expectation values
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2022 ◽  
Vol 8 (1) ◽  
pp. 252-256
Author(s):  
Aulia Riski Pratikha ◽  
Bambang Supriadi ◽  
Rif’ati Dina Handayani

The purpose of this study is to determine the electron’s position expectation values and energy spectrum on the Li2+ ion on the principal quantum number n≤3. This research using literature study methods on quantum mechanics. The expectation values of the electron position and the energy spectrum of the Li2+ ion uses numerical calculations using the Matlab 2019a program. The steps in this research method include: preparation; theory development; simulation; validation of the results of theory development; results of theory development; discussion and conclusion. The results obtained in this study are the electron’s position expectation values and energy of the Lithium ion. The electron’s position expectation values indicates the presence of electrons that often appear around the x-axis by relying on the interval used. The larger the interval, the more constant the electron’s position expectation values will be and towards an almost constant value. From the analysis results, the expectation value varies in positions from  0,0001a0 to 0,1637a0. The electron energy spectrum of the Li2+ ion is inversely proportional to the square of the principal quantum number (n),E1= -122,4 eV ; E2= -30,6 eV ; E3= -13,6 eV


Quantum ◽  
2022 ◽  
Vol 6 ◽  
pp. 617
Author(s):  
David Plankensteiner ◽  
Christoph Hotter ◽  
Helmut Ritsch

A full quantum mechanical treatment of open quantum systems via a Master equation is often limited by the size of the underlying Hilbert space. As an alternative, the dynamics can also be formulated in terms of systems of coupled differential equations for operators in the Heisenberg picture. This typically leads to an infinite hierarchy of equations for products of operators. A well-established approach to truncate this infinite set at the level of expectation values is to neglect quantum correlations of high order. This is systematically realized with a so-called cumulant expansion, which decomposes expectation values of operator products into products of a given lower order, leading to a closed set of equations. Here we present an open-source framework that fully automizes this approach: first, the equations of motion of operators up to a desired order are derived symbolically using predefined canonical commutation relations. Next, the resulting equations for the expectation values are expanded employing the cumulant expansion approach, where moments up to a chosen order specified by the user are included. Finally, a numerical solution can be directly obtained from the symbolic equations. After reviewing the theory we present the framework and showcase its usefulness in a few example problems.


2022 ◽  
Vol 2022 (1) ◽  
Author(s):  
Horacio Casini ◽  
Javier M. Magán ◽  
Pedro J. Martínez

Abstract The entropic order parameters measure in a universal geometric way the statistics of non-local operators responsible for generalized symmetries. In this article, we compute entropic order parameters in weakly coupled gauge theories. To perform this computation, the natural route of evaluating expectation values of physical (smeared) non-local operators is prevented by known difficulties in constructing suitable smeared Wilson loops. We circumvent this problem by studying the smeared non-local class operators in the enlarged non-gauge invariant Hilbert space. This provides a generic approach for smeared operators in gauge theories and explicit formulas at weak coupling. In this approach, the Wilson and ’t Hooft loops are labeled by the full weight and co-weight lattices respectively. We study generic Lie groups and discuss couplings with matter fields. Smeared magnetic operators, as opposed to the usual infinitely thin ones, have expectation values that approach one at weak coupling. The corresponding entropic order parameter saturates to its maximum topological value, except for an exponentially small correction, which we compute. On the other hand, smeared ’t Hooft loops and their entropic disorder parameter are exponentially small. We verify that both behaviors match the certainty relation for the relative entropies. In particular, we find upper and lower bounds (that differ by a factor of 2) for the exact coefficient of the linear perimeter law for thin loops at weak coupling. This coefficient is unphysical/non-universal for line operators. We end with some comments regarding the RG flows of entropic parameters through perturbative beta functions.


Entropy ◽  
2021 ◽  
Vol 23 (12) ◽  
pp. 1652
Author(s):  
Margret Westerkamp ◽  
Igor Ovchinnikov ◽  
Philipp Frank ◽  
Torsten Enßlin

Knowledge on evolving physical fields is of paramount importance in science, technology, and economics. Dynamical field inference (DFI) addresses the problem of reconstructing a stochastically-driven, dynamically-evolving field from finite data. It relies on information field theory (IFT), the information theory for fields. Here, the relations of DFI, IFT, and the recently developed supersymmetric theory of stochastics (STS) are established in a pedagogical discussion. In IFT, field expectation values can be calculated from the partition function of the full space-time inference problem. The partition function of the inference problem invokes a functional Dirac function to guarantee the dynamics, as well as a field-dependent functional determinant, to establish proper normalization, both impeding the necessary evaluation of the path integral over all field configurations. STS replaces these problematic expressions via the introduction of fermionic ghost and bosonic Lagrange fields, respectively. The action of these fields has a supersymmetry, which means there exists an exchange operation between bosons and fermions that leaves the system invariant. In contrast to this, measurements of the dynamical fields do not adhere to this supersymmetry. The supersymmetry can also be broken spontaneously, in which case the system evolves chaotically. This affects the predictability of the system and thereby makes DFI more challenging. We investigate the interplay of measurement constraints with the non-linear chaotic dynamics of a simplified, illustrative system with the help of Feynman diagrams and show that the Fermionic corrections are essential to obtain the correct posterior statistics over system trajectories.


Author(s):  
Alessandro Pesci

In this paper, we consider a specific model, implementing the existence of a fundamental limit distance [Formula: see text] between (space or time separated) points in spacetime, which in the recent past has exhibited the intriguing feature of having a minimum-length Ricci scalar [Formula: see text] that does not approach the ordinary Ricci scalar [Formula: see text] in the limit of vanishing [Formula: see text]. [Formula: see text] at a point has been found to depend on the direction along which the existence of minimum distance is implemented. Here, we point out that the convergence [Formula: see text] in the [Formula: see text] limit is anyway recovered in a relaxed or generalized sense, which is when we average over directions, this suggesting we might be taking the expectation value of [Formula: see text] promoted to be a quantum variable. It remains as intriguing as before the fact that we cannot identify (meaning this is much more than simply equating in the generalized sense above) [Formula: see text] with [Formula: see text] in the [Formula: see text] limit, namely, when we get ordinary spacetime. Thing is like if, even when [Formula: see text] (read here the Planck length) is far too small to have any direct detection of it feasible, the intrinsic quantum nature of spacetime might anyway be experimentally at reach, witnessed by the mentioned special feature of Ricci, not fading away with [Formula: see text] (i.e. persisting when taking the [Formula: see text] limit).


2021 ◽  
Vol 2021 (12) ◽  
Author(s):  
Alex Kehagias ◽  
Hervé Partouche ◽  
Nicolaos Toumbas

Abstract We determine the inner product on the Hilbert space of wavefunctions of the universe by imposing the Hermiticity of the quantum Hamiltonian in the context of the minisuperspace model. The corresponding quantum probability density reproduces successfully the classical probability distribution in the ħ → 0 limit, for closed universes filled with a perfect fluid of index w. When −1/3 < w ≤ 1, the wavefunction is normalizable and the quantum probability density becomes vanishingly small at the big bang/big crunch singularities, at least at the semiclassical level. Quantum expectation values of physical geometrical quantities, which diverge classically at the singularities, are shown to be finite.


2021 ◽  
Vol 2021 (12) ◽  
Author(s):  
Masahide Manabe ◽  
Seiji Terashima ◽  
Yuji Terashima

Abstract We construct 3D $$ \mathcal{N} $$ N = 2 abelian gauge theories on $$ \mathbbm{S} $$ S 2 × $$ \mathbbm{S} $$ S 1 labeled by knot diagrams whose K-theoretic vortex partition functions, each of which is a building block of twisted indices, give the colored Jones polynomials of knots in $$ \mathbbm{S} $$ S 3. The colored Jones polynomials are obtained as the Wilson loop expectation values along knots in SU(2) Chern-Simons gauge theories on $$ \mathbbm{S} $$ S 3, and then our construction provides an explicit correspondence between 3D $$ \mathcal{N} $$ N = 2 abelian gauge theories and 3D SU(2) Chern-Simons gauge theories. We verify, in particular, the applicability of our constructions to a class of tangle diagrams of 2-bridge knots with certain specific twists.


2021 ◽  
pp. 1-2
Author(s):  
T. Amano

Jensen (Can. J. Phys. 98, 506 (2020). doi: 10.1139/cjp-2019-0395 ) presents theoretical justification for the claim that linear triatomic molecules are necessarily observed to be bent. The basis of the assertion is that the expectation value of the supplement of the bending angle, [Formula: see text] used in Jensen’s paper, is calculated to be positive. In this comment, we examine the interpretation of the expectation values of [Formula: see text] in stationary states, and indicate that Jensen’s claim contradicts a basic principle of quantum mechanics that the energy and geometrical variables cannot have definite values at the same time.


Author(s):  
Feng Zhang ◽  
Zhuo Ye ◽  
Yong-Xin Yao ◽  
Cai-Zhuang Wang ◽  
Kai-Ming Ho

Abstract We present a random-sampling (RS) method for evaluating expectation values of physical quantities using the variational approach. We demonstrate that the RS method is computationally more efficient than the variational Monte Carlo method using the Gutzwiller wavefunctions applied on single-band Hubbard models as an example. Non-local constraints can also been easily implemented in the current scheme that capture the essential physics in the limit of strong on-site repulsion. In addition, we extend the RS method to study the antiferromagnetic states with multiple variational parameters for 1D and 2D Hubbard models.


Author(s):  
H. Itoyama ◽  
Katsuya Yano

The lowest critical point of one unitary matrix model with cosine plus logarithmic potential is known to correspond with the [Formula: see text] Argyres–Douglas (AD) theory and its double scaling limit derives the Painlevé II equation with parameter. Here, we consider the critical points associated with all cosine potentials and determine the scaling operators, their vacuum expectation values (vevs) and their scaling dimensions from perturbed string equations at planar level. These dimensions agree with those of [Formula: see text] AD theory.


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