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Eigenvalue problem for arbitrary linear combinations of a boson annihilation and creation operator.

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Ann. Physik 1 (1992) 181 -197
Annalen
der Physik
0 Johann Ambrosius Barth 1992
Eigenvalue problem for arbitrary linear combinations
of a boson annihilation and creation operator
Alfred Wiinsche
Arbeitsgruppe “Nichtklassische Strahlung” der Max-Planck-Gesellschaft, Rudower Chaussee 5 ,
0-1 199 Berlin, Germany
Received 5 December 1990, resubmitted 7 February 1992, accepted 3 March 1992*
Dedicated to the 60th birthday of Prof. Dr. Harry Paul
Abstract. The eigenvalue problem for arbitrary linear combinations Ka + put of a boson annihilation
operator (I and a boson creation operator ut is solved. It is shown that these operators possess
nondegenerate eigenstates to arbitrary complex eigenvalues. The expansion of these eigenstates into the
basic set of number states 1 n ) , (n :0, I , 2, . . . ), is found. The eigenstates are normalizable and are
therefore states of a Hilbert space for I ( I < I with [ = P / K and represent in this case squeezed coherent
states of minimal uncertainty product. They can be considered as states of a rigged Hilbert space for
1 [ I 2 1. A completeness relation for these states is derived that generalizes the completeness relation
for the coherent states 1 a ) . Furthermore, it is shown that there exists a dual orthogonality in the entire
set of these states and a connected dual completeness of the eigenstates on widely arbitrary paths over
the complex plane of eigenvalues. This duality goes over inlo a selfduality of the eigenstates of the
hermitian operators K U + K* u’ to real eigenvalues. The usually as nonexistent considered eigenstates
of the boson creation operator at are obtained by a limiting procedure. They belong to the most singular
case among the considered general class of eigenstates with ( = P / K as a parameter.
Keywords: Boson operator; Eigenvalue problem; Squeezed coherent states; Dual orthogonality and
completeness.
1 Introduction
Undoubtedly, the two most important sets of basic states of a single boson system in
quantum optics are formed by the occupation number states (or Fock states) 1 n ) , ( n =
0, 1, 2, . . .), and by the coherent states (or Glauber states) I a), (a arbitrary complex
numbers). It is well known that these two sets of states can be constructed by the solution
of two interesting eigenvalue problems. The number states I n ) are the normalized
eigenstates of the (occupation) number operator N = a’a to eigenvalues n, where 4
denotes the boson annihilation operator and at the corresponding boson creation
operator satisfying the commutation relation [a, a’] = I ( I unity operator in a Hilbert
space 2”).They were introduced by Fock [ l ] as the basic quantum-mechanical states in
the second quantization of boson systems for the representation of states with variable
or indefinite numbers of identical particles (Fock representation). By a well-known
analogy of a single boson system to a quantum-mechanical harmonic oscillator, they are
* Editor’s note: The regrettable delay in publication is due to the extensive reorganization of the journal.
182
Ann. Physik I (1992)
also the energy eigenstates of the harmonic oscillator. Glauber [2, 31 was the first who
realized that the coherent states I a ) which were known in another form from the work
of Schrodinger [4] as displaced wave packets of the fundamental Gaussian wave function
of the harmonic oscillator can be constructed as the normalized eigenstates of a boson
annihilation operator a to eigenvalues a. Apart from the great importance of the coherent
states for the description of the fields of laser modes [ 5 ] , they provided a first example
for the complete explicit solution of an eigenvalue problem for a nonhermitian (and also
nonunitary) operator in quantum mechanics which had attracted little attention before.
On the other side, it was proved soon after the cited works of Glauber that the eigenvalue
problem of a boson creation operator at cannot be solved with (normalizable) states in
the Hilbert space W in case of finite eigenvalues (Miller and Mishkin [6], Klauder and
Sudarshan [7] chap. 7, $ 2 A , PeTina [8] chap. 13.1).
The single boson system or the harmonic oscillator can equivalently be described also
by t w o hermitian operators Q and P that form a pair of canonical operators (generalized
position and momentum operator, respectively) given by linear combinations of the
operators a and at and satisfying the commutation relation
PI = i h l . Each of the
operators Q and P has a continuous spectrum of real eigenvalues to eigenstates which are
not normalizable in the usual way and, therefore, do not belong to the Hilbert space X
However, they are mutually orthogonal for different eigenvalues and normalizable by
means of the delta function (see, e. g., Messiah [9]). These sets of eigenstates form also
complete sets of states for the representation of arbitrary states.
It seems to be difficult to understand not only formally why the operators a, at, Q and
P have so different kinds of solution of the eigenvalue problems. A step towards a deeper
understanding of this intriguing behaviour can be made by solving more general
eigenvalue problems for operators which depend on continuous parameters and comprise
the numbered operators for specified values of the parameters. In the present paper, we
solve and discuss the eigenvalue problem of the operator ~a + pat with arbitrary
complex parameters K and p and specify the solution to the corresponding solutions for
the operators a, at, Q and P and derive completeness and orthogonality relations for the
eigenstates.
A particular problem of this kind is solved in the theory of squeezing by a unitary
transformation of the annihilation operator a with unitary squeezing operators [lo- 151.
The corresponding eigenstates are squeezed coherent states obtainable from the coherent
states by applying these squeezing operators. However, the mentioned unitary
transformation of the operator a leads to a new operator K U + pat with the restriction
K K * - yy * = 1 and, therefore, cannot provide the solutions of the eigenvalue problem,
for example, for the hermitian operators Q and P and for the creation operator at in a
unique manner. For this purpose, it is necessary to use a nonunitary formalism leading
to another and even more simple parametrization of the corresponding eigenstates than
the parametrization of the squeezed coherent states. The connection between these two
different approaches will be established in Section 4 of the present paper.
[a
2 Novel solution of the posed eigenvalue problem
The set of eigenstates to arbitrary linear combinations ~a + pat of boson operator a
and at with complex coefficients K and p is essentially the same as for the operators a
+ ~ U / K at. We solve the considered eigenvalue problem therefore in the following
standardized form
A. Wiinsche, Eigenvalue problem for arbitrary linear combinations. . .
183
that is equivalent to
The most singular case K = 0 will be dealt with as a limiting case. The coherent states
I a ) must be proportional to the states I a; 0) of our more general eigenvalue problem.
Now, we make an expansion of the states I a; c) into the basic set of Fock states I n )
using their orthonormality and completeness (e. g. [ 2 , 31)
Taking into account the well-known relations for the action of the boson operators onto
the Fock states (e.g. [ 2 , 3]), one obtains the following recurrence equation for the
coefficients of this expansion
1/.+1(n +
1 1 a;
c) + f i c ( n
-
I
1
a; [)
=
a ( n / a; 0.
The only solution of this equation for the coefficients ( n I a;
coefficient (0 1 a; 5 ) is (proof by complete induction)
c) expressed
(2.4)
by the
where H,,(z) denotes the Hermite polynomials and where the roots l/z3 must be taken
with the same but arbitrary sign. The arbitrariness of the coefficient (0 I a; [)in Eq. (2.5)
can be used for the normalization of the solution in Eq. (2.3) by suitable choice, but this
is only possible for I [ I < 1 as we will later see. For our purposes it is better to renounce
on the normalization and to choose
(0 1 a;
c) = 1 .
(2.6)
The solution for the eigenstates I a; 4') of the considered eigenvalue problem can then be
represented in the forms
(2.7)
where we used the well-known generation of the Fock states I n ) from the vacuum state
10) by applying the operator l/Fat". Now, by using the generating function of the
Hermite polynomials (see, e.g. [17] chap. 10.13, formula (19)), one obtains from
Eq. (2.7) the following representation of the solution
184
Ann. Physik 1 (1992)
This representation can more directly be obtained from the operator identity
by applying it to the vacuum state I 0) after multiplication from the left with the operator
exp(aat - C/2 at’). The operator identity in Eq. (2.9) can be derived as a consequence
of the basic commutation relations for boson operators. Another very useful
representation of the states 1 a; 5) can be obtained from the relation of the Hermite
polynomials to derivatives of a Gaussian function (see, e. g., [ 171 chap. 10.13 formula (7))
( 2 . 10)
Let us now discuss the solution.
The above formulae determine to arbitrary complex eigenvalues a and complex values
of the parameter < a nondegenerate solution I a; <>ofthe eigenvalue problem in Eq. ( 2 .I ) .
According to Eq. (2.8), this solution can be generated from the vacuum state 10) by
applying the nonunitary operator exp(arrt - [ / 2 at2).This is the main difference to the
usual approach in the theory of squeezed coherent states where unitary squeezing
operators are applied to coherent states. We come back to this point in Section 4 and
accomplish the connection between both approaches. The present approach leads to a
more simple parametrization of the eigenstates and gives the possibility to go beyond the
condition of normalizability of these states. One further advantage of our convention is
that the states a; 5) depend only on the complex variables a and (‘ and do not depend
on the complex conjugate variables a* and [* as it is the case with normalized solutions.
From Eq. (2.8) it follows
( 2 .I 1)
The states I a ; 0) are nonnormalized coherent states I a ) according to
(2.12)
that are sometimes denoted by I 1 a ) (see [2, 31). The states 10;
nonnormalized squeezed vacuum states (see also Section 4).
5) are for I ( 1 <
1
3 Basic properties of the states I a; [)
The states I a; 5) are regular (normalizable) states of the Hilbert space k of the single
boson system if their norm
exists, i.e. if the scalar product under this
root has a finite positive value. One can calculate more generally the following scalar
)‘vu
A. Wunsche, Eigenvalue problem for arbitrary linear combinations. . .
185
product by applying a formula of Mehler (see, e. g. Bateman and Erdelyi [17] chap. 10.13,
formula (22), see also remarks in Section 6 of this paper for a possible proof of this
formula)
The sum in Eq. (3.1) converges in a regular sense to the right-hand side of this equation
only for I [['* I = I [ I I 5' I < 1. Hence, the states I a; [> are normalizable only for I [ I
< 1. Nevertheless, one can give the states I a; C>a well-founded sense also for I [ I I I
as states of a rigged Hilbert space as we will discuss in Section 5 . As the coherent states,
the states I a; [>
are nonorthogonal to each other for different a and the same value of
the parameter c.
The action of the creation operator at onto the states I a; [> can be substituted by the
differentiation of these states with regard to the variable a as can be seen from Eq. (2.8).
In connection with Eq. (2. l), one finds
and the corresponding adjoint relations
The last relations give rise to the following representation of the basic boson operators
a and at
a
a a*
a+-,
at+.*
-
a
C* aa* , [a: at] = I-+ 1,
in the sense
where I w) denotes arbitrary states for which the scalar product (a; [ I
differentiable function of a* for fixed values of C.
From Eqs. (2.1 l ) , (3.2) and (3.3) it follows
I a; [ > ( a ; [ I
= exp
(
-
c
-a2 -
2
aa2
'*
w> exists as a
)
-a2
l a ; o>(a;0 1 .
2
aa*2
The well-known completeness relation for the coherent states 1 a> [2, 31 can now be
converted into a completeness relation for the states I a; [> in the following way (i/2 d a
A
da*
E
d2a)
186
I
Ann. Physik 1 (1992)
t
i
= - f - da A
=
7 T 2
i
Sda
2
A
da*
d a * exp(-aa *) I a; 0 ) (a; 0 I
I a; [)(a; { I
exp
(3.7)
It can be proved by complex Fourier transformation that the action of the operator
exp(c/2 a2/aa2 + &*I2 a2/aa*2) onto functions of a and a* is equivalent to a
convolution with a certain Gaussian function. Inserting the result of this convolution into
Eq. (3.7)’ one obtains the following completeness relation for the states I a; c)
1
1
f-da
2
A
da*exp
2 a a * - (c*a2
2(1 -
+
cr*)
As in the case of coherent states that is involved in Eq. (3.8) for [ = 0, the states I a; [>
form not only a complete set of states but an overcomplete set of states. Each vanishing
nontrivial linear combination of the coherent states can be turned into a corresponding
combination of the states I a; c).
The expectation values of normally ordered powers of the boson operators can be
calculated by using Eqs. (3.2) and (3.3) in the following way
Thus, one obtains for the expectation values of the boson operators a and at
(a; [ l a l a ; i>
a - [a*
(a; i l a + I a ; i>
- a* - i * a
(a; iI a; C>
1 - i ~
’ *
(a; [ l a ; i>
1 - ci* ’
(3.10)
and for the expectation values of products of two boson operators
We introduce the following pair of canonical conjugated hermitian operator Q(v,) and
P ( p ) depending on an angle v,
A. Wunsche, Eigenvalue problem for arbitrary linear combinations. . .
187
(3.12)
P(q) = - i
8
- (aefiP -
at e - j v ) = Q(0) sinp
+ P(0)cosq.
Then, one finds from Eqs. (3.10)- (3.12) the following uncertainty product
(3.13)
c*/[
Choose the angle q by the condition ei4v =
with four possible solutions, and the
uncertainty product becomes the minimal possible one equal to h2/4. This means a
definite choice of the axis positions in the phase plane. The states I a; () are for I I <
1 nonnormalized squeezed coherent states of minimal uncertainty product. The
uncertainty product is in general not an invariant under rotations of the axes in the phase
plane.
r
4
Connection of I a; 4) to squeezed coherent states in the usual parametrization
c
We establish now shortly the connection of the states I a; 4') for I I < 1 to squeezed
coherent states in the common treatment and with the common parametrization (e. g.
[lo- 161). Introduce the following in general nonunitary squeezing operators S ( ( , q, r )
depending on three complex parameters (, q and
r
s((,q, r )
=
a2 + iq
exp
1
+ ata) -
- (sat
2
(4.1)
The totality of these operators forms a Lie group that is the complex extension of the twodimensional unimodular group or equivalently of the two-dimensional symplectic group
( S L (2, C) Sp(2, C)). The fundamental two-dimensional representation of this group
in the basis of the boson operators a and at is obtained by the following similarity
transformations of these operators
-
S ( ( , rl, r ) (a, at) ( S ( ( , q, a)-'
=
(a, at)
(,": t)
.
= (Ka
+ put, IZa + vat),
(4.2)
where the unimodular matrix of this representation is given by the relations
e
&
r-
sh E
e
, che
e
+ i q JhEJ
with the following inverse connection to the parameters [, q and [
(4.3)
188
Ann. Physik 1 (1992)
The solution of the eigenvalue problem for the operator K a + pat can now be obtained
by means of the coherent states I y > and of the squeezing operator S ( < , rl, r ) in the form
of Eq. (2.2)
By choosing
one obtains the solution of the eigenvalue problem from the above formulae in the form
(4.7)
that is equivalent to the solution of Eq. (2.1) by the state I a; (>in Eq. (2.8) (up to
normalization).
The usual approach with unitary squeezing operators S ( < , q, r) is obtained by the
restrictions
i = <*, rl
= rl*,
I ; ( ;)
+
=
("3
"*)
PlK*
,
K K * - ",Ll*
= 1,
(4.8)
The corresponding Lie group of squeezing operators is now the two-dimensional real
unimodular or symplectic group ( S L ( 2 , R )
Sp(2, R )
SU(1, 1)). An equivalent
eigenvalue problem to Eq. (2.1) solvable with unitary squeezing operators applied to
coherent states can only be obtained by choosing
-
-
where the angle x is a freely chosen real parameter. By comparison of the Fock-state
representation of the states 1 a; [>
given in Eq. (2.7) with the Fock-state representation
of the squeezed coherent states obtainable from the normally ordered representation of
the squeezing operator, we obtained the following connection
(4.10)
A. Wunsche, Eigenvalue problem for arbitrary linear combinations. . .
189
with the following abbreviations in the arguments of the unitary squeezing operator in
accordance with Eq. (4.4)
(4.1 1)
On the left-hand side of Eq. (4.10) it is written the normalized state I a; (>according to
Eq. (3.1) that is only possible for I ( I < 1. The state I y ) on the right-hand side of
Eq. (4.10) with y according to Eq. (4.9) means a coherent state. The above equations
simplify a little bit by special choice of the angle x with the possibilities x = 0, x = n,
x = k n/2 and sinx = ? 1 (1.
5 The states I a; 6 ) as states of a rigged Hilbert space
The states I a; [) are elements of the Hilbert space ,R of the single boson system that
is here the Fock space only for 1 [ 1 < 1 because only in this case the scalar products (a;
[ I a; [) are finite and positive. However, they can be considered as elements of a certain
rigged Hilbert space for arbitrary (. The idea of the rigged Hilbert space was elaborated
by Gelfand and Vilenkin [ 181 and can be found in a short and readable form in [I 91 and
more detailed in [20]. We will sketch here only some basic ideas in application to our
problem.
As is well-known, the Hilbert space is a countable (finite or infinite-dimensional) linear
space with a scalar product definition (a, I w) linear in the elments I y ) € 2and antilinear
in the elements (a, I E # *. The dual space (or cospace) T'* to 2 can be considered as
the space of linear functional over A? In a Hilbert space there is a one-to-one
correspondence between elements I w ) E 2' and linear functional (w 1 E 2Y *, and one
can consider the s p a c e 2 ' with the same right also as the space of antilinear functionals
over P'*.In a rigged Hilbert space, this symmetry between elements and linear or
antilinear functionals is disturbed. The basic idea of the introduction of a rigged Hilbert
space is to extend the usual Hilbert space to a more general space by definition of its
elements as linear or antilinear functionals over a more narrow space than the Hilbert
space but with compatible topologies of the more narrow space and of the usual Hilbert
space according to the scheme of Gelfand triplets of spaces
The second line in this relation is necessary only because we do not consider a bilinear
scalar product as mostly in the theory of generalized functions but a scalar product linear
in one and antilinear in the other factor as usually in quantum theory. A state I ty) can
be considered as an element of the rigged Hilbert space .3" if the antilinear functional
190
Ann. Physik 1 (1992)
(u, 1
w ) exists for arbitrary elements (w 1 E 3’*. The corresponding costate (w 1 E A!’ * ‘ to
I w) E .W ’ can unambiguously be constructed by the corresponding linear functional
(w I p) for arbitrary elements I u,) E XThe concept of the rigged Hilbert space is usually
applied to give a legitimate status to the eigenstates of hermitian operators with
continuous spectrum as, for example, the position and momentum operators, but it is
flexible enough to be applied to other or to more general situations. In the conventional
theory of Hilbert spaces, awkward constructions are necessary to deal with the eigenvalue
problem of hermitian operators with continuous spectrum.
A rigged Hilbert space must be constructed in our case to give a definite sense to the
states I a; I!Jwith I [ I 1 1. We describe roughly, how one can construct an appropriate
rigged Hilbert space. We use that the scalar products (p; C I a; [) exist for arbitrary
complex a and p and for I [ I I C I < 1 (see Eq. (3.1)). In principle, one can construct for
every 1 C 1 I1 a maximal possible space 2’ (1 [ 1) C JY in such a way that the antilinear
function (p I a; [) exists for arbitrary I p) E X (I [ I). This gives a minimal possible
appropriate rigged Hilbert space X I ( I [ I ) for arbitrary I [ I I1. However, one can also
construct a rigged Hilbert space 2’that is appropriate for all values [ with finite I [ I.
We use that the scalar product @; 0 1 a; C) exists for arbitrary [ with finite 1 [ 1 where 1 p;
0) denotes arbitrary (nonormalized) coherent states and compare the following Fock-state
expansions
Let us define that I u,) belongs to the space 2’ of basic elements if there exists a certain
value 1 /3 1 and a constant C so that the following inequality holds for the expansion
coefficients in Eq. (5.2)
This means that the coefficients of the Fock-state expansion of I u,) must decrease with
n -+ 03 so fast or faster as the coefficients of the Fock-state expansion of a certain coherent
state. We do not investigate wether this definition is sufficient with all mathematical
finesses or wether it must be improved. Because the scalar products (u, I u,) exist, the
are compatible. Usually, a counttopologies of X and 2‘defined by the norm i@l-$
able set of compatible norms is considered to separate from the Hilbert space 2f those
states that must not belong to the more narrow space X
6 Dual orthogonality and dual completeness of the states I a; 4)
The totality of states I a; [) with fixed [ under the condition I [ I < 1 forms a nonorthogonal overcomplete set of states when a goes over the entire complex plane. A
corresponding completeness relation was obained in Eq. (3.8). Now, we will show that
the states I a; [)with fixed [form already a complete set of states when a goes over certain
paths in the complex plane. This is connected with a kind of duality relations in the
totality of states I a; c).
First, let us write down the hermitian conjugate equation to Eq. (2.1)
A. Wiinsche, Eigenvalue problem for arbitrary linear combinations. . .
Dividing this equation by c* and making then the substitutions c*
one obtains
191
+
I/[ and a*
-+
p/c,
By forming the scalar product of Eq. (2.1) with @*/[*, l/c* 1 and of Eq. (6.2) with
1 a; 0,
from the difference of the obtained equations it follows
(a -
(;: *; 1
p ) -' *
a; [) = 0.
This means that the scalar product @*/?; 1/c* I a; 1;) can be different from zero only for
a = p and, therefore, must be a function of a and /3 singularly concentrated on a - /3
= 0. From zd(") (2) = - nd("-') (z), (z complex variable), it follows that @*/c*;
l/c* I a; c> must be proportional to the one-dimensional delta function 6(a - p ) of the
complex variable a - p with a proportionality factor depending on a or p and 1;. Let us
determine this proportionality factor.
From the representation of 1 a; 1;) in Eq. (2.10) it follows'
Hence, these both equations yield
The application of the operator exp (1; a2/a a ap) to functions f ( a , p ) is equivalent to a
convolution of this function with a certain Gaussian function. One can calculate this
convolution by means of Fourier transformation and its inversion or more directly by
using the following results for specified convolutions in intermediate steps
'
Latest here but already when writing down, for example, Eqs. (6.2) or (3.1) and (3.8) one feels that
it is awkward to denote the adjoint state to I a; [>by (a; I and not by the more appropriate (a*; <* 1
because it really depends only on a* and [*. However, the last notation would bring confusion with
the generally accepted rule in quantum mechanics to denote the adjoint state to an arbitrary state I ty)
by (ty 1 and not by (ty* I that is probably also not acceptable for all situations.
<
192
Ann. Physik 1 (1992)
The application of these two formulae gives also a possibility to prove the formula of F. G.
Mehler ( I 835 - 1895) that was used for the calculation of the scalar product in Eq. (3.1).
The right-hand side of the second formula in Eq. (6.6) goes over into the delta function
d@) in the limiting case q + [ 0 and the convolutions have to be understood as
convolutions of generalized functions or, more exactly, analytic functionals because the
variables are in general complex. Thus we obtain from Eq. (6.5) the following result for
the scalar product
+
This equation expresses a kind of duality of the states I a; [) and I p*/[*; 1/[*) to each
other. We will call the orthogonality of these states for a =k p a dual orthogonality.
Eq. (6.7) can be used to establish the following dual completeness of the states 1 a; [)
and I a*/[*; l/[*)
where L' denotes an integration path through the complex plane that can be widely
deformed under the following conditions. There are two opposite angular sectors of the
variable a in dependence on the fixed value of [, where the real part of a2/2[ has a
positive value and, consequently, exp( - a2/2[) is vanishing in the infinity. The
integration path C'must begin in the infinity of one of these sectors and must end in the
infinity of the opposite sector and can be chosen widely arbitrary in the complex plane
under these conditions. This can be also seen from the following formula that is
equivalent to Eq. (6.8) and can be obtained from the Fock-state expansions of the states
in Eq. (6.8) according to Eq. (2.7) by using the completeness of the Fock states
Hn (L)
m /5 da exp(- $)H , (mL)
m
1
~
=
2" n ! a
q n .
(6.9)
The chosen direction on the integration pathcdefines the sign of the complex root [
that must be taken in Eqs. (6.8) and (6.9) (for example, the positive sign of
for real
positive [ and integration from minus to plus infinity).
From Eq. (6.8) it follows that one can make expansions of arbitrary states I w) in the
following way
w[
(6.10)
This formula shows that the overcomplete set of states I a; () for arbitrary fixed [ and
when a goes over the entire complex plane is already complete on widely arbitrary paths
{'through the complex plane if they satisfy the described weak conditions. In case of the
coherent states I a; 0), where this is also valid, one has to do with the most singular case
of dual states ([ = 0, l/[* = m) that separately will be dealt with in Section 7.
In case of ([* = 1, the sets of states I a; [) and dual states I p*/[*; 1 /[*) belong to
the same value of the fixed parameter [. The eigenvalue problem in Eqs. (2.1) or (2.2) and
A. Wiinsche, Eigenvalue problem for arbitrary linear combinations. . .
193
(6.2) is now equivalent to the following eigenvalue problem for a hermitian operator K a
+ K*Ut
(6.11)
(Ka
K
K
+ K*U+)
= &
From Eq. (6.7) it follows by obvious substitutions
K*
K*
(6.12)
2KK*
and from Eq. (6.8)
’
yzizF
dyenp(---;) Y 2
LTzti,
2KK
1
IL;--)($;K*
K*
=I.
K
K
(6.13)
K
The transformed integration path 4” = K t’can now begin in the sector around minus
infinity and end in the sector around plus infinity and can widely be deformed otherwise
in the complex plane, where the positive sign of the root
must be chosen. In
particular, the integration path 6” in Eq. (6.13) can be the entire real axis in the complex
plane of the variable y. In this case, one has y = y*, and Eq. (6.13) expresses the
completeness of the eigenstates of the hermitian operator K U + K * U ~to real eigenvalues
y. One can choose such normalization factors that the eigenstates of the hermitian
operator K ( I + K* at to real eigenvalues y are orthonormalized with the delta function.
These are, for example, the following states (not to be confused with coherent states!)
i m
(6.14)
On the other side, the Fock states I n) can be represented by the states 1 y ) in the following
way
The above equations can immediately be applied to the eigenvalue problems of the
canonical operators Q and P defined by
(6.16)
Ann. Physik 1 (1992)
194
and y = p , K
by the substitutions y = q, K 3
the displacement operator D ( p , /3 * ) defined by
= -i
m,respectively. In case of
one has the hemitian operator @*a - @at in the exponent. The eigenstates of D@,P*)
to eigenvalues exp(iy) with real y can be obtained from the above formulae by the
substitution K = iP*. However, the eigenvalues are now degenerate because all values y
modulo 2 II n, ( n = 0, t 1, t 2, . . .), lead to the same eigenvalue exp (i y ) but to different
eigenstates.
As in the more general case, the hermitian operators K U + K * U + possess
nondegenerate eigenstates to arbitrary complex eigenvalues y. The real eigenvalues are
distinguished by the property that they belong to selfdual eigenstates in the introduced
sense. The totality of eigenstates to real eigenvalues y forms already an orthogonal and
complete set of states.
7 The eigenstates of the boson creation operator
Making in Eq. (2.1) the substitution a =
equation after division by [
where the nonnormalized eigenstates
in the following way
TP, one obtains the following eigenvalue
I [P; 0 can be represented according to Eq. (2.10)
It is not possible immediately to go in Eq. (7.2) to the limiting case + M, and, therefore,
to obtain the eigenstates of the boson creation operator ut in Eq. (7.1) to eigenvalues 8.
However, one can multiply Eq. (7.2) by
x exp(- 4'P2/2) and can use that the
one-dimensional delta function is representable as an analytic functional by the following
limiting procedure
Introducing the notation
A. Wiinsche, Eigenvalue problem for arbitrary linear combinations. . .
195
we have obtained a complete set of eigenstates I p'; 03) of the boson creation operator at
to arbitrary complex /3 according to the eigenvalue equation
The states I 03) as eigenstates of the boson creation operator at are in a great analogy
to the nonnormalized coherent states I a; 0) as eigenstates of the boson annihilation
operator a. These two sets of states are connected to each other by one-dimensional
complex Fourier transformation. By applying the identity
one can directly check that the states given in Eq. (7.4) satisfy the eigenvalue equation
for the operator at to eigenvalues p. There are also some other heuristic ways not
presented here to obtain this result.
It is clear that the states I p:, 0 0 ) do not belong to the Hilbert space 2 because the
coefficients in the expansion in Eq. (7.4) are generalized functions. They also do not
belong to the rigged Hilbert space X ' described in Section 5 because the scalar product
does not give a complex number but a generalized function of a (or more exactly analytic
functional). One has to consider such superpositions of coherent states with analytic
functions c(a) as multipliers that the following integral over a path 4'in the complex
plane
is well defined. Eq. (7.7) and its conjugation can be used for expansions of a class of
states I w) into a set of nonnormalized coherent states I a; 0) over paths 8 in the complex
plane. The here considered most singular case 4' = 03 is the only case, where the
corresponding rigged Hilbert space cannot be constructed as a space of linear or antilinear functionals over a space of basic functions containing the coherent states. This
means that it must be a more narrow space of basic elements.
The eigenstates of the creation operator at can be also represented by using the twodimensional delta function S(P, p*) and its derivatives (6(z, z*) = 6 ( z + z*/2)
S ( z - z*/i2)means the complex representation of the two-dimensional delta function).
By applying the following limiting procedure
S@, p*)
=
lim
-
1il-m
71
2
2
(7 * 9)
that can be proved by two-dimensional complex Fourier transformation and its inversion
and introducing the notation
Ann. Physik 1 (1992)
196
(7.10)
fi.
(7.1 1)
This can also be checked by applying the identity
for k = 0, I = 1 and m = 0. There is no possibility to obtain the two-dimensional delta
function 6(J, p * ) by a limiting procedure from a function depending only on P and not
on p*. In this sense, the states I depend on P and P* in analogy to the normalized
coherent states I a) depending on a and a*.
p)
8 Conclusion
It was found that the operators K U + p a t with arbitrary complex numbers K and p as
parameters possess the totality of complex numbers as nondegenerate eigenvalues. The
corresponding eigenstates have explicitly been obtained in form of an expansion in the
basis of the Fock states and depending on the parameter c p / or~ as squeezed coherent
states for I I < 1 . They form an overcomplete set of states of a Hilbert space for I I
< 1 and must be understood as elements of a rigged Hilbert space for I I 2 1 . A dual
orthogonality and a dual completeness of the eigenstates belonging to a pair of
parameters and
= l/<* was established. The real eigenvalues of the hermitian
operators Ka + K* ut are distinguished by selfduality of the corresponding eigenstates.
They are often considered as the only eigenvalues of these operators and can be separated
from the totality of complex eigenvalues by an orthonormalization condition expressible
by means of the one-dimensional delta function.
The solution of the general eigenvalue problem for the operator K U + p u t gives a
deeper insight into the connections between the very different types of solution of the
eigenvalue problems for the particular cases of these operators as a, Q, P and at. It
seems to us that it would be very useful to solve the more general eigenvalue problem for
arbitrary linear combinations of the quadratic boson operators u2, aut + utu and ut2in
a similar manner.
<
<
I like to express my gratitude to Prof. Dr. Harry Paul from our working group in Berlin for valuable
discussions.
A. Wiinsche, Eigenvalue problem for arbitrary linear combinations. . .
197
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