Table Of ContentQuantum-mechanical Landau-Lifshitz equation
Y.Yerchak (a), D.Yearchuck (b)
(a) - Belarusian State University, Nezavisimosti Ave.4, Minsk, 220030, RB; [email protected],
(b) - Minsk State Higher College, Uborevich Str.77, Minsk, 220096, RB; [email protected]
(Dated: February 2, 2008)
Quantum-mechanical analogue of Landau-Lifshitz equation has been derived. It has been estab-
lished that Landau-Lifshitz equation is fundamental physical equation underlying the dynamics of
spectroscopictransitionsandtransitionalphenomena. Newphenomenonispredicted: electricalspin
waveresonance (ESWR) being tobe electrical analogue of magnetic spin wave resonance.
PACSnumbers: 78.20.Bh,75.10.Pq,42.50.Ct
8
0
Correct formal description of dynamics of spectro- is consequence of the attempt to preserve the only po-
0
2 scopic transitions as well as a number of transitional ef- lar symmetry properties for the vectors of electric field
fects, that is, for instance, Rabi-oscillations, free induc- strengthandelectric polarizationofthe medium. At the
n
tion and spin echo effects in magnetic resonance spec- same time the vectors P~, E~ in eq.2 should be axial vec-
a
J troscopy and their optical analogues in optical spec- tors. Tobuilttheaxialvectors,satysfyingeq.2,thethird
9 troscopy is achieved in the frame of a gyroscopic model, componentsofpolarvectorsofelectricfieldstrengthand
see, e.g., [1, 2]. Mathematical base for gyroscopic model electricpolarizationischangedartificiallyintoquantities
] isLandau-Lifshitz(L-L)equation. L-Lequationwaspos- whicharebelievedtobesomemathematicalabstractions
h
tulated by Landau and Lifshitz for macroscopic classical inthe sence,thatthey cannotbe refferedto electricfield
p
description of the motion of magnetization vector in fer- strengthandelectricpolarizationcorrespondingly. So, it
-
t romagnetsasearlyas1935[3]. L-Lequationisasfollows: isgenerallyaccepted,thatthecomponentsofthevectors
n
a dS~ P~, E~ in eq.2 are consisting from various physical quan-
u =[γ H~ ×S~], (1) tities. This situation seems to be incorrect, if to pro-
q dt H ceed from the assumption of the identity of the nature
[ where S~ is magnetic moment, H~ is effective magnetic of the spectroscopic transitions in optical and radiospec-
3 field, γ is gyromagnetic ratio. L-L equation was sub- troscopy regions.
H
v stantiated quantum-mechanically in magnetic resonance
8 The aim of given work is to obtain quantum-
theory, but as the equation, describing the only the mo-
5 mechanicalequationsfordescriptionofdynamicsofboth
tion of magnetic moment in external magnetic field. At
0 magnetic resonance and optical transitions with clear
the same time L-L equation and Bloch equations, which
1 physicalsenceofallthe quantities(forthecaseofsimple
. are based on L-L equations, were in fact postulated for
0 1D model of quantum system).
description of dynamics of magnetic resonance transi-
1
tions, see, e.g., [4]. The gyroscopic model for optical Let us consider the general properties of electromag-
7
0 transition dynamics and for description of optical tran- netic field to clarify the sense of the quantities in (2). It
: sitional effects was introduced formally on the base of is well known, that electromagnetic field can be charac-
v
analogy with gyroscopic model, developed for magnetic terized by both contravariant tensor Fµν (or covariant
i
X resonance. However,theopticalanalogueofL-Lequation F ) and contravariant pseudotensor F˜µν which is dual
µν
r was obtained quantum-statistically by means of density to Fµν (or covariant F˜ , which is dual to F ). For
a operator formalism [1, 2]: µν µν
instance, F˜µν is determined by the following relation:
dP~ =[P~ ×γ E~], (2) tF˜isµyνm=m21eterαiβcµuνnFiµtν4,-wtehnesroer.eαTβµhνeiussLeeovfi-fiCehldivitteansfuolrlsyaannd-
dt E
pseudotensorsseemstobeequallypossiblebydescription
whereP~,E~ arevectors,whichareconsideredto be some of electromagnetic field and its interaction with a mat-
mathematical abstractions, since their components rep- ter. So, for example, by using of both field tensor and
resent themselves various physical quantities. So, in [1] pseudotensorthefieldinvariantsareobtained[5,6]. Fur-
is emphasized that the only Px, Py and Ex, Ey compo- ther in the practice of treatment of experimental results
nents of vectors P~, E~, correspondingly, characterize the is generally accepted, that electric field strength, dipole
genuine electromagnetic properties of the system, at the moment and polarization vectors are polar vectors but
same time the components P and E cannotbe reffered magnetic field strength, dipole moment and magnetiza-
z z
toelectromagneticcharacteristics. Further,γ wascalled tionvectorsarealwaysaxialvectors. Howeverforthe ef-
E
gyroelectric ratio the only tentatively, its analogy with fects, describing the interaction of electromagnetic field
gyromagnetic ratio was suggested. The formal charac- in optical experiments the picture seems to be reverse.
ter of optical gyroscopic model is indicated also in mod- It follows directly from the structure of algebraic linear
ern quantum optics theory, see, e.g., [2]. This situation space, which produce field of tensors and pseudotensors.
2
Really, if the structure of Fµν is Inother words,the mappingΦ:Fµν,F˜µν →P˜ is taking
place, where P˜ is pseudoscalar field. It is clear, that Φ
0 −E −E −E
1 2 3 on the space F is linear functional. Really, let α,β ∈P,
Fµν = E1 0 −H3 H2 (3) then, taking into account the properties (8) to (10) of
E2 H3 0 −H1 hF,+,·,∗i, we have
E −H H 0
3 2 1
hFµν|αΦ +βΦ i=αhFµν|Φ i+hFµν|Φ i, (13)
1 2 1 2
then the structure of F˜ will be:
µν
0 −H1 −H2 −H3 F˜µν|αΦ +βΦ =α F˜µν|Φ + F˜µν|Φ . (14)
1 2 1 2
F˜µν =H1 0 E3 −E2 . (4) D E D E D E
H −E 0 E
H2 E3 −E 01 Consequently, the set Φ(Fµν,F˜µν) of linear functionals
3 2 1
on the space hF,+,·i represents itself linear space over
So we have for contravariant and covariant electromag- the field P(since itis evident, thatthe conditions liketo
netic field pseudotensors the expressions: (6) and (7) hold true), which is dual to space hF,+,·i.
Therefore, we have
F˜µν =(−H~,E~), F˜ =(H~,E~). (5)
µν ×
hΦ,+,·i= F ,+,· . (15)
We considerfurther formally the same vectorsH~ andE~, (cid:10) (cid:11)
×
It is substantionally, that F is not self-dual. Actually,
that is, the vectors consisting of the components, deter-
minedbythestructureofF˜µν,butnowvectorH~ ispolar the ”vector” (in algebraic sense), which can be built on
basis ”vectors” of space F with the projections, which
andvectorE~ is axial. The possibility ofgivenconsidera-
are functionals, corresponding, in accordance with rela-
tion requires the validation, which can be obtained from
tionships(11),(12),togivenbasis”vectors”ofF,cannot
the following. Let us define the space F of sets of con-
belong to F. Really, its components are pseudoscalars
travariant tensors {Fµν} and pseudotensors F˜µν and and, being to be consideredas coefficients in linear com-
n o
correspondingtothemsetsofcovarianttensorsandpseu- bination of the elements of vector space F, cannot be-
dotensors of electromagnetic field over the field of scalar long to field P (properties (6) (7) do not hold true). At
values P. It is evident that all the axioms of linear space the sametime full physicaldescriptionofdynamicalsys-
hold true, i.e., if F µν and F µν ∈F, then tem requires to take into consideration both the spaces
1 2
(morestrictly,Gelfandthreeshouldbeconsidered,ifcor-
F1µν +F2µν =F3µν ∈F, (6) responding topology is determined in these spaces). On
other hand, the description with help of Gelfand three
and, if Fµν ∈F, then
willbeequivalenttothefollowingdescription. Wedefine
α Fµν ∈F (7) starting space F over the P + P˜. Then resulting func-
tionalspace will be self-dual. In extended by such a way
for ∀α ∈P. Let us define the linear algebraF by means spacewecanchoose4physicallydifferentsubspaces: two
of definition in above definited vector space hF,+,·i of subspaces 1) {Fµν} and 2) F˜µν over the scalar field
transfer operation (∗) to dual tensor, using the convolu- n o
tion with Levi-Chivita fully antisymmetric unit 4-tensor P and two subspaces 3) {Fµν} and 4) F˜µν over the
n o
eαβµν. Itisevident,thatinalgebrahF,+,·,∗ithefollow- pseudoscalar field P˜. The second case differs from the
ing axioms of linear algebra hold true: if F1αβ and F2αβ first case by the following. Symmetry properties of E~
∈F, then andH~ remainthe same,i.e.,E~ ispolarvector,since itis
e (F αβ +F αβ)=(F˜) +(F˜) ∈F, (8) dual vector to antisymmetric 3D pseudotensor, and H~,
µναβ 1 2 1 µν 2 µν respectively, is axial. At the same time, the components
ofvectorE~ correspondnowtopurespacecomponentsof
(F αβ+F αβ)e =F αβe +F αβe ∈F, (9) fieldtensorF˜µν,thecomponentsofvectorH~ correspond
1 2 αβµν 1 αβµν 2 αβµν
to time-space mixed components. Arbitrary element of
the third subspace
(e λFµν)=λ(e Fµν)=(e Fµν)λ (10)
αβµν αβµν αβµν
αFµν(x )+βFµν(x ), (16)
for∀λ ∈P. WecandeterminenowonthespacehF,+,·i 1 2
the functional Φ as follows: whereα,β ∈P˜,x ,x arethepointsofMinkowskispace,
1 2
Φ(Fµν)≡hFµν|Φi=FµνF˜ , (11) represents itself the 4-pseudotensor. Its space compo-
µν
nents, being to be the components of antisymmetric
and 3-pseudotensor, determine dual polar vector H~, mixed
components are the components of 3-pseudovector E~.
Φ(F˜µν)≡ F˜µν|Φ =F˜µνFµν. (12) Therefore, the symmetry properties of the components
D E
3
ofvectorsE~ andH~ relativelytheimproperrotationswill the equation of dynamics of magnetic resonance transi-
be inverse to the case 1. It is evident, that in the 4- tions (in which magnetic field components are parts of
th case the symmetry properties of the components of genuine tensor Fµν). In other words, mathematical ab-
vectors E~ and H~ relatively the improper rotations will stractionsineq.2becomerealphysicalsense: E~ isvortex
be inverse to the case 2. Given consideration allows to part of intracrystalline and external electric field, P~ is
suggest, that free electromagnetic field is 4-fold degen- electrical moment, which should be defined like to mag-
erated. The interaction with device (or, generally, with netic moment. So, both the vectors are axial vectors,
some substance) relieves degeneracy and leads to forma- that is, they have necessary symmetry properties (rel-
tionofresonancestate: field+device(substance),which atively reflection and inversion), in order to satisfy the
has finite lifetime. It is reasonable to suggest, that the eq.2. It should be noted that the statement on”equality
field in the resonance state becomes nondegenerate. Re- inrights”ofgenuinefieldtensorandpseudotensorbyde-
alization of concrete field state (one of 4 possible) will, scriptionofelectromagneticphenomenafollowsfromgen-
evidently, be determined by symmetry characteristics of eral consideration of the geometry of Minkowski space,
registering device (interacting substance). So, we sug- which determine unambiguously the full set of 3 possi-
gest, that, in principle, various symmetry properties of ble kinds of geometrical objects. Really, the tensors and
the same fieldcanbe obtainedby registrationofinterac- pseudotensors are equally possible geometrical objects
tionwiththe samesubstance,butwithvariousmethods, among them for any pseudo-Euclidean abstract space,
e.g., with ESR and optical absorption. to which Minkowski space is isomorphic [7].
We will consider the case 4 in more detail. The com- Quantum-mechanicaldescription of dynamics of spec-
ponents ofH~ andE~ aredeterminedinthis casebyother troscopic transitions on the example of 1D system inter-
potentials, accordingly,by dual scalar ϕ˜ and dual vector acting with electromagnetic field confirm given general
A~˜potentials, whichcanbe obtainedfrom the solutionof conclusion. Really, let us consider the system represent-
the following set of differential equations: ing itself the periodical ferroelectrically (ferromagneti-
cally) ordered chain of n equivalent elements, interact-
∇ϕ˜(~r,t)=−[∇×A~(~r,t)], ing with external oscillating electromagnetic field. It is
[∇×A~˜(~r,t)]=∂A~(~r,t) − ∂A0(~r,t), (17) atasrsyumuenditst)haotftthheeicnhtaeirnacctaionnbbeetdwesecernibeeledmbeynttshe(eHleammenil--
∂x0 ∂~r tonianof quantum XYZ Heisenberg model in the case of
a chain of magnetic dipoles and by corresponding opti-
where A (~r,t) ≡ ϕ(~r,t) is scalar potential and A~(~r,t)
0 cal analogue of given Heisenberg model in the case of a
is vector potential, which determine the components of
chainofelectricdipoles. Wewillconsiderforthesimplic-
genuinefieldtensorsFµν,F . Itcanbeshown,thatthe
µν ity the case of isotropic exchange. Each elementary unit
solution of eq.17 is viewed as follows:
of the chain will be considered as two-level system like
to one-electron atom. Then Hamiltonians for the chain
~r
ϕ˜=− [∇×A~]d~r, A~˜=A~˜ +A~˜ , (18) of electrical dipole moments and for the chain of mag-
Z 1 2 netic dipole moments will be mathematically equivalent.
o We will use further the rotating wave approximation[2].
Then the chain for distinctness of electrical dipole mo-
where
ments can be described by the following Hamiltonian:
A~˜1 =−41π∇Z |~r−Qρ~|d3ρ⇒A~˜1 = 4Qπ|~r~r|2, (19) Hˆ = ~ω20 Xn σˆnz −pαEβXn E1n(σˆn+e−iωt+σˆn−eiωt) (21)
(ρ~) 1
+ [J (σˆ+σˆ− +σˆ−σˆ+ + σˆzσˆz )+H.c.].
E n n+1 n n+1 2 n n+1
X
n
A~˜ =∇× 1 ∂∂xA~0 − ∂∂Aρ~0 d3ρ. (20) whereσˆnz =|αnihαn|−|βnihβn|is socalledσˆz-operator,
2 4π Z |~r−ρ~| observable quantity for which is population difference of
(ρ~) the states of n-th element, σˆn+ = |αnihβn| and σˆn− =
|β ihα | are transition operators of n-th element from
n n
It is takeninto account by derivationof expressions (19)
eigenstate |α i to eigenstate |β i and vice versa, corre-
n n
and (20), that dual vector potential satisfies to calibra-
spondingly. It is suggested in the model, that |α i and
tion condition: ∇·A~˜(~r,t) = Q, where Q is const. It is |β i are eigenstates, producing the full set for enach of
n
evident,thatinthecaseofusuallyusedCoulombcalibra- n elements. It is evident, that given assumption can be
tiondualvectorpotentialwillbedeterminedthe onlyby realized strictly the only by the absence of the interac-
relationship (20). Thus the simple analysis shows, that, tion between the elements. At the same time proposed
ifelectricfieldcomponentsarecomponentsofpseudoten- model will rather well describe the real case, if the in-
sor, the equation of dynamics of optical transitions will teraction energy of adjacent elements is much less of the
havemathematicallythesamestructurewiththatonefor energyofthesplitting~ω =E −E betweentheenergy
0 β α
4
levels, corresponding to the states |α i and |β i. This tions [σˆ+,σˆ−] = δ σˆz, [σˆ−,σˆz] = 2δ σˆ−, [σˆz,σˆ+] =
n n k n kn n n k kn n k n
case includes in fact all known experimental situations. 2δ σˆ+, we obtain
kn n
Further, pαβ is matrix element of dipole transitions be-
E ∂σˆz
tween the states |αni and |βni, which along with energy k =2iΩ e−iωtσˆ+−eiωtσˆ− (24a)
difference of these states E −E are suggested to be in- ∂t E k k
β α (cid:0) (cid:1)
dependent on n, E1n is amplitude of electric component +2iJE σˆ− ,(σˆ+ +σˆ+ ) − σˆ+ ,(σˆ− +σˆ− ) ,
of electromagnetic wave on the n-th element site, ~ is ~ k k+1 k−1 k k+1 k−1
(cid:0)(cid:8) (cid:9) (cid:8) (cid:9)(cid:1)
Planck’sconstant,JE isopticalanalogueoftheexchange ∂σˆk+ =iω σˆ++iΩ eiωtσˆz (24b)
interaction constant. Here, in correspondence with the ∂t 0 k E k
suggestion, J = Jx = Jy = Jz. In the case of the
chain of magEnetic dEipole mEomenEts E1n in Hamiltonian +iJ~E σˆk+ ,(σˆkz+1+σˆkz−1) −{σˆkz ,(σˆk++1+σˆk+−1) ,
(21) is replaced by H1n, i.e., by amplitude of magnetic (cid:0)(cid:8) ∂σˆ− (cid:9) (cid:9)(cid:1)
component of electromagnetic wave on the n-th element k =−iω σˆ−−iΩ e−iωtσˆz (24c)
site, J is replaced by the exchange interactionconstant ∂t 0 k E k
qJuHe,nmcyEatωri0xiselreempelanctepdαEbβyis~1rgeHpβlaHcHed0b=yγpHαHβH0a,nwdhtehreefHre0- −iJ~E (cid:0)(cid:8)σˆk− ,(σˆkz+1+σˆkz−1)(cid:9)+{σˆkz ,(σˆk−+1+σˆk−−1)(cid:9)(cid:1),
is external static magnetic field, β is Bohr magneton,
H where expressions in braces {,} are anticommutants.
g is g-tensor, which is assumed for the simplicity to be
isHotropic. The first term in Hamiltonian (21) character- Here ΩE and γE is Rabi frequency and gyroelectric ra-
tio. They are determined, correspondingly, by relations
izes the total energy of all chain elements in the absence E1pαβ pαβ
of external field and in the absence of interaction be- ΩE = ~E =γEE1, γE = E~ , which are replaced by
tween chain elements. The second item characterizes an H1pαβ pαβ
relations Ω = H = γ H , γ = H in the case
interactionofachainwithanexternaloscillatingelectro- H ~ H 1 H ~
of the chain of magnetic dipole moments. The equations
magnetic field in dipole approximation. Matrix elements
(24) can be represented in compact vector form, at that
of dipole transitions pαβ and pβα between couples of the
the most simple expression is obtained by using of the
E E
states (|α i, |β i) and (|β i, |α i), respectively, are as-
n n n n basis~e+ =(~ex+i~ey),~e− =(~ex−i~ey),~ez. So, we have
sumedtobeequal,i.e.,spontaneousemissionisnottaken
itnotnoiacnonosfidqeuraantitounm. THheeistehnibredrgiteXmXiXs,-minoedseslenincet,hHeacmaisle- ∂~σˆk = ~σˆk×G~ˆk−1,k+1 , (25)
∂t h i
ofmagnetic versionandits electricalanaloguein electric
version of the model proposed. Let us define the vector where~σˆ is givenby(22),butwiththe components,cor-
k
operator: respondingtonewbasis,k =2,N −1,andvectoropera-
~σˆk =σˆk−~e++σˆk+~e−+σˆkz~ez. (22) tor G~ˆk−1,k+1 =Gˆ−k−1,k+1~e++Gˆ+k−1,k+1~e−+Gˆzk−1,k+1~ez,
where its components are
It seems to be the most substantial for the subsequent
panroadlyuscies,stahlagtebaras,etwohficσˆhkmisoipsoermaotorprsh,icwhtoerSe m= i1s/z2,P+a,u-l,i Gˆ−k−1,k+1 =ΩEe−iωt− 2J~E(σˆk−+1+σˆk−−1), (26a)
matrix algebra, i.e., mappings fk : σˆkm →σPm realize iso- Gˆ+ =Ω eiωt− 2JE(σˆ+ +σˆ+ ), (26b)
morphism. Here k is a number of chain unit, σm is the k−1,k+1 E ~ k+1 k−1
P
set of Pauli matrices for the spin of 1/2. Consequently, 2J
Gˆz =−ω − E(σˆz +σˆz ). (26c)
fromphysicalpointofview~σˆ representsitselfsomevec- k−1,k+1 0 ~ k+1 k−1
k
tor operator, which is proportional to operator of the
It should be noted, that vector product of vector oper-
spin of k-th chain unit. Vector operators σˆm produce
k ators in (25) is calculated in the correspondence with
also linear space over complex field, which can be called
known expression
transition space. It is 3-dimensional (in the case of two-
lftoeoifovrtnehlceoosyfmqsuoptapeonmtnitceusan)m,ltsttohrpoaaftntisc~iσˆisstki3coainoslcpsu.nelreaTactteyiosopsrniacseraqoylufaiftnotrairaocnnccssouitrroiraofecntcyhtdeiydnmneasmocmtraiiionpcnys- h~σˆk×G~ˆk−1,k+1i=(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)~e~~ee+−z×××~e~e~e+−z σσˆˆσˆkk−+kz GGGˆˆˆ+z−kkk−−−111,,,kkk+++111 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12), (27)
(cid:12) (cid:12)
is connected with obliteration of correct dimensionality (cid:12) (cid:12)
however, by its calculation one should take anticommu-
oftransitionspace. Theequationofthe motionfor~σˆ is:
k tants of corresponding components instead their prod-
∂~σˆ ucts. Givendefinitionseemstobe naturalgeneralization
i~ ∂tk =[σˆk−,Hˆ]~e++[σˆk+,Hˆ]~e−+[σˆkz,Hˆ]~ez. (23) of vector product for the case of operator vectors, since
the only in this case the result will be independent on
We will consider the case of homogeneous excitation a sequence of components of both the vectors in their
of the chain, that is En is independent on unit num- products like to that one for usual vectors. Naturally,
1
ber n (En ≡ E ). Then, using the commutation rela- the expressions like to (27) can be used for calculation
1 1
5
of vector product of common vectors. Taking into ac- equation in continuous limit, in fact it is operator equa-
count the physical sense of vector operators ~σˆ we con- tion, which argues the correctness of eq.(2), that is, the
k
clude, that (25) represent themselves required quantum- physical correctness of gyroscopic model for description
mechanical difference-differential equations (the time is of optical transitions and transitional optical analogues
variedcontinuously,thecoordinatesarevarieddiscretely) of magnetic resonance phenomena. If J 6= 0 we have
for the description of the dynamics of the spectroscopic quantum-mechanical optical analogue of classical L-L
transitions (in the frames of the model proposed). From equation, which was introduced by Kittel for SWR de-
here in view of isomorphism of algebras of operators ~σˆ scription [8]. Therefore, the results obtained allow to
k
predictanewphenomenon-electricspinwaveresonance
and components of the spin it follows that the (25) is
(ESWR). The equation (29) (if put aside the operator
equivalent to L-L equation (in its difference-differential
symbols) and equation introduced by Kittel are coincid-
form). Consequently we have proved the possibility to
ing mathematically to factor 2 in the second term. This
use L-L equation for the description of the dynamics
differenceisliketowellknowndifferenceofgyromagnetic
of spectroscopic transitions, as well as for the descrip-
ratios for orbital and spin angularmoments. Thus along
tion of transitional effects. To obtain the continuous
with magnetic resonance methods we can detect a spin
approximation of (25) for coordinate variables too, we
valueofparticles,quasiparticles,impuritiesorothercen-
have to suggest that the length of electromagnetic wave
tersinsolidsbyopticalmethods: bystudyoftransitional
λ satisfies the relation: λ >> a, where a is 1D-chain-
optical analogues of magnetic resonance phenomena or
lattice constant. Then the continuous limit is realized
ESWR. It should also be noted that above considered
if to substitute all the operators, which depend on dis-
theoretical description of ESWR allow predict the dif-
crete variable k, for the operators depending on con-
tinuous variable z, that is: σˆ± → σˆ±(z),σˆz → σˆz(z). ference in splitting constants which characterize ESWR
k k
by its experimental detection with using of one-photon
Thus, we obtain, taking also into account the relations
σˆz,± +σˆz,± −2σˆz,± →a2∂2σˆz,±(z), the equation,which, methods like to IR-absorption or IR-reflection and with
k+1 k−1 k ∂z2 using of two-photon methods like to Raman scattering.
like to (25), in compact vector form is:
It is evident that equations (28), (29) can immediately
be used for single transition methods, for instance, for
∂~σˆ(z) 2a2J
= ~σˆ(z)×γ E~ − E ~σˆ(z)×∇2~σˆ(z) , (28) IR-absorption. By Ramanscattering we havetwo subse-
∂t h E i ~ h i quent transitions. Then operator ~σˆ(z), which character-
whereE~ =E1eiωt~e−+E1e−iωt~e++(cid:16)−γωE0(cid:17)~ez. Thestruc- iczeesss,thheastrtaonsbiteiocnondsyisntainmgicfsrobmy Rtwaomaconmspcaotnteenrtinsg~σˆp1(rzo)-
tureofvectorE~ clarifiesitsphysicalmeaning. Twocom- and~σˆ (z)characterizingboth the transitions,takensep-
ponentsE+, E− areright-andleft-rotatorycomponents aratel2y, that is ~σˆ(z) =~σˆ (z)+~σˆ (z). Consequently, the
of oscillating external electric field, third component Ez 1 2
equation for transition dynamics of second component,
is intracrystalline electric field, which produces two level
which will determine experimentally observed ESWR-
energy splitting for each of the unit of a chain system
spectrum, is
with value, equal to ~ω . It means that the following re-
0
liasteiolenctirsictaaklianngaplolagcuee:oωf0B=oh~1rgmEβaEgnEe0to=n,γEgEE0is,welheecrtericβaEl ∂~σˆ∂2t(z) =h~σˆ2(z)×γEE~i− 2a~2J h~σˆ2(z)×∇2~σˆ2(z)i
analogueofmagnetic g -tensor,whichis assumedfor the
2a2J
simplicity to be isotropic. In other words, by means of − ~σˆ (z)×∇2~σˆ (z) . (30)
~ 2 1
given relation the correspondence between an unknown h i
intracrystalline electric field E and observed frequency
0 The second and the third items in eq.19 are practically
ω is set up. Further, we take into consideration, that
0 equal to each other, since we are dealing with interact-
physicalsenseofoperators~σˆ(z)incontinuouslimitisre- ing electric dipole moments of the same chain, that is,
mained, i.e., for each point of z the components σ±(z), ∇2~σˆ (z) and ∇2~σˆ (z) have almost equal values. Then
σz(z) are satisfying to algebra, which is isomorphic to 1 2
weobtain,thatthe valueofsplittingconstantbyRaman
b
algebra of the set of spin components. Then by means
scattering detection of ESWR in the same sample is al-
b
of relation, which has mathematically the same form for
mostdoubleincomparisonwiththatonebyIRdetection
both the types of the systems studied S~ˆ(z) ∼ ~~σˆ(z), the of ESWR. The observation of doubling in the splitting
2
equationofthemotionforoperatorsofmagneticandelec- constant by Raman ESWR-studies is additional direct
tric spin moments are obtained. So, e.g., the equationof argument in ESWR identification.
the motion for electric spin moment operator is: Therefore, quantum-mechanical analogue of Landau-
Lifshitz equation has been derived with clear physical
∂S~ˆ∂(tz) =hS~ˆ(z)×γEE~i− 4a~22JE hS~ˆ(z)×∇2S~ˆ(z)i (29) streonssceoopfy.thIetqhuaasntbietieens efosrtabbolitshherdadtihoa-tanLdanodpatuic-aLlifsspheitcz-
equationisfundamentalphysicalequationunderlyingthe
Equation (29) gives for the case J = 0 quantum- dynamics of spectroscopic transitions and transitional
mechanical optical analogue of classical Landau-Lifshitz phenomena. New phenomenon- electricalspin waveres-
6
onance and its main properties are predicted. ofSciencesofRBL.Tomilchickforthehelpfuldiscussions
The authors are thankfull to Doctor V.Redkov and to of the part of results.
Professor, Corresponding Member of National Academy
[1] J. D. Macomber, The dynamics of spectroscopic transi- [5] L.D.LandauandE.M.Lifshitz, The Classical Theory of
tions (John Wiley and Sons, New York, London, Sydney, Fields (Butterworth-Heinemann,1981).
Toronto, 1976). [6] H. Stephani, An Introduction to Special and General Rel-
[2] M. O. Scully and M. S. Zubairy, Quantum Optics (Cam- ativity (Cambridge University Press, Cambridge, 2004).
bridge University Press, Cambridge, 1997). [7] P.K.Rashevskii,RiemannianGeometryandTensorAnal-
[3] L.D.LandauandE.M.Lifshitz,PhysikalischeZeitschrift ysis (Editorial URSS,Moscow, 2003).
der Sowjet Union 8, 153 (1935). [8] C. Kittel, Phys.Rev. 110, 1295 (1958).
[4] C. P. Slichter, Principles of Magnetic Resonance
(Sprlnger-Verlag, Berlin, Heidelberg, NewYork,1980).
