system Hamiltonian are both autonomous. This recursion corresponds to a denumerable infinite number of algebraic equations whose solutions can or can not be obtained analytically. This idea is not completely original. There are many recursive relations amongst the expectation values of the natural number powers of position operator. However, those recursions may not be always efficient to get the system energy values and especially the eigenstate wavefunctions. The present approach is somehow improved and generalized form of those expansions. We focus on this issue for a specific system where the Hamiltonian is defined on the coordinate of a curved space instead of the Cartesian one.
We have recently shown that the operator expectation values for the quantum dynamical systems of singular Hamiltonians, cannot be expanded into temporal Maclaurin series. Depending on the nature of the system under consideration and the relevant operator, either a finite number of terms can exist while the others become infinite or the series can convergence only at the beginning of the evolution even if the coefficients of the series all exist. This however happens when the initial wave packet has some features leading us to nonintegrabilities in certain expectation values. The reason why this happens underlies beneath the fact that the certain images of the initial wave packet under certain natural number powers of the Hamiltonian become nonintegrable. This negativity can be bypassed by using certain weight functions in the expectation value definition and corresponds to working on an extended space constructed from the Hilbert space in which the initial and any time wave function lay. We have used a two parameter weight function in our most recent publication. Here, in this work, we extend the weight function to multiparameter structures where the number of parameters is more than two and may climb even up to infinity.
This proceedings paper aims to show the efficiency of an expectation value identity for a given algebraic function operator which is assumed to be depending pn only position operator. We show that this expectation value formula becomes enabled to determine the eigenstates of the quantum system Hamiltonian as long as it is autonomous and an appropriate basis set in position operator is used. This approach produces a denumerable infinite recursion which may be considered as revisited but at the same time generalized form of the recursions over the natural number powers of the position operator. The content of this short paper is devoted not only to the formulation of the new method but also to show that this novel approach is capable of catching the eigenvalues and eigenfunctions for Hydrogen-like systems, beyond that, it can give a hand to us to reveal the wavefunction structure. So it has also somehow a confirmative nature.
For a quantum system with Hilbert space H of dimension N and a set S of n Hermitian operators Oi, a basic question is to understand the set ES⊂Rn of points e where ei=tr(ρOi) for an allowed state ρ. A related question is to determine whether a given set of expectation values e lies in ES and in this case to describe the most general state with these expectation values. In this paper, we describe various ways to characterize ES, reviewing basic results that are perhaps not widely known and adding new ones. One important result (originally due to E. Wichmann) is that for a set S of linearly independent traceless operators, every set of expectation values e in the interior of ES is achieved uniquely by a state of the form ρ(β)=e−∑iβiOi/tr(e−∑iβiOi) for Oi∈S. In fact, the map β→E(β)=tr(Oρ(β)) is a diffeomorphism from Rn to the interior of ES with symmetric, positive Jacobian; using this fact, we provide an algorithm to invert E(β) and thus determine a state ρ(β(e)) with specified expectation values e provided that these lie in ES. The algorithm is based on defining a first order differential equation in the space of parameters β that is guaranteed to converge to β(e) in a precise way, with ∣E(β(t))−e∣=Ce−t.
This work is a new extension to our a very recent work whose paper will appear in the proceedings of a very recent international conference. What we have done in the previous work is the use of a weight operator to suppress the singularities causing nonexistence of some of temporal Maclaurin expansion coefficients. The weight operator has been constructed in such a way that certain number of expectation values of position operator’s first positive integer powers with and without the chosen weight operator match. Therein this match has not been considered for the momentum operator’s corresponding power expectation values and a finite linear combination of the spatial variable’s first reciprocal powers has been used in the construction of the weight operator. Here, that approach is extended to the case where matches for both position and momentum operators are considered and the weight operator involves finite linear combinations of the spatial variable’s both positive integer powers and their reciprocals.