# The Gross-Pitaevskii Model of Spinor BEC

Authors: Dongming Yan; Tian Ma
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

The Gross-Pitaevskii model of spinor Bose-Einstein condensates
is studied. Using the abstract results obtained for infinite-dimensional
Hamilton system, we establish the mathematical theory for the model of
spinor BEC. Furthermore, three conservative quantities of spinor BEC, that is, the energy, total particle number, and magnetization intensity, are also proved.


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## Body

## 1. Introduction

After the first remarkable experiments concerning the observation of Bose-Einstein condensate (BEC) in dilute gases of alkali atoms such asRb87 [1], Na23 [2], and Li7 [3] the interest in this phenomenon has revived [4, 5]. On the mathematical side, most of the work has concentrated on the Gross-Pitaevskii (GP) model of BEC, which is usually referred to as nonlinear Schödinger equation (NLSE) (cf. [6–14] and references therein). There are also many pieces of the literature on the spinor BEC ([15–19]). In the spinor BEC case, the constituent bosons have internal degrees of freedom, such as spin, the quantum state, and its properties becomes more complex [20]. What has made the alkali spinor BEC particularly interesting is that optical and magnetic fields can be used to probe and manipulate the system.In [15], Ho shows that in an optical trap the ground states of spin-1 bosons such as Na23, K39, and Rb87 can be either ferromagnetic or polar states, depending on the scattering lengths in different angular momentum channels. In [17], Pu et al. discuss the energy eigenstates, ground and spin mixing dynamics of a spin-1 spinor BEC for a dilute atomic vapor confined in an optical trap. Their results go beyond the mean field picture and are developed within a fully quantized framework. In [19], Zou and Mathis propose a three-step scheme for generating the maximally entangled atomic Greenberger-Horne-Zeilinger (GHZ) states in a spinor BEC by using strong classical laser fields to shift atom level and drive single-atom Raman transition. Their scheme can be directly used to generate the maximally entangled states between atoms with hyperfine spin 0 and 1.In this paper, we want to establish the mathematical theory of the GP model of spinor BEC in which the internal degrees of freedom of atoms are also under consideration. Furthermore, three conservative quantities of spinor BEC, that is, the energy, total particle number, and magnetization intensity are also proved.


## 2. Gross-Pitaevskii Model for Spinor BEC

In this section we derive GP equation for spinor BEC, that is, the following equation:(2.1)iℏ∂ψ1∂t=-ℏ22mΔψ1+V(x)ψ1+gn|Ψ|2ψ1+gsψ-1*ψ02+gs(|ψ1|2+|ψ0|2-|ψ-1|2)ψ1,iℏ∂ψ0∂t=-ℏ22mΔψ0+V(x)ψ0+gn|Ψ|2ψ0+2gsψ1ψ-1ψ0*+gs(|ψ1|2+|ψ-1|2)ψ0,iℏ∂ψ-1∂t=-ℏ22mΔψ-1+V(x)ψ-1+gn|Ψ|2ψ-1+gsψ1*ψ02+gs(|ψ-1|2+|ψ0|2-|ψ1|2)ψ-1.We consider the GP model forF=1 spinor BEC. Particles of F=1 have three quantum states: magnetic quantum number m=1,0,-1. The corresponding wave function of these three quantum states are denote by(2.2)Ψ=(ψ1,ψ0,ψ-1).
Here the physical meaning of |ψi|2 is the density of m=i particles (i=1,0,-1). The corresponding Hamilton energy functional of F=1 spinor BEC is as follows:(2.3)E(Ψ,Ψ*)=∫Ω[ℏ22m|∇Ψ|2+V(x)|Ψ|2+12gn|Ψ|4+12gs|Ψ*SΨ|2]dx,
where m is the boson mass, ℏ is Planck constant, V(·) is the external trapping potential, |Ψ|2 is the density of dilute bosonic atoms, gn is the interaction constant between atoms, and gs is the spin exchange interaction constant(2.4)gn=4πℏ2ma0+2a23,gs=4πℏ2ma2-a03,a0 and a2 are the scattering length, and S=Sxi⃗+Syj⃗+Szk⃗ is the spin operator:(2.5)Sx=12(010101010),Sy=12(0-i0i0-i0i0),Sz=(10000000-1).
We adopt the following notations:(2.6)Ψ*SΨ=(ψ1*,ψ0*,ψ-1*)[Sxi⃗+Syj⃗+Szk⃗](ψ1ψ0ψ-1),|Ψ*SΨ|2=|Ψ*SxΨ|2+|Ψ*SyΨ|2+|Ψ*SzΨ|2=12|ψ1*ψ0+ψ0*(ψ1+ψ-1)+ψ-1*ψ0|2+12|-ψ1*ψ0+ψ0*(ψ1-ψ-1)+ψ-1*ψ0|2+(|ψ1|2-|ψ-1|2)2.
By calculation we can get(2.7)|Ψ*SΨ|2=|ψ1|4+|ψ-1|4-2|ψ1|2|ψ-1|2+2|ψ0|2|ψ1|2+2|ψ0|2|ψ-1|2+2ψ02ψ1*ψ-1*+2ψ0*2ψ1ψ-1.By using Lagrange multiplier theorem, from the Hamilton energy functionalE (see (2.3) and the total particle number(2.8)N=∫Ω|Ψ|2dx,Ψ=(ψ1,ψ0,ψ-1)
is conservative, and we can obtain the steady state GP equation of spinor BEC as follows:(2.9)μψk=δδψk*E(Ψ,Ψ*),k=1,0,-1,
where μ is the chemical potential. Furthermore, according to general rules of quantum mechanics from steady state GP equation, we can get the dynamical model as follows:(2.10)iℏ∂ψk∂t=δδψk*E(Ψ,Ψ*),k=1,0,-1,
where i=-1. From ((2.3) and (2.7), we can obtain the concrete expression of (2.10) as (2.1).In the spinor BECgn and gs we have following physical meaning:(2.11)gn{>0,correspondingtotherepulsiveinteractionbetweenatoms,<0,correspondingtotheattractiveinteractionbetweenatoms,gs{>0,correspondingtotheantiferromagneticstates,<0,correspondingtotheferromagneticstates.


## 3. Equivalent Form of Spinor BEC

Letψk=ψk1+iψk2. In this section we will show that GP equation (2.1) is equivalent to the following quantum Hamilton systems (see [21]):(3.1)∂ψk1∂t=δF(ψ1,ψ2)δψk2,∂ψk2∂t=-δF(ψ1,ψ2)δψk1,k=1,0,-1,
where ψ1=(ψ11,ψ01,ψ-11),ψ2=(ψ12,ψ02,ψ-12), and the energy functional defined as(3.2)F(ψ1,ψ2)=12ℏE(Ψ,Ψ*),(Esee(2.3)).In fact, on the one hand, splitting real and imaginary parts of (2.1), we can obtain
(3.3)ℏ∂ψ11∂t=(-ℏ22mΔ+V(x)+gn|Ψ|2+gs(|ψ1|2+|ψ0|2-|ψ-1|2))ψ12+gs(2ψ-11ψ01ψ02-ψ-12(ψ01)2+ψ-12(ψ02)2),ℏ∂ψ12∂t=(ℏ22mΔ-V(x)-gn|Ψ|2-gs(|ψ1|2+|ψ0|2-|ψ-1|2))ψ11-gs(2ψ-12ψ01ψ02+ψ-11(ψ01)2-ψ-11(ψ02)2),ℏ∂ψ01∂t=(-ℏ22mΔ+V(x)+gn|Ψ|2+gs(|ψ1|2+|ψ-1|2))ψ02+2gs(ψ11ψ-12ψ01+ψ-11ψ12ψ01-ψ11ψ-11ψ02+ψ12ψ-12ψ02),ℏ∂ψ02∂t=(ℏ22mΔ-V(x)-gn|Ψ|2-gs(|ψ1|2+|ψ-1|2))ψ01-2gs(ψ11ψ-11ψ01-ψ-12ψ12ψ01+ψ11ψ-12ψ02+ψ12ψ-11ψ02),ℏ∂ψ-11∂t=(-ℏ22mΔ+V(x)+gn|Ψ|2+gs(|ψ-1|2+|ψ0|2-|ψ1|2))ψ-12+gs(2ψ11ψ01ψ02-ψ12(ψ01)2+ψ12(ψ02)2),ℏ∂ψ-12∂t=(ℏ22mΔ-V(x)-gn|Ψ|2-gs(|ψ-1|2+|ψ0|2-|ψ1|2))ψ-11-gs(2ψ12ψ01ψ02+ψ11(ψ01)2-ψ11(ψ02)2).
On the other hand, it is easy to check that
(3.4)δF(ψ1,ψ2)δψ12=1ℏ[(-ℏ22mΔ+V(x)+gn|Ψ|2+gs(|ψ1|2+|ψ0|2-|ψ-1|2))ψ12+gs(2ψ-11ψ01ψ02-ψ-12(ψ01)2+ψ-12(ψ02)2)(-ℏ22mΔ+V(x)+gn|Ψ|2+gs(|ψ1|2+|ψ0|2-|ψ-1|2))],δF(ψ1,ψ2)δψ11=1ℏ[(-ℏ22mΔ+V(x)+gn|Ψ|2+gs(|ψ1|2+|ψ0|2-|ψ-1|2))ψ11+gs(2ψ-12ψ01ψ02+ψ-11(ψ01)2-ψ-11(ψ02)2)(-ℏ22mΔ+V(x)+gn|Ψ|2+gs(|ψ1|2+|ψ0|2-|ψ-1|2))],δF(ψ1,ψ2)δψ02=1ℏ[(-ℏ22mΔ+V(x)+gn|Ψ|2+gs(|ψ1|2+|ψ-1|2))ψ02+2gs(ψ11ψ-12ψ01+ψ-11ψ12ψ01-ψ11ψ-11ψ02+ψ12ψ-12ψ02)(-ℏ22mΔ+V(x)+gn|Ψ|2+gs(|ψ1|2+|ψ-1|2))],δF(ψ1,ψ2)δψ01=1ℏ[(-ℏ22mΔ+V(x)+gn|Ψ|2+gs(|ψ1|2+|ψ-1|2))ψ01+2gs(ψ11ψ-11ψ01-ψ-12ψ12ψ01+ψ11ψ-12ψ02+ψ12ψ-11ψ02)(-ℏ22mΔ+V(x)+gn|Ψ|2+gs(|ψ1|2+|ψ-1|2))],δF(ψ1,ψ2)δψ-12=1ℏ[(-ℏ22mΔ+V(x)+gn|Ψ|2+gs(|ψ-1|2+|ψ0|2-|ψ1|2))ψ-12+gs(2ψ11ψ01ψ02-ψ12(ψ01)2+ψ12(ψ02)2)(-ℏ22mΔ+V(x)+gn|Ψ|2+gs(|ψ-1|2+|ψ0|2-|ψ1|2))],δF(ψ1,ψ2)δψ-11=1ℏ[(-ℏ22mΔ+V(x)+gn|Ψ|2+gs(|ψ-1|2+|ψ0|2-|ψ1|2))ψ-11+gs(2ψ12ψ01ψ02+ψ11(ψ01)2-ψ11(ψ02)2)(-ℏ22mΔ+V(x)+gn|Ψ|2+gs(|ψ-1|2+|ψ0|2-|ψ1|2))].
Consequently, GP equation (2.1) is equivalent to (3.1).


## 4. Infinite Dimensional Hamilton System

In this section, we consider the following infinite-dimensional Hamilton system(4.1)dudt=-DvF(u,v),dvdt=DuF(u,v),u(0)=φ,v(0)=ψ,((u,v)∈X1×X2)
where X⊂Xi⊂H(i=1,2) is dense, X is linear space, X1,X2 is reflexive Banach space, H is Hilbert space, F:X1×X2→R1  is C1 functional, and DF=(DuF,DvF) is derived operator.Remark 4.1.
Infinite-dimensional Hamilton system (4.1) not only has some kind of beauty in its own form, but also many equations can be written as (4.1). For example, Schödinger equation, Weyl equations, and Dirac equations can be written as (4.1). Hence, it is worth to study the infinite-dimensional Hamilton system (4.1), also see [21].Definition 4.2.
One says(u,v)∈X1×X2 is a weak solution of Hamilton system(4.1), provided(4.2)〈u,ũ〉H+〈v,ṽ〉H=∫0t[〈DuF(u,v),ṽ〉-〈DvF(u,v),ũ〉]dt+〈φ,ũ〉H+〈ψ,ṽ〉H,
for every ũ∈X2,ṽ∈X1.
LetF:X1×X2→R1 satisfy
(4.3)||(u,v)||X1×X2⟶∞⟺F(u,v)⟶∞(or-F(u,v)⟶∞).
Then we have the following existence theorem.Theorem 4.3 (see [21]).
Assume thatF satisfies condition (4.3) and  DF:X1×X2→(X1×X2)* is weakly continuous, then for any  (φ,ψ)∈X1×X2, there exists one global weak solution of equation (4.1)
(4.4)(u,v)∈L∞((0,∞),X1×X2).
Furthermore, F(u,v) is a conservative quantity for weak solution (u,v), that is,
(4.5)F(u(t),v(t))=F(φ,ψ),∀t>0.Proof.
We prove the existence of global solution for (4.1) in L∞((0,∞),X1×X2) by standard Galerkin method. Choose
(4.6){ek∣k=1,2,…}⊂X
as orthonormal basis of space H. Set Xn,X̃n as follows:
(4.7)Xn={∑k=1nαkek∣αk∈R1,1≤k≤n},X̃n={∑k=1nβk(t)ek∣βk(⋅)∈C1[0,∞),1≤k≤n}.
Consider the ordinary equations as follows:(4.8)dxk(t)dt=〈-DvF(un,vn),ek〉,dyk(t)dt=〈DuF(un,vn),ek〉,xk(0)=〈φ,ek〉H,yk(0)=〈ψ,ek〉H,(k=1,…,n),
where un=∑k=1nxk(t)ek,vn=∑k=1nyk(t)ek.
By the theory of ordinary equations, there exists only one local solution of (4.8):
(4.9){x1(t),y1(t),…,xn(t),yn(t)},0≤t≤τ.
From (4.8) we can obtain the equality
(4.10)〈un,ũn〉H+〈vn,ṽn〉H=∫0t[〈DuF(un,vn),ṽn〉-〈DvF(un,vn),ũn〉]dt+〈φ,ũn〉H+〈ψ,ṽn〉H
holds true for any ũn,ṽn∈Xn. Moreover, equality
(4.11)∫0t[〈dundt,ũn〉H+〈dvndt,ṽn〉H]dt=∫0t[〈DuF(un,vn),ṽn〉-〈DvF(un,vn),ũn〉]dt
holds true for any ũn,ṽn∈X̃n.
Putting(ũn,ṽn)=(-dvn/dt,dun/dt) in (4.11), we obtain that
(4.12)0=∫0t[〈DuF(un,vn),dundt〉+〈DvF(un,vn),dvndt〉]dt=∫0tddtF(un,vn)dt,
which implies
(4.13)F(un,vn)=F(φn,ψn),
where
(4.14)φn=∑k=1n〈φ,ek〉Hek,ψn=∑k=1n〈ψ,ek〉Hek.
From (4.3) and (4.10), we deduce that {(un,vn)}n=1∞ is bounded in L∞((0,∞),X1×X2). Therefore there exists a subsequence; we still write it as {(un,vn)}n=1∞, such that
(4.15)(un,vn)⇀(u,v)inX1×X2,a.e.t∈(0,∞).
According toDF:X1×X2→(X1×X2)* being weakly continuous and (4.10), (4.15), we know the following equality
(4.16)〈u,ũ〉H+〈v,ṽ〉H=∫0t[〈DuF(u,v),ṽ〉-〈DvF(u,v),ũ〉]dt+〈φ,ũ〉H+〈ψ,ṽ〉H
holds true for any ũ,ṽ∈⋃n=1∞Xn. Since ⋃n=1∞Xn is dense in X1 and X2, equality (4.16) holds true for all (ũ,ṽ)∈X1×X2, which implies that (u,v)∈L∞((0,∞),X1×X2) is a global weak solution of (4.1).
Next, we proveF(u,v) is a conservative quantity for weak solution (u,v). From (4.16), for all h>0 we have
(4.17)〈u(t+h)-u(t),ũ〉H+〈v(t+h)-v(t),ṽ〉H=∫tt+h[〈DuF(u(τ),v(τ)),ṽ〉-〈DvF(u(τ),v(τ)),ũ〉]dτ.
Putting
(4.18)ũ=-Δhv=-(v(t+h)-v(t)),ṽ=Δhu=u(t+h)-u(t)
in (4.17), we obtain that
(4.19)0=1h∫tt+h[〈DuF(u(τ),v(τ)),Δhu〉+〈DvF(u(τ),v(τ)),Δhv〉]dτ=F(u(t+h),v(t+h))-F(u(t),v(t)).
Therefore, F(u,v) is a conservative quantity for weak solution (u,v). The proof is completed.Theorem 4.4 (see [21]).
LetX1,X2 be Hilbert space and  F:X1×X2→R  be C1 functional. Then a  C1 functional  G:X1×X2→R is a conservative quantity for the infinite-dimensional Hamilton system (4.1) if and only if the following equality
(4.20)〈δG(u,v)δu,δF(u,v)δv〉X1×X2=〈δG(u,v)δv,δF(u,v)δu〉X1×X2
holds true for any (u,v)∈X1×X2.Proof.
Let(u,v) be a solution of (4.1). Then we have
(4.21)ddtG(u,v)=〈δG(u,v)δu,dudt〉X1×X2+〈δG(u,v)δv,dvdt〉X1×X2=-〈δG(u,v)δu,δF(u,v)δv〉+〈δG(u,v)δv,δF(u,v)δu〉,
which imply that (d/dt)G(u,v)=0 if and only if equality (4.20) holds true. The proof is completed.


## 5. The Existence of Global Solution of Spinor BEC

In this section we consider the Gross-Pitaevskii equation of spinor BEC (2.10) under the Dirichlet boundary condition, to wit the following initial boundary problem:(5.1)iℏ∂Ψ∂t=δδΨ*E(Ψ,Ψ*),x∈Ω,Ψ|∂Ω=0,Ψ(x,0)=Ψ0(x),
where Ω⊂Rn(1≤n≤3) is a domain. When Ω=Rn, then (5.1) become Cauchy problem. By applying Theorem 4.3, we can obtain the following theorem.Theorem 5.1.
Assume thatV∈L2(Ω) and  gn>max{0,-2gs}, then for any  Ψ0∈H1(Ω,𝒞3), there exists one global weak solution of problem (5.1)
(5.2)Ψ∈C0([0,∞),L2(Ω,C3))∩L∞((0,∞),H1(Ω,C3)).Remark 5.2.
Ifgs=0, then (5.1) reduce to the GP equation of BEC. Theorem 5.1 is also consistent with the experiments in repulsive case. In the situation of repulsive interaction, solutions to the GP equation of BEC are well defined for all times [12, 13, 20], which corresponds to the emergence of the BEC.Proof.
LetH=L2(Ω,R6),H1=H1(Ω,R6). Firstly, we need to verify condition (4.3) in Theorem 4.3. From Section 2, we know that
(5.3)F(ψ1,ψ2)=12ℏ∫Ω[ℏ22m|∇Ψ|2+V(x)|Ψ|2+12gn|Ψ|4+12gs|Ψ*SΨ|2]dx,
where Ψ=(ψ1,ψ0,ψ-1),ψk=ψk1+iψk2(k=1,0,-1), and
(5.4)|Ψ*SΨ|2=|ψ1|4+|ψ-1|4-2|ψ1|2|ψ-1|2+2|ψ0|2|ψ1|2+2|ψ0|2|ψ-1|2+2ψ02ψ1*ψ-1*+2ψ0*2ψ1ψ-1≤2(|ψ1|4+|ψ0|4+|ψ-1|4).
Hence, when gn>max{0,-2gs}, we have
(5.5)∫Ωgn|Ψ|4+gs|Ψ*SΨ|2dx≥λ∫Ω|Ψ|4dx,
where λ=gn-max{0,-2gs}>0. Therefore, we deduce
(5.6)F(ψ1,ψ2)⟶∞⟺‖(ψ1,ψ2)‖H1⟶∞,
which implies that condition (4.3) holds true.
Next we need to verify the continuous condition in Theorem4.3. Let operator DF:H1→H1 be defined by
(5.7)〈DF(Ψ),Ψ̃〉=1ℏ∫Ω[ℏ22m∇Ψ⋅∇Ψ̃+V(x)Ψ⋅Ψ̃+gn|Ψ|2Ψ⋅Ψ̃+gs(Ψ*SΨ)Ψ̃*SΨ̃[ℏ22m∇Ψ⋅∇Ψ̃+V(x)Ψ⋅Ψ̃]dx.
For any Ψ̃∈C0∞(Ω,R6) and Ψn⇀Φ in H1, we have
(5.8)lim⁡n→∞〈DF(Ψn),Ψ̃〉=〈DF(Φ),Ψ̃〉.
Since C0∞(Ω,R6) is dense in H1, equality (5.8) holds true for all Ψ̃∈H1, which implies that DF:H1→H1 is weakly continuous.
Therefore, according to Theorem4.3, there exists a global weak solution of (5.1). The proof is completed.


## 6. The Conservative Quantities of Spinor BEC

In this section we will discuss the conservative quantities of spinor BEC. LetE be defined as ((2.3), N,M as follows:(6.1)N=∫Ω[|ψ1|2+|ψ0|2+|ψ-1|2]dx,M=∫Ω[|ψ1|2-|ψ-1|2]dx.
Then by using the same method as the proof of Theorem 4.4, we will prove the following theorem.Theorem 6.1.
Hamilton energyE, the total particle number  N, and magnetization intensity  M are conservative quantities for problem (5.1).Proof.
Firstly, from (3.1) and (3.2) we can get
(6.2)12ℏdE(Ψ,Ψ*)dt=dF(ψ1,ψ2)dt=〈δF(ψ1,ψ2)δψ1,∂ψ1∂t〉+〈δF(ψ1,ψ2)δψ2,∂ψ2∂t〉=∑k=1,0,-1[〈δF(ψ1,ψ2)δψk1,∂ψk1∂t〉+〈δF(ψ1,ψ2)δψk2,∂ψk2∂t〉]=∑k=1,0,-1[〈-∂ψk2∂t,∂ψk1∂t〉+〈∂ψk1∂t,∂ψk2∂t〉]=0,
which imply that the energy E is a conservative quantity for problem (5.1).
Secondly, by using (3.3) we can get the following equalities:
(6.3)dNdt=ddt∫Ω∑k=1,0,-1[|ψk1|2+|ψk2|2]dx=2∫Ω∑k=1,0,-1[ψk1∂ψk1∂t+ψk2∂ψk2∂t]dx=0,
which imply that the total particle number N is a conservative quantity for problem (5.1).
At last, we showM is a conservative quantity for problem (5.1). Let X1=X2=H1(Ω,R3), then
(6.4)E:X1×X2⟶R1definedby(2.3),M:X1×X2⟶R1definedby(6.1)
are both functional. Let ψ1=(ψ11,ψ01,ψ-11),ψ2=(ψ12,ψ02,ψ-12). It is easy to check that
(6.5)δMδψ1=(2ψ11,0,-2ψ-11),δMδψ2=(2ψ12,0,-2ψ-12),δEδψ1=(δEδψ11,δEδψ01,δEδψ-11),δEδψ2=(δEδψ12,δEδψ02,δEδψ-12).
From ((2.3) and (2.7), we have
(6.6)δEδψ12=[-ℏ22mΔ+V(x)+gn|Ψ|2+gs(|ψ1|2+|ψ0|2-|ψ-1|2)]ψ12+gs(2ψ-11ψ01ψ02-ψ-12(ψ01)2+ψ-12(ψ02)2),δEδψ11=[-ℏ22mΔ+V(x)+gn|Ψ|2+gs(|ψ1|2+|ψ0|2-|ψ-1|2)]ψ11+gs(2ψ-12ψ01ψ02+ψ-11(ψ01)2-ψ-11(ψ02)2),δEδψ-12=[-ℏ22mΔ+V(x)+gn|Ψ|2+gs(|ψ-1|2+|ψ0|2-|ψ1|2)]ψ-12+gs(2ψ11ψ01ψ02-ψ12(ψ01)2+ψ12(ψ02)2),δEδψ-11=[-ℏ22mΔ+V(x)+gn|Ψ|2+gs(|ψ-1|2+|ψ0|2-|ψ1|2)]ψ-11+gs(2ψ12ψ01ψ02+ψ11(ψ01)2-ψ11(ψ02)2).
Combining (6.5), (6.6) with (3.1), (3.2), we can get following equalities:
(6.7)dMdt=〈δM(ψ1,ψ2)δψ1,∂ψ1∂t〉X1×X2+〈δM(ψ1,ψ2)δψ2,∂ψ2∂t〉X1×X2=〈δM(ψ1,ψ2)δψ1,δFδψ2〉X1×X2-〈δM(ψ1,ψ2)δψ2,δFδψ1〉X1×X2=12ℏ∫Ω2[ψ11∂E∂ψ12-ψ-11∂E∂ψ-12-ψ12∂E∂ψ11+ψ-12∂E∂ψ-11]dx=gsℏ∫Ω[[ψ11∂E∂ψ12-ψ-11∂E∂ψ-12-ψ12∂E∂ψ11+ψ-12∂E∂ψ-11]ψ11(2ψ-11ψ01ψ02-ψ-12((ψ01)2-(ψ02)2))-ψ-11(2ψ11ψ01ψ02-ψ12((ψ01)2-(ψ02)2))-ψ12(2ψ-12ψ01ψ02+ψ-11((ψ01)2-(ψ02)2))+ψ-12(2ψ12ψ01ψ02+ψ11((ψ01)2-(ψ02)2))[ψ11∂E∂ψ12-ψ-11∂E∂ψ-12-ψ12∂E∂ψ11+ψ-12∂E∂ψ-11]]dx=0,
which imply that the magnetization intensity M is a conservative quantity for problem (5.1). The proof is completed.



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*Source: 757616-2012-04-17.xml*