Physics Letters 161 (1991) 202-206 North-Holland PHYSICS LETTERS A

Floquet theory in phase space quantum mechanics for an electron in a rotating magnetic field

David J. Fernandez C . l and Luis M. Nieto Departamento de Fisica Te6rica, Universidad de Valladolid, 47011 Valladolid, Spain

Received 22 July 1991; revised manuscript received 29 October 1991; accepted for publication 31 October 1991 Communicated by J.P. Vigier

Some techniques of the Floquet theory are incorporated in the study of periodic Hamiltonians in the context of phase space quantum mechanics. The case of an electron trapped in a rotating magnetic field is solved explicitly.

The phase space formulation of non-relativistic quantum mechanics is a well established theory widely stud- ied in recent years [ 1-7 ]. In this approach the states and observables of the system are represented by functions defined on the appropriate phase space. The "twisted product" is defined between them to take into account the noncommutativity of the quantum observables. From these functions, all the physical information can be obtained.

We are going to apply the formalism to the study of a spinning charged particle in an external electromagnetic field with a periodic time dependence. The phase space that describes that system is ~:~6X S 2, parametrized by the coordinates (q, p, n ) = (u, n), ueR 6, n being the coordinates of a point on the sphere S 2.

In this case, the twisted product of f and g is

(fxg)(u,n)= ~ ~ f(u',n')g(u",n").~(u,n;u',n'; u", n") du'dn' du" dn", (1) ]q6 X S2 ~6X82

where the integral kernel LP is

LP(u, n; u', n'; u", n")

f 1 "~ 6 f2 i )/~J_~) 2 = [ ~-~,] exp~,~ (uVu'+u'Vu"+u"Vu)__ _ {1 +3(nn'+n'n"+n"n)+3x/~i[n, n', n"]} (2)

and J = ( _o / ), I being the 3 X 3 identity matrix. All the relevant physical information is contained in the "Moyal propagator" 3,(u, n; t), the analogue of the

evolution operator in the Hilbert space formalism. The dynamical evolution of the system is governed by the equivalent of the Schr6dinger equation,

O _ i -~--(u,n;t)=- ~H(t)X~(u,n;t), S(u, n ; 0 ) = 1 , (3)

H( t ) being the Hamiltonian function of our system. The time evolution of a general observable is given by

f(u, n; t)=3,*(u, n; t ) × f ( 0 ) × ~ ( u , n; t ) . (4)

On leave of absence from Departamento de Fisica, CINVESTAV IPN, Mexico.

202 0375-9601/91/$ 03.50 © 1991 Elsevier Science Publishers B.V. All rights reserved.

Volume 161, number 3 PHYSICS LETTERS A 30 December 1991

One can define the Fourier transform of E(u, n; t) with respect to time,

P(u, n; E) = ~ h 3(u, n; t) exp( i tE /h) d t , (5)

we call it the "spectral projector". It can be shown [3 ] that, for a time-independent Hamiltonian, the support on the variable E of this function provides the energy spectrum.

As it has been said, that formalism will be applied to an interesting example: an electron placed in a ho- mogeneous rotating magnetic field of the form

B( t )=B(cosco t , sincot, O), B > 0 . (6)

The Hamiltonian which describes the evolution of the particle is the Pauli Hamiltonian, which we can write in the phase space formalism as [ 4 ]

2

1 (p e 4 ( q , t ) ) _ ~ c B ( t ) W ( n ) = H o r b ( 1 ) + H s ( 1 ) (7) H ( t ) = ~ - c

where W ( n ) = ½x/~ n is the function associated to the spin operator S = ½or [1 ]. The orbital and spin coor- dinates are not mixed in (7). Therefore, the Moyal propagator can be expressed as the ordinary product of the orbital and spin propagators, which we will denote as ~(u; t) and ~(n; t) respectively:

~(u, n; t) =.E(u; t).V(n; t ) . (8)

Due to this fact, we can perform the analysis of the orbital and spin propagators separately. Let us study first the orbital contribution, .7(u; t). As a suitable vector potential generating (6), we choose

q3 sin cot ) A (q, t) = ½B - - q 3 COS cot .

\ - - q t sin cot+q2 cos cot

An explicit calculation shows that Horb(t) has the quadratic form

Horb(t) = ½utN(t)u,

with N(t) the 6 × 6 periodic matrix

N =

mb z sin2cot - mb 2 sin cot cos cot 0 0 0 - b sin cot\ /

- mb 2 sin cot cos cot mb 2 cos2cot 0 0 0 b cos cot

o 0 0 mb 2 b sin cot - b cos cot 0 0 b sin cot 1/m 0 '

0 0 - b cos cot 0 1/m - b sin cot b cos cot 0 0 0 l / m I

(9)

(10)

(II)

where b= I e lB/2mc. For a quadratic Hamiltonian there exist explicit formulae to compute the propagator [ 3 ]. The quantum motion (4) of a point in phase space follows the classical trajectory, which is expressed as a linear transformation on the initial conditions u(t)=27(t)Uo. The matrix 27(t) is symplectic and, from Ham- ilton's equations of motion, obeys the equation

~ r ( t ) = J N ( t ) S ( t ) , Z'(0) = I . (12)

It can be shown [ 3 ] that the solution to (12) leads to the explicit expression for ~(u; t):

( i ) ~=(u; t) = (det{½ [~r(t) + i ] } ) - , / 2 exp ~ utGu , (13)

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and

G=J(-V+ I ) - l ( ~ - - l ) = J - - 2 J ( . ~ + I) - t . (14)

To obtain (13) for our periodic Hamiltonian (10), we refer to the Floquet theory [8,9]: i f J N ( t ) has period T, then the solution Z'(t) to (12) admits the decomposition

S ( t ) = (9(t)e F' , ( 15 )

where (9 (t) is a periodic matrix such that (9 (t = n T) = (9 (t = 0) = L n~q and F is time-independent. From ( 15 ), an analogous decomposition can be found for .7(u; t):

-(u; t) =---~, (u; t) ×St(u; t ) , (16)

with Se (u ; t) and SF(U; t) obtained when both (9(t) and e F~ are substituted instead of 27(t) in (13) and (14). Note that S~,(u; t) is also periodic, while ,TF(U; t) represents a physical scenario described by the time- independent Hamiltonian HF= -½utjFu.

In our particular case, ( 11 ) has period T = 2n/to and the requirements to perform the Floquet decomposition are fulfilled. Due to the fact that the eigenvalues of the matrix J N ( t ) are constant, { - 2ib, 2ib, 0}, the Jordan form o f J N ( t ) is time-independent. We eliminate the time dependence o f J N ( t ) making the "transition to the rotating frame" [ 10,11 ], choosing as matrix (9:

:cos -sin 0 0 0 i) sin tot cos tot 0 0 0 0

0 0 0 1 0 0 6 o ( t ) = [ 0 0 0 c o s t o t - s i n t o t "

\00 0 0 sin tot cos tot 0 0 0 0

(17)

We define Z'~ ( t ) = (9-t ( t )X(t) , and using (12), we can show that Z'~ (t) satisfies the following differential equation,

S , ( t ) = F S ~ ( t ) , S t ( 0 ) = / , (18)

t ( t ) J N ( t ) (9 ( t ) - (9 - ~ ( t ) ~ ( t ) is a constant matrix of the form

to 0 1 / m 0 0

0 - b 0 1 / m 0

b 0 0 0 1 / m

0 0 0 to 0

- m b 2 0 - t o 0 - b

0 - mb 2 0 b 0

where F = (9-

0

- - t o

0 F =

0 0 0

(19)

The solution of ( 18 ) provides an easy Floquet decomposition in the form given by ( 15 ). Now we proceed to the spectral analysis of F and find the following behaviour for its eigenvalues: (1) Below a "critical value" for the parameter or---b/to, a < acr ~ 0.579982 [ 10], the six eigenvalues of F

are all distinct and imaginary: {_+ito~, +-ito2, +i to3} , 0 < t o l <0)2<093 • (2) For a > act, the six eigenvalues are distinct again, but two of them are imaginary while the other four

are complex of the form {2= +_p_+i% -+itoa}, with p > 0 and 0>0. (3) For ot = acr, there are just four imaginary eigenvalues { _+ ito2, -ito3}, the first couple with multiplicity

two and the last one being single roots. We will focus, for simplicity, on just the diagonalizable case a < act. The spectral form of e F' is

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3

eFt=S~(t)= ~ [exp(iCOfl)Pj+exp(--iCOfl)P~], (20) j = l

where Pj is the projector onto the subspace of R 6 spanned by the eigenvalue iCOj and P~' denotes the complex conjugated matrix to Pj. We have evaluated

3 3 GF=J(ert+I)-l(ert--I)= ~ [--tan(½COfl)Nj], {det[½(eVt+I)]} -'/2= 1-[ sec(½cofl), (21)

j = l j = l

where Nj= J (P j -P j ' ) / i are 6 X 6 real matrices. Inserting these results in (13), we obtained

3 ~F(U; t) = 1--[ see(½COjt) exp[ --i tan( ½cojt)Ajl , (22)

j = l

with Aj= utNju/h, A I, A3 being positive definite quadratic forms while A2 is a negative definite quadratic form. We consider now the spin part of the problem. The "transition to the rotating frame" means that, in (7),

Hs=hcoc( Wl cos cot+ W2 sin cot) can be expressed as

Hs =hcoc~3(n; t ) × IV, X ~ ( n ; t ) , (23)

where coc = I el B~ mc, we have used the relations Wk × WI= ½ iekt,~ Wm+ ~ 8kt [ 1 ] and

33(n; t) = ½ [exp(½icot) ( 1 --2 W3) +exp( -- ½iCOt) ( 1 +2 W3)] (24)

represents the rotation around the z-axis in the spin-space. If we express the Moyal symbol as

S(n; t) =~3(n; t)XY--.F(n; t ) , (25)

and use the Schr~dinger equation for ~(n; t), we find that NF(n; t) obeys

OF-F(n;t)=--i(COcWI--COW3)X~.F(n;t), ~F(n; 0) = 1 . (26)

TO solve this equation, we use the property that ~F(n; t) must be a linear combination of the Wj:

3 F.F(n;t)=go(t)+ ~ gj(t)Wj. (27)

j = l

Formula (26) leads to a matrix equation for the four-vector g t= (go, gl, g2, g3):

0 0 g=Fg, F = -iCOc 0 ½CO

0 -½CO 0 -½COo/' iCO 0 ½COc o /

with initial conditions g t ( t = 0 ) = (1, 0, 0, 0). The solution to (28), inserted in (27), gives

(28)

- 0 - - - ) - ) ~.F(n,t)=exp(½Wgot) coc Wl+ 09 W3 +exp(-½icoot) + coc W~- co W3 COO COO COO COO

(29)

where O9o = ~ + 092. Taking into account eqs. (8), (16), (25), we can express the complete propagator as

~(u, n; t) =~e (u; t)33 (n; t) ×3F(U; t)-Te(n; t) . (30)

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An in te res t ing a p p l i c a t i o n o f ( 3 0 ) is to o b t a i n the spec t ra l p r o j e c t o r PF(u , n; E ) a s soc ia t ed to the n o n - p e r i o d i c con t r i bu t i on , 3 r ( u ; t ) ~ r ( n ; t ) . Indeed , it is poss ib le to see tha t

S(u, n; t) ×Pr(u, n; E) = e x p ( -iEt/h)S,e.(u; t),E3 (n; t ) XPF(U, n; E ) . ( 3 1 )

F o r the set o f d i sc re te t imes t = nT, (31 ) r educes to

S,(u, n; nT) XPF(U, n; E) = e x p ( - iEnT/h )PF(U, n; E) . (32 )

In the case we are cons ide r ing , an expl ic i t ca l cu la t ion gives

PF(U,n;E)= ~ 8a, ~ (__l)l+,n+nexp[__(Al__A2+h3)]tl(2Aj)tm(__2A2)tn(2A3) ~ = + , - - h I,m,n=O

×d(E/h+ ½EOJo -o9~ (l+ i ) + 0)2 ( m + ½ ) -0)3(/ ' / - t- ½ ) ) , ( 3 3 )

where

The s up po r t on the va r i ab l e E o f P r ( u , n; E ) is

E=ho9, (l+ ½ ) -ho92(rn+ ½ ) +hfo3 (n + ½ ) ___ ½ho9 o . ( 3 4 )

No t i ce tha t this resul t is, essent ia l ly , the s ame as tha t o b t a i n e d in ref. [ 10 ]. We be l i eve this e x a m p l e i l lus t ra tes c lear ly an easy way to take in to a c c o u n t s o m e expl ic i t ly t i m e - d e p e n d e n t p r o b l e m s in the phase space a p p r o a c h o f q u a n t u m mechan ic s .

We acknowledge s u p p o r t f r om C I C Y T o f Spa in a n d Ca j a Sa l amanca . O n e o f the au tho r s ( D . J . F . C . ) wishes a lso to acknowledge C O N A C y T , Mexico , for f inanc ia l suppor t .

References

[ 1 ] J.M. Gracia-Bondia and J.C. V~irilly, Ann. Phys. 190 (1989) 107. [ 2 ] J.M. Gracia-Bondia and J.C. V~lrilly, J. Phys. A 21 ( 1988 ) L879. [ 3 ] M. Gadella, J.M. Gracia-Bondia, L.M. Nieto and J.C. V~irilly, J. Phys. A 22 (1989) 2709. [4] L.M. Nieto, J. Phys. A 24 ( 1991 ) 1579. [ 5 ] D.J. Fern~lndez C. and L.M. Nieto, Phys. Lett. A 157 ( 1991 ) 315. [6] G. Garcia-Calder6n and M. Moshinsky, J. Phys. A 13 (1980) L185. [7] M. Moshinsky and C. Quesne, Ann. Phys. 148 (1983) 474. [8] S.R. Barone, M.A. Narcowich and F.J. Narcowich, Phys. Rev. A 15 (1977) 1109, and references therein. [ 9 ] E.A. Coddington and N. Levinson, Theory of ordinary differential equations (Tata McGraw-Hill, New Delhi, 1972 ).

[ 10] B. Mielnik and D.J. Fern~indez C., J. Math. Phys. 30 (1989) 537. [ 11 ] B. Mielnik and D.J. Fern~indez C., Lett. Math. Phys. 17 (1989) 87.

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