G. DATTOLI (1), S. LORENZUTTA(2), G. MAINO (2) and A. TORTE (1). (1) ENEA~ Dipartimento Innovazione, Divisione Fisica Applicata - Frascati, Ronur Italy.
IL NUOVO CIMENTO
VOL. 111B, N. 12
Dicembre 1996
NOTE BREVI
Biorthogonal bases for coupled harmonic oscillators G. DATTOLI(1), S. LORENZUTTA(2), G. MAINO(2) and A. TORTE(1) (1) ENEA~ Dipartimento Innovazione, Divisione Fisica Applicata - Frascati, Ronur Italy (2) ENEA, Dipartimento Innovazione, Divisione Fisica Applicata Via Don G. Fiammelli 2, 1-40129 Bologna, Italy (ricevuto il 14 Febbraio 1996; approvato il 30 Agosto 1996)
Summary. - - We show that a generalized class of harmonic-oscillator wave functions can be exploited in order to study physical problems involving coupled harmonic oscillators. We further discuss the biorthogonal properties of these states, the relevant creation and annihilation operators and some physical consequences of the formalism. PACS 02.30.Gp - Special functions. PACS 03.65.Ca - Formalism.
Biorthogonal bases have been introduced to deal with physical problems where use is made of non-Hermitian operators, usually within the context of open dissipative systems [1]. We discuss a non-dissipative conservative problem, which can be naturally treated by means of a biorthogonal set of suitable eigenfunctions. It will be proved that quantum systems described by coupled harmonic-oscillator Hamiltonians can be diagonalized using a new set of generalized harmonic-oscillator functions belonging to a biorthogonal basis. It is worth recalling that, given two sets of vectors vm and w~ (m, n = 0, 1), respectively, they form a biorthogonal system if
(1)
v , . " W n = 5m,,~ ,
where the dot refers to the ordinary scalar product between vectors and 5~, ~ is the Kronecker symbol. The usual Gram-Schmidt orthogonalization procedure [2] can be generalized in a suitable way to construct a biorthogonal basis. Considering two sets of linearly 1529
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G. DATTOLI, S. LORENZUTTA, G. MAINO
and A. TORRE
independent vectors, x.. and y . . the following sets can be defined: Vl ----X1 , ,
W l ----Yl ,
m ~ l Xm .W~ ~ - - V jP j = 1 W j 9 Vj
_
U m ----X~,
(2) ,
I
i
W , = y ~ -j=l
, ,
~noV j W ! - - , .7 ' Wj .Vj
and it is easily verified that the vectors
(3)
t
W~
t r! r
Wn ./
form a biorthogonal basis according to eq. (1). After these preliminary remarks, we address the main topic of this work by introducing the following functions:
K , , . . ( x , y) -
X/~
~
exp - l z ~ f / l z
,
(4)
where
(5)
z:(;)
:)
(a, c > O; A = ac - b 2 > 0),
and H.,. ,r and G ~ . . are generalized Hermite polynomials defined as follows [3]:
(6)
H., ~(x, y) = ( - 1 ) "~+'' exp
zt f/lz %x ''~ %Y" exp - - z t f / l z 2 '
G .....~(x, y) -- ( - 1)'" § '~ exp
wtM lw %~mD~" exp - l w ~ M - l w 2
'
Functions of eq. (4) can be viewed as generalized harmonic-oscillator eigenfunc-
BIORTHOGONM~ BASES FOR COUPLED HARMONIC OSCILLATORS
1531
tions and the associated creation and annihilation operators are, respectively,
by)
a l , + = 2 ( a x JF
-
~x
(7) ~ , _ = ~ \C ~x -
+ -2 '
1
~
a2,+= ~(bx+cy)
,
~y
(8) l(b ~
a ~ )
Y
and satisfy the commutation relations (9)
Jar,+,
as,_]
(r, s = 1, 2).
~- - - d r , S
Moreover, they act on K~n,,~ as follows:
{
al, +Km, n = grm + l K m + l , n ,
(10)
g~, +K,..,,~
V~+
a,,
V~K,,,-,,~,
K.~,,~
[t2, - Km, n
1K.,,,.,+I,
~F~ Km, n_ l .
Similarly, the operators
al'* =
3 ~ ~
--
(11) 1 al, -
'~ Z
a
(ax + by) + : - , OX
1 A
(12) 62, _
~-y ] -}- -'2
(
bO 3x
+~, ~y
2
1 A ~ ( bx + cy ) + ~Y
act on F~, n in an analogous way as eqs. (10). It is worth pointing out that operators (7), (8), (11), (12) satisfy the relation (13)
~, _+ = ~r, 9
where the superscript t denotes the operation of Hermitian coniugation. Furthermore, the following number operators can be introduced: (14)
al, + al, - = Kt,
a2, + 62, - =
(r = 1, 2),
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G. DATTOLI, S. LORENZUTTA, G. MAINO ~tlad A. TORRE
and the relation
(15)
(a~. + al,
+ [i2, + a2, -. ) K , . , . = ( M + n ) K,~,.,.
proved to correspond to the partial differential equation
--(~r
(16)
3q)~4 1
K,, , ( m + n + 1)K,,
+ lZt~/lz
~x~th
-(3,.
(17)
3,~)1~ -1
= -~-
c---2b
Dx"
Dx Dy
+a--
3ye
'
being the generalized Laplacian operator. By defining number operators for the 5~ set, we can also prove that F ...... satisfy the same equation (16). Since the term (18)
l ztf/i z = 1 (ax~ + 2 b x y + cy "e) 4 4
in eq. (16) plays the same role as a two-variable coupled harmonic potential, it is possible to exploit the use of both K ...... and Y ...... in order to diagonalize a coupledoscillator Hamiltonian whose eigenvalues are (m + n + 1). Analogously to the ordinary one-dimensional case, m and ~ can be considered excitation numbers. The open problem is whether m and ~ are observables or only their sum is. According to eqs. (13) and (14), we find (~,. + a,. _ )" = &. + s
(19)
_ ,
and we c a n n o t conclude that
(20) On the other side, it is easy to prove that
(21)
(al. + al,
-]- a., + ae, - ) ; K ...... = (m + n)K,.~, ,,
and analogous considerations hold for F,,, ,~. Therefore, one can state that m and n are not Hermitian operators in the canonical sense since they are not individually obse~cables but only their sum, (m + n), is a Hermitian operator commuting with the total Hamiltonian. This conclusion may suggest further speculations about the non-observability of internal degrees of freedom in microscopic systems described by coupled harmonic oscillators. The mathematical by-product of the shown non-hermiticity of the 5 and operators is the fact that K,,, ,7 and F,,~,, functions are biorthogonal with respect to
BIORTHOGONAL BASES FOR COUPLED
HARMONIC
OSCILLATORS
1533
each other according to the following relation: (22)
§
+~
f
I dxdy['m'n(X' Y)Kr's(X' y) -~ (~m'r(~n's "
Then, within the Dirac formalism, we label a number state of the coupled harmonic-oscillator system with the symbol, Ira, n). Analogously to the ordinary case, this state can be generated from the quantum vacuum by acting [4] with the creation operators (23)
[m, n} = ~
1
(a~, + )~'~(a2, + )~ (0, 0}.
The states biorthogonai to those of eq. (23) are defined according to the relation (24)
Ira,
n)'-
~
1
(~, + )'~(52, + )~ 10, 0)'.
It is worth remarking that in the (x, y) configuration space states (23) and (24) have the simple form
(25)
(x, Y lm, n},= K~. ~ (x, y), (x, Y lm, n} = Fm,,~ (x, y),
and the vacuum state provided by (26)
(x, ylO, O) = Ko, o(X, y) = (x, ylO, O)'= Fo, o(x, y) =
is self-conjugated as can be immediately verified from eq. (4). Finally, according to the above formalism, it is also possible to define generalized coherent states satisfying the following uncertainty relation: (27)
12:'//} = 1 ~ 4 '
where I is the identity operator and
(28)
-~o -~ + ~
-~
dxdyro, o(X,y)z Ko, o(X,y), +
-zv
These aspects of interest in many optical problems such as the treatment of multimode mixed light, will be deepened in a forthcoming longer paper.
This work has been performed in the frame of ENEA-MURST research project on Nonlinear and Complex Systems.
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G. DATTOLI, S. LORENZUTTA, G. MAINO
and
A. TORRE
REFERENCES [1] KIM Y. S. and Noz M. E., Phase-Space Picture of Quantum Mechanics (World Scientific, Singapore) 1991. [2] See, for instance, KOL~aOGOROVA. and FOmNE S., Elements de la th~orie des fonctions et de t'analyse fonctionelte (MIR Publishers, Moscow) 1974. [4] DATTOL1 G., LORENZVTTA S., MAIN0 G. and TORRE A., Or. Math. Phys., 35 (1994) 4451; DATTOLI G., CHICCOLI C., LORENZUTTAS., MAINO G. and TORRE A., Comp. Math. Apptic., 28 (1994) 71. [4] DATTOLI G., LORENZUTTA S., MAINO G. and TORRE A., Ann. Numer. Math., 2 (1995) 211.
