The Schrödinger equation on time dependent quantum graphs

2025-05-06 0 0 741.78KB 17 页 10玖币

侵权投诉

TIME EVOLUTION AND THE SCHRÖDINGER EQUATION ON

TIME DEPENDENT QUANTUM GRAPHS

UZY SMILANSKY AND GILAD SOFER

Dedicated to Michael Berry on his forthcoming anniversary.

Abstract. The purpose of the present paper is to discuss the time dependent Schrödinger

equation on a metric graph with time-dependent edge lengths, and the proper way to

pose the problem so that the corresponding time evolution is unitary. We show that

the well posedness of the Schrödinger equation can be guaranteed by replacing the

standard Kirchhoﬀ Laplacian with a magnetic Schrödinger operator with a harmonic

potential. We then generalize the result to time dependent families of vertex condi-

tions. We also apply the theory to show the existence of a geometric phase associated

with a slowly changing quantum graph.

1. Introduction

In the present work, we study the time evolution on quantum graphs whose edge

lengths vary in time. A stationary quantum graph consists of a set of vertices con-

nected by edges of prescribed lengths. On each edge, the Schrödinger operator is the

one-dimensional Laplacian. The graph is supplemented with appropriate boundary

conditions at the vertices, which ensure that the operator is self-adjoint and so the

resulting time evolution is unitary. Our goal is to generalize this model by allowing the

edge lengths to be time dependent.

To illustrate the possible diﬃculties this time dependence causes, we start by consid-

ering the simplest example. Namely, a graph that consists of two vertices connected by

an edge of length L(t/T ). Here, T−1measures the rate of change of the edge lengths.

The evolution for this system is dictated by the time-dependent Schrödinger equation:

iℏ∂ψ (x, t)

∂t =−ℏ2

∂2ψ(x, t)

∂x2, x ∈−L(t/T )

2,L(t/T )

2.(1.1)

At the two vertices we impose the standard Neumann boundary condition:

∂ψ (x, t)

∂x = 0 at |x|=L(t/T )

2,∀t≥0.(1.2)

The operator on the right hand side of (1.1) involves only derivatives with respect to

x. The time dependence is due to the boundary conditions, which are imposed at the

Uzy Smilansky, Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot

7610001, Israel. Email: uzy.smilansky@weizmann.ac.il

Gilad Sofer, Faculty of Mathematics, Technion - Israel Institute of Technology, Haifa 32000, Israel.

Email: gilad.sofer@campus.technion.ac.il

arXiv:2210.14652v2 [math-ph] 19 Jan 2024

2 UZY SMILANSKY AND GILAD SOFER

time dependent boundary. If one considers the time tas a parameter, the operator on

the right is referred to as the instantaneous Hamiltonian.

Given some initial condition ψ(x, 0), we let the system evolve according to (1.1). A

straightforward computation then gives the following expression for the time derivative

of the squared L2norm of ψ(x, t):

dt∥ψ∥2=d

dtZL(t/T )

−L(t/T )

2|ψ(x, t)|2dx(1.3)

2T·dL(t/T )

dt"

ψL(t/T )

2, t

+

ψ−L(t/T )

2, t

2#,

This rather unexpected change of the L2norm in time is due to the choice of Neumann

boundary condition, which causes the additional term RL(t/T )

−L(t/T )

dt|ψ(x, t)|2dxto vanish

after integrating by parts.

Recalling the standard interpretation of |ψ(x, t)|2dx as the probability for ﬁnding the

particle in the interval (x, x +dx)at time t, the above result implies that there exists

a probability ﬂux which equals to the product of the density at the boundary and its

speed. Hence, the probability to ﬁnd the particle within the boundaries changes in

time. This implies that the evolution dictated by (1.1) is not unitary. In this context,

it is worth noting:

1. The instantaneous L2norm is constant in the adiabatic limit T→ ∞.

2. For the Dirichlet boundary condition (ψ(x, t)=0), the instantaneous L2norm

is constant. Equation (1.3) shows that the change in norm arises also for the Robin

boundary condition:

(1.4) ∂ψ

∂x (x, t) = Cψ (x, t) at |x|=L(t/T )

2, C ∈R.

In Section 2 we return to this problem, and derive the boundary conditions which

render the time evolution unitary, ensuring probability conservation. We follow previous

studies which treat time dependent domains in diﬀerent contexts. The time dependent

Schrödinger equation on domains in Rnwith time dependent boundary was studied in

[12, 19, 13]. The propagation of electromagnetic waves in vibrating cavities is a closely

related problem studied in [18]. The present work can be considered as an extension of

these studies to quantum graphs. We shall note that the time dependent Schrödinger

equation on a star graph was studied in [17]. However, the vertex boundary conditions

considered there correspond to the Dirichlet condition, and thus do not address the

problem discussed above.

Once the problem is properly reformulated in Section 2, the eﬀect of the moving

boundary is expressed in terms of a gauge ﬁeld which, when included, renders the

Schrödinger operator self-adjoint (Theorem 2.1). This is analogous to the derivation

for slowly varying time dependent domains presented in [13].

In Section 3, we generalize the result for graphs where the vertex conditions depend on

time as well (Theorem 3.4). Moreover, we apply these results by studying the geometric

phase associated with the time evolution in the adiabatic limit. Finally, in Appendix

THE SCHRÖDINGER EQUATION ON TIME DEPENDENT QUANTUM GRAPHS 3

A we derive the secular equation whose zeros give the instantaneous spectrum at each

time.

2. Quantum graphs with time dependent edge lengths and standard

boundary conditions

This section consists of three parts, in which the time dependent Schrödinger equation

for quantum graphs with time dependent edge lengths is formulated and discussed. The

ﬁrst subsection addresses the problem in the simple case of a single interval. It serves

as a primer for the treatment of a general graph which is presented in the second

subsection. In the third subsection we demonstrate the theory by studying the time

evolution on a time dependent equilateral quantum graph.

2.1. A time dependent interval. Before proceeding further, it is worthwhile to

rewrite the Schrödinger equation (1.1) in dimensionless form. Observing that ℏ

2mhas

the dimension of length2

time , then scaling xby Land tby Tresults in the dimensionless

quantity L2

T·ℏ

2m. Using the standard convention in which ℏ

2mtakes the numerical value

1, we remain with a dimensionless L2

T·ℏ

2m. We then get the following Schrödinger

equation:

i∂ψ (x, t)

∂t =−∂2ψ(x, t)

∂x2, x ∈−1

2L(t/T ),1

2L(t/T ).(2.1)

Following [12, 19, 13], we wish to write the Schrödinger equation in terms of the

scaled variables

(2.2) τ=t

T, ξ =x

L(t/T ).

To do this, we introduce the unitary transformation:

ω(ξ, τ ) := L(τ)1/2ψ(L(τ)ξ, T τ),(2.3)

where now τ∈[0,1] and ξ∈−1

2,1

2. The Schrödinger equation (2.1) can be rewritten

in the new coordinates as

∂

∂τ ω(ξ, τ )(2.4)

L(τ)2

− ∂

∂ξ −iξ ˙

L(τ)L(τ)

2T!2

−ξ2 ˙

L(τ)L(τ)

2T!2

ω(ξ, τ ).

In the new Schrödinger equation, the Schrödinger operator includes a repelling har-

monic potential, and the Laplacian appears as a magnetic Laplacian with A(ξ, τ) =

ξ˙

L(τ)L(τ)

2Tplaying the role of a vector potential. This suggests that one can render the

magnetic operator above self-adjoint by replacing the original Neumann condition with

the magnetic Neumann condition:

(2.5) ∂

∂ξ −iξ ˙

2T!ω(ξ, τ ) = 0 at |ξ|=1

4 UZY SMILANSKY AND GILAD SOFER

This indeed gives a self-adjoint operator on L2−1

2,1

2. Using (2.5) and (2.3), we obtain

the corresponding boundary condition for ψ(x, t):

(2.6) ∂ψ (x, t)

∂x =±i˙

4Tψ(x, t) at x=±1

2L(t/T ).

Repeating the computation in (1.3) with the new boundary condition immediately

shows that the resulting time evolution is unitary.

The following alternative explanation for the magnetic boundary condition was sug-

gested by Michael Berry [10]. The movement of the edges induces a nontrivial proba-

bility current at the boundary. One may account for this probability current by intro-

ducing a modiﬁed Robin boundary condition:

(2.7) ∂ψ (x, t)

∂x =Cψ (x, t) at |x|=1

2L(t/T ), C ∈C.

While the usual Robin condition corresponds to C∈R, a straightforward computation

shows that the requirement d

dt∥ψ∥2= 0 gives the condition Im (C) = ˙

4T. We may thus

solve the problem by suggesting the boundary condition (2.6). The associated Robin

parameter may be considered as an eﬀective magnetic ﬁeld.

The Schrödinger equation and boundary conditions in (2.4), (2.5) can be further

simpliﬁed by introducing the gauge transformation

(2.8) ω(ξ, τ ) = ei

2θ(τ)(ξ2−1/4)g(ξ, τ ).

One may then eliminate the magnetic Laplacian in Equation (2.4) if the phase is chosen

to be θ(τ) = ˙

2T. Note that in the units used in this work, θis dimensionless.

The Schrödinger equation is now given by

∂g (ξ, τ )

∂τ =(2.9)

−1

L(τ)2·∂2g(ξ, τ )

∂ξ2+¨

4T2ξ2g(ξ, τ )−¨

LL (τ) + ˙

L2(τ)

16T2g(ξ, τ ),|ξ| ≤ 1

and subject to standard Neumann boundary conditions.

A similar equation was presented in [17, 19]. We derived it here as a prelude to the

treatment of the time dependent Schrödinger operator on a graph. Note that the factor

Tmultiplies the τderivative, implying that the adiabatic limit T→ ∞ is the analogue

of the semi-classical limit.

2.2. Compact metric graphs with time dependent edge lengths. Consider now

the metric graph Γ = (V,E,L), where Vis the vertex set and Eis the edge set (both

assumed ﬁnite). Moreover, L: [0, T ]→R|E| is a family of edge lengths parameterized as

{Le(t/T )}e∈E with Le(t/T )positive and twice continuously diﬀerentiable. A coordinate

xewith xe∈−1

2Le(t/T ),1

2Le(t/T )is assigned to every edge in E. We denote the set

of edges connected to the vertex vby Svand the degree of the vertex by dv:= |Sv|.

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：

TIMEEVOLUTIONANDTHESCHRÖDINGEREQUATIONONTIMEDEPENDENTQUANTUMGRAPHSUZYSMILANSKYANDGILADSOFERDedicatedtoMichaelBerryonhisforthcominganniversary.Abstract.ThepurposeofthepresentpaperistodiscussthetimedependentSchrödingerequationonametricgraphwithtime-dependentedgelengths,andtheproperwaytoposetheproblems...

展开>> 收起<<

The Schrödinger equation on time dependent quantum graphs.pdf

共17页,预览4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！

The Schrödinger equation on time dependent quantum graphs

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: