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Submitted on 1 Jan 1989
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Berry phases for quadratic spin Hamiltonians taken from atomic and solid state physics: examples of Abelian gauge fields not connected to physical particles
C. Bouchiat
To cite this version:
C. Bouchiat. Berry phases for quadratic spin Hamiltonians taken from atomic and solid state physics:
examples of Abelian gauge fields not connected to physical particles. Journal de Physique, 1989, 50
(9), pp.1041-1055. �10.1051/jphys:019890050090104100�. �jpa-00210975�
Berry phases for quadratic spin Hamiltonians taken from atomic and solid state physics: examples of Abelian gauge fields not connected to physical particles
C. Bouchiat
Laboratoire de Physique Théorique, E.N.S. Paris (*), 24 rue Lhomond, 75231 Paris Cedex 05, France
(Reçu le 7 octobre 1988, accepté le 4 janvier 1989)
Résumé.
2014Cet article contient une évaluation de la phase de Berry pour des systèmes de spin gouvernés par la classe suivante de Hamiltoniens non linéaires : H(B, n) =
03B3S S · B + 03B3Q((S · n )2 - S2/3), avec B · n
=0. On donne des exemples de ces Hamiltoniens pris
en Physique Atomique et en Physique du Solide. On calcule exactement les phases de Berry pour S = 1, 3/2 et pour des spins plus élevés, jusqu’au second ordre dans le rapport 03B3Q/(03B3S B ). Les
résultats sont donnés en termes de la circulation le long d’une courbe fermée d’un champ de jauge
Abélien qui peut être considéré comme la généralisation du champ d’un monopole magnétique, le plan complexe projectif P2(C) jouant le rôle de la sphère S2. Dans le cas du spin 1 on explicite la
relation entre la phase quantique cyclique de Aharonov et Anandan et la phase de Berry. On analyse la correspondance entre la limite semi-classique de la phase de Berry et l’angle adiabatique de Hannay, dans le cadre du modèle de spin classique proposé récemment par Nielsen et Rohrlich.
Abstract.
2014This paper contains an evaluation of the Berry phases associated with the following
class of nonlinear spin Hamiltonians : H (B, n )
=03B3S S · B + 03B3Q((S · n)2 - S2/3), with B · n
=0.
Examples of these Hamiltonians are given in Atomic and Solid State Physics. We compute exactly
the Berry phases for S = 1, 3/2 and for S > 3/2, to second order with respect to the ratio
03B3Q/(03B3S B ). The results involve loop integrals of an Abelian gauge field which can be considered
as a generalization of the magnetic monopole field, the projective complex plane P2(C) playing
the role of the sphere S2. In the case of the spin-1 the relation between the Aharonov and Anandan cyclic quantum phase and the Berry adiabatic phase is made explicit. The connection between the semiclassical limit of the Berry phase and the classical adiabatic Hannay angle is analyzed using the classical spin model recently proposed by Nielsen and Rohrlich.
Classification
Physics Abstracts
03.65
-42.50
.Introduction.
After the work of Berry [1], Simon [2], Aharonov and Anandan [3], and mâny others [4],
there has been considerable interest in the physical implications of the non-trivial topology of
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019890050090104100
the space of quantum states. All the physical information concerning an isolated quantum system is contained in the density operator p = 1 t/1) t/11 / t/11 t/1) or equivalently in a given
ray of the state vectors’ Hilbert space. For a quantum system described by a Hilbert space of finite dimension n, JCn, the pure state density operator space E (p ) can be identified with the
complex projective space Pn - 1 (C ), which is endowed with a non-trivial topology : for instance, in the simple case of a spin 1/2 system, E (p ) reduces to Pl (C ) which is isomorphic to
the sphere S2.
When a quantum system evolves in such a way as to describe a closed loop (C) in the space
E (p ), it acquires a path dependent phase j6 (C ). Let 1 t/1 (t ) be a representative of the ray p (t ) and assume that the closed circuit (C) is performed in the time interval 0 -- t * T. We
have 1 t/1 (T) = q 1 03C8 (0 ) > where q is a c-number of phase e : q = 1 q 1 exp (i 0 (If the
circuit is generated by a Hermitian Hamiltonian 1 TJ 1 = 1). The geometrical phase (3 (C) associated with the closed path is given by :
One verifies that (3 (C) is invariant if one performs the following gauge transformation
upon 1«/1 (t» :
where g (t ) is an arbitrary complex function of t.
In this paper we shall be concerned, for the case of spin systems, by a particular class of quantum cycles, namely the adiabatic cycles, which were the subject of the original work of Berry. We shall consider the adiabatic cycles generated by Hamiltonians containing quadratic spin terms, which occur in various problems of Atomic and Solid State Physics. The nonlinearity introduces an essential complication and the quantum adiabatic phase is no longer given by a loop integral involving a magnetic monopole field, as in the linear case.
The paper is organized as follows.
In section 1, we begin by rewieving some elementary notions of linear fiber bundle theory
which is the natural mathematical framework to study the quantum cyclic phase. We give an expression of (3 (C) as a loop integral of an Abelian gauge defined on the pure state density
matrix space E (p ).
In section 2, we study, in the case of spin-1 systems, the relation between the quantum cyclic phase {3 (C) and the Berry adiabatic phase for a class of non-linear Hamiltonians which appear in Atomic and Solid State Physics :
If the condition B . n
=0 is satisfied, the parameter space of H(B, n) is isomorphic, up to
an overall scale transformation, to the space E (p ) of the pure state density operators, namely
the complex projective plane P2(C). As a consequence, the adiabatic cycles generated by H(B, n) are the most general quantum cycles for spin-1 systems.
In section 3, we evaluate the Berry phase associated with adiabatic cycles of H(B, n) for spins higher than 1. The case S
=3/2 is treated exactly and a second order perturbative expansion with respect to yQ is used for arbitrary high spins. Finally we study the classical adiabatic Hannay angle associated with the Hamiltonian H(B, n ), using the classical spin
model of Nielsen and Rohrlich. On this particular example, the connection between the
semiclassical limit of the Berry phase and the adiabatic Hannay angle is clearly exhibited.
1. The quantum cyclic phase and the geometry of the linear fiber bundle associated with the quantum states space.
The Hilbert space JC, can be considered as a linear fiber bundle constructed over the base space E (p ) ; the fiber is the one-dimensional space
-the ray
-associated with a given pure state density operator.
By a suitable choice of gauge, it is always possible to choose a representative of the ray,
1 Z) having a unit norm and such that it is a uniform function of p. An arbitrary element of the bundle 1 t/J) is then written as 1 03C8r > = e 1 Z > . The variation d 1 t/J) is split into two parts :
the vertical part dv 11/1 > is along the fiber :
where úJ (1) > is a c-number differential one-form and dHII/1) is orthogonal to 11/1) : 1/11 dH 11/1) = O. The one-form w (1) is easily written in terms of e and 1 Z) :
We decompose CI) (1) into its real and imaginary parts :
The ray representative Z) having been chosen with a unit norm, we have :
It follows then :
w1 (1) is an exact differential, while wl1) is not and consequently it can be used to define a
connection on the bundle. The parallel transport of 1 &03C8> will be associated with the vanishing
of the one-form w21). When 1 t/J) is parallel transported during the time interval
0 * t * T, we have :
-With our definition of 1 Z>, we have 1 Z(T» = Z(0)) so we can write :
The parallel transport condition, written in terms of 1 f/1) reads :
It follows that when 1 t/J) is parallel transported {3 (C) coincides with the phase shift and we
have finally :
This result could have been obtained directly from equation (1) by taking 1 «/1 (t) = 1 Z (t ) >
and noting that, according to our definition of Z) , we have 1 Z(T) = Z(0)) . The purpose of the above more elaborate considerations is to show that /3 (C) is closely related to the geometry of the space E (p ) : f3 (C) is the phase of the holonomy 1 x 1 matrix which results from the parallel transport of 1 «/1) in the bundle along a path associated with a closed curve in the base space E (p ). The one-form i (ZldIZ) can be considered as a U(1 ) gauge field A defined on the density operator space E(p). Let (xl ... Xi
...x,) be a set of coordinates of
E (p ) ; we can write :
.Our definition of Z) does not lead to a unique determination and other possible choices
are obtained by performing a U(1 ) gauge transformation 1 Z> --+ 1 Z>’
=exp (if (x» 1 Z>.
The gauge field A transforms then according to the usual rule :
2. Relation between the Aharonov and Anandan quantum phases and the Berry adiabatic phases for nonlinear spin-1 Hamiltonians.
In reference [5], the case of the spin-1 was studied in details. A set of coordinates for the manifold E (p ) was constructed in terms of physical observables. An arbitrary spin-1 state is completely described by the polarization vector p = Iz-l (8) and the alignment tensor Aij = (2 1z2)- 1 {Si’ Sj} >. For a pure state p 2 = p,
@the polarization p and the alignment
tensor A are given in terms of a triad of unit vectors ei, forming an orthonormal basis for
1R3 and an extra angular parameter e. One introduces the rotation matrix R (0, p, a )
=R (2, cp ) . R (y, 0 . R (i, a ), where R (n, cp ) in the rotation of angle cp around
the unit vector n. The unit vectors ei are given by :
where ti for i = 1, 2, 3 stand for the unit vectors along the x, y, z axis respectively. The polarization vector p and the alignment tensor A for a pure state (p 2
=p ) are given by :
Together with the normalization condition Tr p = 1, p and A lead to a complete
determination of the pure state density matrix. The four angular parameters constitute a set of coordinates for the manifold E (p )
=P2(C). The non-trivial topology of P 2 (C) manifests
itself by the existence of coordinate singularities, for instance, when C
=0 or nr/2. A closed
curve (C) is drawn in E (p ) during the time interval 0 - t- T if the four time-dependent
parameters 0 (t ), cp (t ), a (t), (t) satisfy the following boundary conditions :
In reference [5] the following expression was derived for the geometrical phase {3 (C) :
In view of an eventual experimental determination of f3 (C), the authors of reference [5]
have constructed a time-dependent Hamiltonian Hp (t ) which takes a pure state around an
arbitrary chosen closed loop (C) in E (p ) and at the same time parallel transports the phase.
In such a situation, the geometrical phase coincides exactly with the phase shift at the end of
the cycle. In this paper, as in the pioneering work of Berry [1], we shall study a particular class
of quantum cycles, namely the adiabatic cycles. The Hamiltonian of the quantum system is assumed to depend upon a set of slowly time-varying parameters A.(t). More precisely one
assumed that max ( 1 hd/dt X/ 1 X 11 )/min ( 1 E,, (k) - En(A.) 1 ) « 1 where Em(A.), En (X) stand
for two arbitrary non-degenerate energy levels of H(X). If one takes as initial condition
1 «fi (0» = ma (0 ) ) the adiabatic approximation tells us that 1 «fi (t» is given up to a phase, by the eigenstate ImX(t), obtained from [mX(0)) by an analytic continuation from
B=B(0) to A. = A. (t ). Let us consider a situation such that during the time interval 0 _ t _ T, a closed curve (T ) is drawn in the parameter space t(A.), i. e. X (T) = A (0). For
each non-degenerate level of H(a), we introduce the pure state density matrix :
The above equation defines a mapping of the parameter space 6(X) onto the space
E (p ). The Berry phase ym, (F) is nothing but the geometrical phase /3 (C) associated with the closed curve in E (p ) image of the closed curve ( t’ ) by the mapping 9 (k) --> E (p ) defined by equation (10). The mapping is particularly simple for a spin 1/2 system. Up to the addition of a
c-number the most general Hamiltonian can be written as : H(B )
=ys S ’ B (t ). The density
matrix associated with the two eigenstates labelled by cr = ± 1 is :
Let ( 0, cp ) be the spherical coordinates of p, the ray representative 1 Z ( (J, cp )) is then given by :
Using equation (4) one recovers easily the well known result :
where 03A9 (B ) is the solid angle drawn on the unit sphere by the unit vector B (t ).
Let us now concentrate on studying the Berry phase of a spin-1 system described by a
Hamiltonian non linear in the spin operator S. The most general spin-1 Hamiltonian can be considered as an element of the SU(3) Lie algebra and consequently depends upon eight real parameters X. In order to get the Berry phase we will have to deal with a mapping of R8 onto P2(C). The general case is too complicated to be physically illuminating ; we have
then decided to restrict ourselves to the following class of non linear spin-1 Hamiltonian :
where B is a time dependent magnetic field and n a unit vector. The quadrupole term in
H(B, n) can be associated with various physical situations, provided the parameter yQ is suitably defined. Let us give three physical systems which can be described by H(B, n) :
Exemple 1 : a nuclear spin, having a quadrupole moment Q, in a uniaxial crystal ;
VZZ is the second order z-derivative of the electronic potential with the z-axis along the crystal symmetry axis, evaluated at the position of the nucleus. Note that since ILS remain finite in the classical limit, the quadrupole parameter yQ has also a finite classical limit.
It should be said that the restriction to uniaxial symmetry is not essential. The formulas
given below for the Berry phases can be easily extended to the general case provided B is
taken along one of the principal axis of the field gradiant tensor Vij.
Exemple 2 : the second order Stark effect for an atomic level :
d is the electric dipole operator, G (Em ) the atomic Green function and Eo the static electric field. The quantum average ( > is taken upon an atomic state of energy Em with J2 = h2 S(S + 1) and J,
=hS.
Exemple 3 : the triplet state of an isolated proton pair in a single crystal :
where m is the distance between the protons.
In reference [5] the ray representative Z) was written in terms of two orthogonal real 3-
vectors a and b with a 2+ b2 = 1. Let A the complex 3-vector a + ib, then 1 Z) reads as
follows :
Another way to state this result is to say that the set of pairs of 3-vectors (a, b ) constitutes a
realization of the P2(C) manifold. This remark suggests to make a further restriction upon
H(B, n) by imposing the orthogonality of B and n. Since a change of scale of H can be accounted for by a change of the time variable, the space of the Hamiltonian parameters with
n. B
=0 is isomorphic to the space E (p ). Consequently the most general spin-1 quantum cycle can be generated by our restricted Hamiltonian H(B, n) with B . n
=0.
A remarkable property of H(B, n) when B. n
=0 is the fact that it is symmetric under the
space reflexion r(B ) with respect to a plane perpendicular to B. Let pm be the polarization of
the eigenstates of H(B, n). If B is non zero the eigenstates are in general non-degenerate and consequently invariant, up to a phase, under the unitary transformation U(r(B)) associated
with r (B ). The polarizations p, being invariant upon r(B), have no component perpendicular
to B. Similarly, with the z and x axis taken along B and n respectively, the components AZx and Azy of the alignment tensor vanishe. H(B, n) is also invariant upon another symmetry, namely the product of the space reflexion r (n ) by the time reflexion T. As a consequence the component Axy is also vanishing, so that the principal axis of the alignment tensor coincide
with B, n, B A n. Note that all the preceding properties are valid for any spin S > 1.
The rotation R ( o, cp, a ) appearing in the construction of 1 Z) can be identified with the
one which brings the z, x axis along B and n respectively :
It is convenient to introduce the unitary U(R) operator associated with R :
We define the rotated state 1 ;; > = U- 1 (R ) 11/1 ) and the rotated Hamiltonian
fi
=U-1 (R ) . H (B, n). U (R )
=H (’zB, X ). We rewrite fi in terms of the 3 x 3 matrices which were introduced in reference [5] :
They can be obtained from the 2 x 2 Pauli matrices by adding a line and a column of zeros.
One gets :
We tentatively define the angular variable Ç by :
One can anticipate that equations (16) and (19) define the mapping of the parameter space
e (A) into the density operator space E (p ). We are now going to show that it is indeed the
case.
Using the angular variable Ç the rotated Hamiltonian FI can then be put under the following
form :
where :
Finally the diagonalization of Fi is achieved by performing a unitary transformation, already introduced in reference [5] :
The transformed Hamiltonian H = V - 1 (03BE ) ÈV (C ) reads :
The three eigenstates of H, with the corresponding eigenvalues are readily obtained :
The polarizations pm are given by :
Although the Berry phase could be obtained directly from equation (9), it is instructive, in
view of possible generalizations to higher spins, to calculate directly the Berry phase starting
from equation (1). The eigenstates of H(B, n) are obtained from the states Il, m) by application of the product of the unitary operators U(R ) . V (03BE) :
We have to compute the differential one-form .pm 1 d 1 «Pm) :
where the two operator one-forms dP and dQ are given by :
Using the definition of V ((03BE)
-see equation (22)
-one gets :
One sees immediatly that (l,mldPll,m) = 0.
Using the results of the appendix of the reference [5] one writes dQ in terms of the left invariant Maurer-Cartan one-forms A defined on the group SO(3) :
Noting that V (03BE) changes the values of Sz by zero or two units of fi, only A, S, contribute to the diagonal element of dQ :
Using the explicit expression of Az, (Eq. (A5) of Ref. [5]) we obtain the final expression :
The quantum adiabatic cycle is defined by the boundary conditions given in equation (8).
We deduce the following relation between 1 t/1m(T» and 03C8m/1m(O» :
From the above expression one gets the phase shift : T
=km7r. We arrive at the final
expression for the Berry phase associated with an adiabatic cycle of the Hamiltonian
H(B, n) :
Apart from the factor m
=± 1 which gives the relative orientation of B and the polarization
p, the right hand side of equations (9) and (27) are identical.
If we take the limit yQ
=0, i.e. cos, = 1, we recover the well known formula :
where 11i (B ) is the solid angle defined by the curve drawn by the unit vector B on the unit
sphere. It has been noted that h does not appear explicitly in the above expression ; it is
consistent with the fact that the Berry phase for a spin system described by an Hamiltonian linear in the spin operator S can be obtained by purely classical considerations. The situation is different when a quadrupole coupling, quadratic in S, is present like in the Hamiltonian
H(B, n) studied in this paper. As we have already noted, the parameter yQ remains finite in the classical limit. As a consequence tg Ç -> 0 when h
-->0. It is convenient to isolate in
ym (T ) the term y (1)(t) associated with the linear part of the Hamiltonian :
In order to exhibit the quantum nature of ym2>(T ) let us compute the mean square deviation of the spin component along its average direction B :
The computation is most conveniently performed with the rotated Hamiltonian 7? :
For the state I 03C8m > with m
=± 1 we have :
One gets the following expression for the relative mean square fluctuation :
One sees that in the limit of the small yQ, ym2>( T ) is proportional to the mean square
fluctuation of S - B.
3. Berry phases for nonlinear spin Hamiltonians for S > 1 and the classical adiabatic Hannay angle for spin systems.
In this section we would like to show first that the Berry phase for a spin 3/2 described by the
Hamiltonian H(B, n) can also be readily obtained. We note first that, since H(B, n)
commutes with U(r(B )), it can be written as a direct sum of two Hamiltonians acting in
2-dimension Hilbert subspaces associated with the eigenvalues ± 1 of U (r (B ) ) :
To see that let us perform the unitary transformation U(R ) associated with the rotation
R ( 0, cp, a ) which brings the z, x axis along B, n respectively. Then we write the rotated
Hamiltonian fi
=U-1 (R ) . H(B, n). U (R )
=H(tB, i) in terms of S+: = Sx ± i Sy :
H is clearly block-decomposable into two subspaces Jc (E>, defined by the sets of vectors
Introducing the projectors P (£) onto the subspaces Je (’), we derive the following identities :
where ai i are the ordinary 2 x 2 Pauli matrices ; one obtains immediatly the following expression for FI( £) :
As before we introduce the angular variable £ defined by :
together with the unitary operator V (03BEê)
=exp (- i /2 U2 C,). The transformed Hamiltonian
H/( ê) = V - 1 (03BEe,) I-I(E) V (e,) reads :
with :
The eigenstates of H(B, n) can then be written as :
where e is given in terms of the magnetic number m by :
The corresponding eigenvalues are given by :
To get the Berry phase, we follow the procedure used previously for the spin-1 case :
We finally arrive at the following expression for the Berry phase relative to a spin 3/2
described with the Hamiltonian H(B, n) :
In order to study the classical limit of the adiabatic phase, let us derive an approximate expression of 1’m(T) valid for an arbitrary spin, to second order with respect to the ratio
1’Q 1l/1’s B. We first note that the rotated Hamiltonian A given by equation (30) can be represented by a real matrix. It implies that the eigenvectors of H can be chosen to be real. As
a consequence the one-form lm (03C8m 1 d 1 «frm) vanishes for any spin. So we can write :
Using, once more, the results of the appendix of reference [5] we get :
We shall compute the average value (03C8m 1 S, 03C8m) using a well known theorem of quantum
mechanics :
It is a straightforward matter ’to compute Em to second order in yQ. One gets :
The computation of the phase shift, performed previously for S = 1 and 3/2, is readily
extended to an arbitrary spin :
The Berry phase relative to an arbitrary spin system performing an adiabatic cycle generated by the Hamiltonian H(B, n) is given, to second order in yQ, by the following expression :
ym (1’)(F) is the Berry phase relative to the linear part of the Hamiltonian given by equation (28). y(2)(r) is the contribution associated with the nonlinear part of the Hamiltonian :
The above formula has been shown to agree with the second order expansion of the exact
results obtained previously for S = 1 and S
=3/2. When m
=± S, Ym (2)(r) is of the order of
h with respect to yml (T ) when the classical, limit is taken :
It is of interest to study in the present case the classical counterpart of the Berry phase, namely the adiabatic angle introduced by Hannay [6] and Berry [7]. To describe the classical
spins we shall use the model proposed recently by Nielsen and Rohrlich [8], which can be quantized by a path integral method. In order to simplify the analysis we shall assume that the
direction of B is fixed along the z axis. In such a situation the Berry phase reduces to
ymJ(2)(r) with dcp
=0 :
Following reference [8] we write the classical spin Lagrangian as :
The space components of the classical spin S are given by :
Nielsen and Rohrlich have shown that the classical spin dynamics can be described by a one degree of freedom Hamiltonian H(p, q ) where the conjugate variables p and q are defined
as :
If one expresses the spin components in terms of p and q, one finds that they obey the expected Poisson brackets relations :
The path integral method leads to the following quantization conditions for the par- ameters A and Z :
A
=0 for bosons, A
=h /2 for fermions and £ = h (S + 1/2 ). In the following we shall ; for
the sake of simplicity, limit ourselves to integer spin 1. e. à
=0. The classical spin Hamiltonian with B along the z axis and n rotated by an angle a with respect to the x axis reads as follows :
For a given energy E, p can be expressed as a function of E and q by writing H(p (q, E ) ) = E. Although an exact formula could be obtained we shall rather use an
expansion in powers of yQ. Let us write down the result when a
=0.
where the dimensionless quantities e, and 7Jr are given by :
We introduce the action I given by the integral :
E can be readly obtained as function of I in the form of a series expansion in
The study of the simple case yQ
=0 suggests the semiclassical quantization rule :
I
=mh, with m integer. Using for Z the value obtained by the path integral method we arrive
at an expression of the energy which coincides with the purely quantum result up to corrections of the order 1z2, as it is obtained in the usual W.K.B. approximation. This result confirms the validity of the classical model of the spin proposed in reference [8].
Let us now concentrate on the classical adiabatic angle. We recall first that the action I is an
adiabatic invariant. One is led to introduce the angular variable X, conjugate to the action I,
which is given for a one degree of freedom system by :
For fixed external parameters, the angular variable X is a linear function of the time t :
We assume now that the parameters « and B are slowly varying with time and perform an
adiabatic cycle during the time interval 0 -- t -- T such that a (0) = a (T) + k7T and B(O)
=B(T). The action angle is given at time T by :
where Ox (1 ; r) is the adiabatic geometric angle of references [6] and [7].
To write down the formula giving AX (1 ; T ) it is convenient to represent the extemal parameters by a 3-vector X
=(ÀI’ À2, À3) with ÀI
=..!’YQ cos a, À2
=XYQ sin a,
À3
=ys B. (The space 6 (X ) is different from the physical space since X does not transform like a space vector.)
The classical adiabatic angle AX (1 ; T ) is given in reference [7] in terms of the flux of an
Abelian gauge field strength W (1 ; A ) through a surface S whose boundary is r :
where dS is the area element in the parameter space. In the case of one degree of freedom system the field strength W (1; A ) is given by :
Let po (X , I ; B ) and qo (X , I ; B ) be the dynamical variables corresponding to the limit
a =
0. The variables for the case a # 0 are readily obtained by writing :
To evaluate W we shall use cylindrical coordinates in the space t (A.) :
A scale transformation of B and yQ, B --> gB, -yo --> IL V’0’ can be accounted for if Vo and B are non zero by a change of scale of the time variable, t
-IL -1 t. So with no loss of generality we can keep p
=V q M fixed.
A convenient choice for the surface S is the cylinder with base Tand generatrixes parallel to
the 3-axis, truncated by the plane A 3
=ys B, where B, is an arbitrary large magnetic field.
When yQ is kept constant, the projection of rupon the plane A3
=0 is a circle of radius p, so
that the unit vector normal to S is the vector n. The scalar product W . n reduces to the simple expression :
The evaluation of the flux of W through the surface S is then readily performed and ones
arrives at the following expression :
When B, is arbitrary large with yQ fixed, the spin motion reduces to the simple Larmor precession. In such a case po is time independent and equal to I. So we can rewrite
,
