fulltext

NOTE BREVI

Lie algebraic treatment of a charged oscillator

in the presence of a constant magnetic field

M. SEBAWEABDALLA

Department of Mathematics, College of Science, King Saud University P.O. Box 2455, Riyadh 11451, Saudi Arabia

(ricevuto il 12 Novembre 1966; revisionato il 17 Luglio 1997; approvato il 21 Agosto 1997)

Summary. — From the Lie algebraic approach the problem of a charged particle in

the presence of a constant magnetic field is treated. The most general solution for the wave function has been obtained. Some comments on the advantage of this technique compared with the classical method are given.

PACS 03.65.Ge – Solutions of wave equations: bound states.

1. – Introduction

The problem of a charged oscillator in the presence of a constant or variable magnetic field represents one of the most important problems which have attracted much attention in both quantum mechanics and electromagnetic theory [1-6]. Effort was concentrated on calculating the propagator in order to find the Bloch density matrix for such a system, either by using a Feynman path integral or by considering a semi-classical approach [7, 8]. In the meantime some authors have extended their discussions to include a damping factor in the Hamiltonian or in the Lagrangian to investigate the thermodynamic properties, by invoking the propagator to calculate the response function in Boltzmann statistics [9]. The problem has also been taken further to discuss some aspects related to the field of quantum optics [10, 11]. For example, the author of ref. [11] considered the photon numbers as well as bunching and anti-bunching besides some statistical investigations for the quasi-probability distribution function. Finally, we may refer to the usual classical approach which has been used to calculate the wave function for this problem in Schrödinger picture [8, 12]. In fact the general solution of the wave function using the classical method does not provide us with the most general solution. Therefore, one can immediately try to find another approach which leads to this solution. The aim of the present paper is to consider the Lie algebraic approach which represents a useful alternative to more classical methods, to find this general solution. To do so we shall write the Hamiltonian which represents the time-independent charged harmonic oscillators in the presence of

a constant magnetic field in terms of Dirac operators, thus [11] H ˇ 4 w(a † a 1b†b 11)1 (l/4 ) 2 w (a †2 1 a21 b†21 b2) 1i(l/2)(a†b 2ab†) , (1)

where w 4 (w1l2 O8w) and l is the coupling constant.

The above Hamiltonian can be regarded as a representation of the interaction between two modes a and b coupled together with the coupling parameter l (the cyclotron frequency). Note that we have considered the above system to be at exact resonance, so that the frequencies of each mode are equal.

As we have mentioned above, the main purpose of the present work is to find the most general solution of the Schrödinger equation

Hc 4iˇ¯c

¯t , c f c at t 40 , (2)

by employing the Lie algebraic approach as used in refs. [13-18]. For this reason let us rewrite the Hamiltonian (1) as follows:

H

ˇ 4 w K11 2 m(K21 K3) 1 (l/2)K4, (3)

where we have defined

K14 (a†a 1b†b 11) , (4a) K24 1 2 (a †2 1 b†2) , (4b) K34 1 2 (a 2 1 b2) , (4c) K44 i(a†b 2ab†) , (4d) and m 4 (l/4)2

Ow. In the following and for convenience the constant ˇ will be taken equal to unity.

The operators K1, K2, K3and K4are generators of su( 1 , 1 ) and form a closed set of

the commutation relations, such that

[K1, K2] 42K2, (5a) [K1, K3] 422K3, (5b) [K2, K3] 42K1 (5c) and [Ki, K4] 40 , i 41, 2, 3, 4 . (5d)

Here we can say that, since su( 1 , 1 ) is a non-compact Lie algebra, then all finite-dimensional representations are non-unitary representations of Lie algebra, and hence we find that the simplest solution of the above commutation relations is

K14 . ` ´ 1 0 0 21 ˆ ` ˜ , K24 . ` ´ 0 0 1 0 ˆ ` ˜ , K34 . ` ´ 0 21 0 0 ˆ ` ˜ , K44 . ` ´ 1 0 0 1 ˆ ` ˜ . (6)

These matrices are linearly independent, and can be regarded as a subgroup of some complex Lie group, which in fact are real. This can be realized through the matrix representation of K2and K3, for more details see ref. [19]. In the meantime we find the

operator K4 is actually a constant of the motion and has been represented by the unit

matrix. This in fact would enable us to find the matrix image of the partial differential equation (2) and to overcome the difficulty of finding the factorization of the exponent operator exp [2itH]. This will be seen in the subsequent calculations. Using the matrix representation given by eq. (6) we can write the matrix image of the partial differential equation (2) in the form

i¯c ¯t 4 . ` ´ w 1l/2 22 m 2 m 2 w 1l/2 ˆ ` ˜ c (7) with c 4.` ´ c1 c2 ˆ `

˜. After straightforward calculations we find that the general solution for

eq. (7) is

c1(t) 4exp [2ilt/2]

k

g

cos Vt 2i

w

Vsin Vt

h

c1( 0 ) 2 2 im

V sin Vtc2( 0 )

l

, (8a)

c2(t) 4exp [2ilt/2]

k

g

cos Vt 1i

w

Vsin Vt

h

c2( 0 ) 1 2 im

V sin Vtc1( 0 )

l

, (8b)

where the frequency V is given by V 4 (w2

1 l2/4 )1 /2_.

(9)

Now notice that the Hamiltonian operator H belongs to the Lie algebra spanned by K1, K2, K3and K4. Therefore, the operator exp [2itH] can be expressed as

exp [2itH] 4exp [a(t) K1] exp [b(t) K2] exp [g(t) K3] exp [d(t) K4] ,

(10)

where a(t), b(t), g(t) and d(t) are functions of t to be determined. It is not easy task to find the functions a(t), b(t), g(t) and d(t) in the Lie algebraic method. However, with the aid of ref. [15, 16] we get closed-matrix representation for exp [a(t) K1],

exp [b(t) K2], exp [g(t) K3] and exp [d(t) K4] as follows:

exp [a(t) K1] 4 . ` ´ exp [a(t) ] 0 0 exp [2a(t) ] ˆ ` ˜ , (11a) exp [b(t) K2] 4 . ` ´ 1 0 b(t) 1 ˆ ` ˜ , (11b) exp [g(t) K3] 4 . ` ´ 1 2g(t) 0 1 ˆ ` ˜ , (11c) exp [d(t) K4] 4 . ` ´ exp [d(t) ] 0 0 exp [d(t) ] ˆ ` ˜ . (11d)

From eqs. (8), (10) and (11) we may obtain the following expressions for a(t), b(t), g(t) and d(t): a(t) 4ln

g

cos Vt 1i w Vsin Vt

h

21 , (12a) b(t) 422imsin Vt V

g

cos Vt 1i w Vsin Vt

h

, (12b) g(t) 422imsin Vt V

g

cos Vt 1i w Vsin Vt

h

21 (12c) and d(t) 42ilt/2 . (12d)

Now consider the actions of the one-parameter subgroups exp [2itK1], exp [2itK2],

exp [2itK3] and exp [2itK4] on functions fi(q1, q2, t), i 41, 2, 3, 4 such that

exp [2itKi] fi(q1, q2, 0 ) 4fi(q1, q2, t) .

(13)

The above actions of one-parameter subgroups are equivalent to the solutions of four initial-value problems which will be treated as follows.

The functions fi are in fact solutions to the following system of partial differential

equations:

Kifi4 i

¯fi

¯t i 41, 2, 3, 4; (14)

using the definitions of the operators Kias in eq. (4) we obtain the solutions for eq. (14):

f1(q1, q2, t)4

!

j, k40 Q CjkHj(kw q1) Hk(kw q2) exp

k

2 w 2 (q1 2 1q22) 2 i 2 ( j1k11) t

l

, (15a) where Cj , k4 (15b) 4 w p 2 2( j 1 k)_{( j! k! )}21



2Q Q f1(q1, q2, 0 ) Hj(kw q1) Hk(kw q2) exp

k

2 w 2 (q1 2 1 q22)

l

dq1dq2

and H(Q) stands for the Hermite polynomial, f2(q1, q2, t)4

!

m, n42Q Q Cmnexp

k

w 2 (q1 2 1 q2 2 ) 1i kw (mq11 nq2) 1 i 4(m 2 1 n2) t

l

, (16a) where Cmn4 (16b) 4 w ( 2 p)2



2p/kw p/kw



2p/kw p/kw f2(q1, q2, 0 ) exp

k

2 w 2 (q1 2 1 q22) 1i kw (mq11 nq2)

l

dq1dq2.

The function f3(q1, q2, t) has been calculated to take the form f3(q1, q2, t) 4

!

r , s 42Q Q Crsexp

k

2 w 2 (q1 2 1 q22) 1i kw (rq11 sq2) 1 i 4(r 2 1 s2) t

l

, (17a) where Crs4 w ( 2 p)2



2p/kw p/kw



2p/kw p/kw f3(q1, q2, 0 ) exp

k

w 2 (q1 2 1 q22) 1i kw (rq11 sq2)

l

dq1dq2. (17b)

Finally, the function f4(q1, q2, t) can be written as

f4(q1, q2, t) 4

!

l 40 Q Clexp

[

2l[ tan21(q2/q1) 1t]

]

, (18a) where Cl4 1 p



2Q Q

f4(q1, q2, 0 ) exp [l tan21(q2/q1) ] exp [2(q121 q22) ] dq1dq2.

(18b)

Now the most general soluton for the wave function “Schrödinger representation” can be obtained from eqs. (15)-(18) together with eqs. (12) to take the following expression

c(q1, q2, t) 4

!

l , j , k 40 Q

!

m , n , r , s 42Q Q ClCmnCrsCjkHj(kw q1) Hk(kw q2) 3 (19) 3exp

k

2w 2 (q1 2 1 q22) 1i kw

(

(r 1m) q11 (n 1 s) q2

)

2 l tan21(q2/q1)

l

3 3exp

k

2 i 2( j 1k11) a(t)1 i 4(m 2 1 n2) b(t) 1 1 4(r 2 1 s2) g(t) 1 i 2llt

l

. In finding the above equations we have used the Dirac representation for the operators a and b such that

a 4 (2w)21 /2_(wq 11 iP1) , (20a) b 4 (2w)21 /2_(wq 21 iP2) , (20b)

which satisfies the commutation relations [a , b] 40, and [a, a†

] 41 4 [b, b†_].

The algebraic technique which we have used to derive eq. (19) provided us with a powerful and feasible tool to solve a wide class of partial differential equations, and this can be considered as a useful alternative to more classical methods which do not provide the same generality.

To see that, let us concentrate on solving the wave function in Schrödinger picture classically; this can be done if one uses eqs. (1), (2) and (20). In this case we have

¯2c ¯q12 1 ¯ 2_c ¯q22 2 V2(q121 q22) c 1il

g

q2 ¯c ¯q1 2 q1 ¯c ¯q2

h

4 22 i ¯c ¯t . (21)

After some calculations the result becomes c(q1, q2, t) 4

!

m 8, n 840 Q NH_{n 8}[kV(q1cos lt/2 2q2sin lt/2 ) ] 3 (22a)

3Hm 8[kV(q2cos lt/2 1q1sin lt/2 ) ] exp

k

2

V 2 (q1 2 1 q22)

l

exp [2iV(m 81n 811) t] , where N 4

g

V p

h

(m 8 !n 8 ! ) 21₂2(m 8 1 n 8 )



2Q Q



2Q Q c(q1, q2, 0 ) exp

k

2 V 2 (q1 2 1 q22)

l

3 (22b)

3Hn 8[kV(q1cos lt/2 2q2sin lt/2 ) ] Hm 8[kV(q2cos lt/2 1q1sin lt/2 ) ] dq1dq2.

The comparison between the two eqs. (19) and (22) shows that the wave function in the classical approach does not contain all the possibilities that appeared from the Lie algebraic approach. Also the classical method shows complication in the argument of the Hermite polynomials, while the other method avoids this complication. Therefore, we can state that the result we have obtained here, using Lie algebraic methods, could have been obtained using conventional methods [16, 20]. However, in practice, because of their concise and modular nature, the use of algebraic approaches makes it possible to obtain results well beyond those currently available by any other method.

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