1 INTRODUCTION Quantum cylindrical integrability in magnetic ﬁelds O. Kub uand L. Šnobl

2025-04-28 0 0 187.83KB 8 页 10玖币

侵权投诉

1 INTRODUCTION

Quantum cylindrical integrability in magnetic ﬁelds

O. Kub˚

u?and L. Šnobl

Czech Technical University in Prague, Faculty of Nuclear Sciences

and Physical Engineering, Prague, Czech Republic

* ondrej.kubu@fjﬁ.cvut.cz

October 10, 2022

34th International Colloquium on Group Theoretical Methods in Physics

Strasbourg, 18-22 July 2022

doi:10.21468/SciPostPhysProc.?

Abstract

We present the classiﬁcation of quadratically integrable systems of the cylindrical type with

magnetic ﬁelds in quantum mechanics. Following the direct method used in classical me-

chanics by [F Fournier et al 2020 J. Phys. A: Math. Theor. 53 085203]to facilitate the

comparison, the cases which may a priori differ yield 2 systems without any correction and

2 with it. In all of them the magnetic ﬁeld Bcoincides with the classical one, only the scalar

potential Wmay contain a ħh2-dependent correction. Two of the systems have both cylindri-

cal integrals quadratic in momenta and are therefore not separable. These results form a

basis for a prospective study of superintegrability.

1 Introduction

This article is a contribution to the study of integrable and superintegrable Hamiltonian systems

with magnetic ﬁelds on the 3D Euclidean space E3in quantum mechanics. More speciﬁcally, we

assume a Hamiltonian of the form (using units where e=−1, m=1)

H=1

2~p2+Aj(~x)pj+pjAj(~x) + Aj(~x)2+W(~x), (1)

with implicit summation over repeated indices j=1,2,3 (in the whole paper), ~p=−iħh~

∇is the

momentum operator and ~

A= (A1(~x),A2(~x),A3(~x)) and W(~x)are the vector and scalar potentials

of the electromagnetic ﬁeld.

Integrability then entails the existence of two algebraically independent integrals of motion

X1,X2(further speciﬁed below) mutually in involution, i.e.

[H,X1]=[H,X2]=[X1,X2] = 0. (2)

They are usually considered to be polynomials in the momenta pj, for computational feasibility

usually of a low order (typically 2).

Integrable (and especially superintegrable) systems are rare and distinguished by the possi-

bility to obtain the solution to their equations of motion in a closed form. They are subsequently

arXiv:2210.03468v1 [quant-ph] 7 Oct 2022

2 CYLINDRICAL–TYPE SYSTEM

invaluable for gaining physical intuition and serve as a starting point for modelling more compli-

cated systems. Finding and classifying these systems is therefore of utmost importance.

The case without the vector potential ~

Ahas been widely studied. The quadratic integrable sys-

tems were classiﬁed in 1960s and the 1:1 correspondence with orthogonal separation of variables

of the Schrödinger (or, in classical context, the Hamilton-Jacobi) equation was found [1–3]. This

leads to the 11 classes of scalar potentials Vstudied by Eisenhart [4]. Higher order superintegra-

bility followed, see e.g. [5]and references therein.

Despite its physical relevance, integrability with magnetic ﬁelds was mostly ignored due to

its computational difﬁculty. The ﬁrst systematic result remedying this omission was the article by

Shapovalov on separable systems [6], followed by the articles in E2[7,8]. Subsequent articles

in E3assumed ﬁrst order integrals [9]or separation of variables [10–13]. Marchesiello et al. [9]

found a quadratic superintegrable system with an integral not connected to separation of variables,

which was recently followed up by [14,15].

Here we present in an abridged form the classiﬁcation of quadratically integrable systems of

the cylindrical type (see (9)) in quantum mechanics obtained in O. Kub˚

u’s Master thesis [16],

which closely followed Fournier et al.’s [13]classical analysis to highlight the differences arising

in quantum mechanics.

In Section 2we introduce the differential form formalism for magnetic ﬁelds in cylindrical

coordinates, derive the determining equations for cylindrical–type integrals and reduce them to

a simpler form. The calculations separate into several cases depending on the rank of the matrix

in equation (15). In the case that may a priori differ from the classical one from [13]only ranks

2 and 1 are relevant. We present the corresponding results in Sections 3and 4, respectively. We

draw our conclusions in Section 5.

2 Cylindrical–type system

Before we specify the corresponding integrals X1,X2, we have to introduce the formalism used for

magnetic ﬁeld in curvilinear coordinates in classical mechanics, cf. [13,17].

Deﬁning the cylindrical coordinates

x=rcos(φ),y=rsin(φ),z=Z, (3)

we represent the vector potential Aas a 1-form

A=Axdx+Aydy+Azdz=Ardr+Aφdφ+AZdZ. (4)

Hence, we obtain the following transformations

Ax=cos(φ)Ar−sin(φ)

rAφ,Ay=sin(φ)Ar+cos(φ)

rAφ,Az=AZ. (5)

As a part of the canonical 1-form λ=pjdxJ, the momenta pjtransform in the same way and we

can deﬁne the covariant momenta by pA

j=pj+Ajin both Cartesian and cylindrical coordinates.

Components of the magnetic ﬁeld 2-form B=dAare

B=Bx(~x)dy∧dz+By(~x)dz∧dx+Bz(~x)dx∧dy

=Br(r,φ,Z)dφ∧dZ+Bφ(r,φ,Z)dZ∧dr+BZ(r,φ,Z)dr∧dφ,(6)

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

1INTRODUCTIONQuantumcylindricalintegrabilityinmagneticeldsO.Kubu?andL.noblCzechTechnicalUniversityinPrague,FacultyofNuclearSciencesandPhysicalEngineering,Prague,CzechRepublic*ondrej.kubu@fj.cvut.czOctober10,202234thInternationalColloquiumonGroupTheoreticalMethodsinPhysicsStrasbourg,18-22July2022...

展开>> 收起<<

1 INTRODUCTION Quantum cylindrical integrability in magnetic ﬁelds O. Kub uand L. Šnobl.pdf

共8页,预览2页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

1 INTRODUCTION Quantum cylindrical integrability in magnetic ﬁelds O. Kub uand L. Šnobl

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: