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On the change of boson number operator functions by antinormal ordering
Volume 102A, number 8
PHYSICS LETTERS
4 June 1984
ON THE CHANGE OF BOSON NUMBER OPERATOR FUNCTIONS BY ANTINORMAL ORDERING R. BALTIN Department of Theoretical Chemistry, University of Ulm, D-7900 Ulm, bed. R ep. Germany Received 7 March 1984
Previous work on normal and antinormal ordering of boson number operator functions f(d+d) is extended. Further formulae for the effect of ordered expansions of f on number eigenstates are derived. Several examples of operator functions having infinite series expansions show that the antinormal ordering procedure can change operator functions though the commutation relation [d, d+l = i is taken into account strictly.
1. Normal and antinormal ordering of functions of boson operators d, d + is a useful technique when quantum statistical averages have to be calculated occurring in the treatment of systems of harmonic oscillators and of quantized fields, see e.g. refs. [1,2]. Suppose that an operator function f(d, d +) is given by its series expansion in powers of d and d +. The mathematical procedure of ordering then means shifting the d and the d + such that all d stand to the right (left) of all d + in the case of normal (antinormal) ordering taking into account the commutation relation [d, ~+] = i strictly throughout. Usually, it is assumed that the operator function changes only its form by these ordering processes, but not its effect when applied to any state belonging to the domain of definition off. This assumption has been investigated for functions depending, because of mathematical simplicity, on the number operator d+d only, but not on d and c~÷ separately, first for the exponential exp (-/a~+d) 0a complex number) [3], and later on for general functionsf(d+d) [4]. It turned out that the assumption is correct for normally ordered functions. However, antinormal ordering of f may produce a new operator function with properties different from those o f f . In ref. [4] general expressions of normally and antinormally ordered forms of operator functions have been presented, along with formulae describing the effect of these expressions on number operator eigenstates. In the present letter alternative relations on this subject are derived and applied to some examples. Let f(x) be a c-number function having a power series expansion
f(x)= ~ b~xv,
(1)
v=O
so that f(d+d) may be defined, at least formally, by replacing x by d+d • f(a+d)
=
* v ~ . b v . ( a * ÷ a) .
(2)
v=0
The coefficients %, dv of the normally and antinormally ordered forms a , f(n)(~+~) = ~ 0 e,,a.+~Av
f(a)(d+~) = ~
=
dvd~d +v,
(3,4)
v=O
are related to the Fourier transform 332
0.375-9601/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 102A, number 8
1
g(~.) = ~
f
PIIYSICS LETTERS
4 June 1984
(5)
f(x)ei?'Xdx
o f f ( x ) by the following expressions oo
1 f g(X)(e - i x - 1)vdk cv=--~,
and
1 f dv =-~.
g(X)ei?'(l - eiX)vdX.
(6,7)
--oo
--oo
Furthermore, if the normally ordered form (3) is applied to an occupation number eigenstate Im)(m = 0, 1, 2 .... ) we always find that
(8)
f(n)(d+d) Im > =f(m) Im), which means that f(n)(d+d) and f(d+d) are merely different forms of the same operator. If, however,f(a)(d+d)acts upon Im)the result
f(a)(~+d)[m)=I~im :
g(X) eihSn,m(iX)dX][m),
(9)
with n +m
(io) v=0
equalsf(m) Irn)under rather restrictive conditions only. Relations (6)-(10) have been proved in ref. [4]. 2. When expression (5) is inserted into (6) and (7) the integration over A can be performed. Interchanging the order of integrations and using the binomial formula we obtain for cv
if
%= 2rw!
f(x)
1
=--v! ~ (--1) k=0
dXeiXX ~ (_l)V_ k v e_iX k dx k=O k
v-k v k
f(x)8(x-k)dx
V!k=o(-l)
v-k v k f(k).
(ll)
-~
This relation has also been derived by another method by Gluck [5]. For the case of antinormal ordering we get in quite the same manner 17
v! k=0 Let us first verify that (8) follows from (3) and (11). From ~lm) = .v/-m Im
1),
d+ Ira)= ~-m-+ 11m + 1),
(13a,b)
we obtain
333
Volume 102A, number 8
PHYSICS LETTERS
f(n)(d÷8)lm)=lm)v~aOV!( m)c v= Ira) ~ ( m] ~v =
V
v=0\
(-1)v-k
4 June 1984
( kV) f(k)__
~'1 k = 0
[ m ) ~m k=0
f(k) v~=k( - 1 ) =
v-k ( m ) ( k ) i
14)
l)
Since (k) = 0 for v < k the inner sum becomes (-1)v-k(mv)(k)
(-l'k
-
~
= ~
v= k
(-l)V(7)v(v-1)...(v-k+l)
v=0
(--l)k[dk ~ (_l)v(mluv 1 = ( - 1 ) k F dk k! dukv=o \Vl _lu= 1 k! Ld---~ ( 1 - u ) m
] u =1
=
(
) [(I mk -
u)m-klu= 1 = 6 k , m
(15)
Hence only the term with k = m contributes to the outer sum of (14) yielding eq. (8). On the other hand, we get for antinormal ordering
90
v
k=0
k
from eqs. (4), (12), (13a,b). The outer sum over v which, contrary to eq. (14), does not terminate, must not be interchanged with the summation over k because in the resulting expression
k=0
=
v
the inner sum would be evidently divergent. Therefore, we start from the Nth partial sum N
v
SN=v~=o(m+v) k~=O(-1)k(k)f(-k-1),
(18)
where it is now allowed to interchange the summations, since they both are finite. N
S~v=
N
(-1)kf(-k
- 1) ~
,,=k
k=0
(m+v~(v~ v ]\g]"
(19)
Substitution of n = v - k in the inner sum yields
m v
=
v=k
m+k+n n=O
=[m+k] I l k
n=0
k +n
)(k+n k
=
(m + 1)(m + k!~. 2) n=O
(
(m+k+n) _ m ;
m k+n = m k~(m+N+l +k ]\m + k + 1 '
m+k n
)
(20)
where use has been made of the relation [6]
~=0
\
l+1
"
When the binomial coefficients are written out and terms are rearranged we get from (20)
334
(21)
Volume 102A, number 8
PHYSICS LETTERS
4 June 1984
IV
~ (m + v'l(v]=(N+m](N]m+ N + l v=k v f \ k f \ m l \ k / m + k + l "
(22)
Since f(a)(d+d)lm)=
Im)
lira N -i, e~
sN,
we obtain from (19) and (22)
.m {,N.m+
N--,,o
k=O
""*
nfYkTl
"/ "
k
(23) • . : N + m
Clearly, whether this limit does exist or not depends u p o n : It is true that (N + m + l)( m ) tends to infinity but the members of the sum over k might cancel as N ~ 0% so the limit possibly exists. Let us now consider some specific functions.
3. Examples. (i) f(x) = x I (1 >10 integer). For the (~+d)I we obtain from eqs. (11), (12)
coefficients
cv, d v of the
normally and antinormally ordered forms of
V
= 1 ~ (-l)V-k(V)kl= cv v! k=0 k
S/v)
(24)
and v+l
(
)k+l "
k=0
=A S
k
-(-1)l+vV+l
)l+k-I
v! k = 1
(_l)V+l_k.+l
( v + l ) ! k= 1
k
(
v
V k .... 1
)
k-i
kl+l
( l)l+vV+l
(
-~ (_l)V+l_ k v+l (v+l)!k= 0 k
)
kl+l=(_l)l+vs(~.;l),
(25)
where S) v) are Stirting numbers of the second kind 16-8]. The result (24) has also been obtained by Katriel [9] using another method. Since S(q) = 0 for q > p the sums (3), (4) terminate, thus yielding 1
[(d+d)l](n)= ~ S(iV)d+vdv
(26)
v=0
and I
[(d+e)l] (a) = ( - 1 ) / ~
(-l)VS(+;1)dvd +v.
(27)
v=O
When the expressions (26), (27) act upon the state Im)we expect that [(d+d)/] (n)Jm) =
[(d+d)t] (a)Im) = mllm),
since only a finite number of interchanges is necessary for ordering well-known relation (loc. cit.)
(28)
(a+d)l. (28)
is readily verified by use of the
l
~-I S(V)x(x -
l ) . . . ( x - v+ l ) = x / .
(29)
v=O
335
Volume 102A, n u m b e r 8
PHYSICS LETTERS
4 June 1984
Thus, when eq. (26) is applied to Im> the result is l
[(a+d)l](n)lm>= Ira> ~ S(V)m(m -
1).., (m - v + 1) =
mtlm).
v=0
Analogously, we obtain from (27) l
[(a+O)l](a)lm)= ( - 1 ) t i m )
(-I)VS]~I)(m + v)... (m + 1)
~ v=0
l+1
-(-1)t+llm>~ t ,~SO~)(rn+ta_l)(m+la_2) m ~=1 ~ - - ~ / I+1
"'"
m.
e(0) _- u n we add the term with/~ = 0, and applying (29) with x = - m and with (I + 1) instead of I we find that Sinc_e ~'t+l (28) holds for antinormal ordering too. (ii) f ( x ) = sin(Trx). Since f(k) = 0 for all integers it follows from (3), (4), (11), and (12) that [sin(Trd+a)] (n) = [sin0r0+a)] (a) = 0.
(30)
Furthermore, sin (nO+a) Im) = sin 0rm) Im) = 0,
(31 )
so that
f(n)(d+d) Ira)= f(a)(o+d) Im) = f(d+a) Im) = 0
(32)
is valid. (iii) f(x)= cos0rx). Noting
f(k)
f(-k -
= cos(Trk) = ( - 1 ) k,
1)= ( - 1 ) k+l
(33a,b)
we obtain v
V! k=O
v
k-
v-----7(-. '
dr=-
v! ~=o
-
(34,35)
~.' '
Clearly, from (8) we must have [cos(Trd+d)] (n)Im) = cos(m rr)Im) = ( - l ) m Im).
(36)
However, (33b) and (23) yield [c°s (Tr0+~)] ( a ) I n ) = - [m)N~lim
{(N+m+l)(N+m)k~= O m
m+k+ll
(N)}=_oolm),k
(37)
evidently divergent, i.e. the antinormal expansion of cos(trY+d) is purely formal and has an empty domain of definition. (iv) f(x)= 0rx)-1 sin0rx). As in examples (ii), (iii), f(x) has a power series expansion convergent everywhere. When k is integer f(k) = 0 except for k = 0 where f(0) = 1. Thus we obtain
cv=(-1)v/v!,
dr=0,
and it follows for all states Im) that
[sinOrd+a)/lr(d+a)]( a ) I m ) 336
= 0.
(38)
Volume 102A, number 8
PHYSICS LETTERS
4 June 1984
[ Iowever, sin(lrm) [ m ) = O, //'Ell = I0),
i f m integer > 0, i f m = 0.
(39)
Thus, antinormal ordering o f sin(rrd+~)/(rtd+~)does not change the effect o f this o p e r a t o r on the basis states Im) except for m = 0. (v) 3"(x) = 1 / F ( x + 1 ) = 1/x!. f(x) has a power series expansion with infinite radius o f convergence [6]. Since f(-k - 1 ) = l / F ( - k) = 0 it follows d v = 0 and therefore ll/r(l
+ d+d)l (a) Im} = 0.
(40)
However, [ I / F ( I + d+d)] Im) = [ I / F ( 1 + m ) ] Im}= In
[I/mP(m)]
general,f(a)(d+d)Ira) 4:f(d+d)Im>i f f ( - k
Im):~ O.
(41)
- 1 ) = 0, but f ( k ) ~: 0 for k = 0, I, 2 .....
Financial support by the " F o n d s der Chemischen lndustrie" is gratefully acknowledged.
References [1 I [21 [31 [41 [51 [6] [7] [81 19]
W.H. Louisell, Radiation and noise in quantum electronics (McGraw-tlill, New York, 1964). W.H. Louisell, Quantum statistical properties of radiation fWiley. New York, 1973). R. Baltin, Phys. Lett. 92A (1982) 110. R. Baltin, J. Phys. A16 (1983) 2721. P. Gluck, Nuovo Cimento 8B (1972) 256. I.S. Gradsteyn and I.M. Ryzhik, Tables of integrals, series and products, fourth Ed. (Academic Press, New York, 1965). Ch. Jordan, Calculus of finite differences, third Ed. (Chelsea, New York, 1965). J. Riordan, Combinatorial identities (Wiley, New York, 1968). J. Katriel, Lett. Nuovo Cimento 10 (1974) 565.
337
PHYSICS LETTERS
4 June 1984
ON THE CHANGE OF BOSON NUMBER OPERATOR FUNCTIONS BY ANTINORMAL ORDERING R. BALTIN Department of Theoretical Chemistry, University of Ulm, D-7900 Ulm, bed. R ep. Germany Received 7 March 1984
Previous work on normal and antinormal ordering of boson number operator functions f(d+d) is extended. Further formulae for the effect of ordered expansions of f on number eigenstates are derived. Several examples of operator functions having infinite series expansions show that the antinormal ordering procedure can change operator functions though the commutation relation [d, d+l = i is taken into account strictly.
1. Normal and antinormal ordering of functions of boson operators d, d + is a useful technique when quantum statistical averages have to be calculated occurring in the treatment of systems of harmonic oscillators and of quantized fields, see e.g. refs. [1,2]. Suppose that an operator function f(d, d +) is given by its series expansion in powers of d and d +. The mathematical procedure of ordering then means shifting the d and the d + such that all d stand to the right (left) of all d + in the case of normal (antinormal) ordering taking into account the commutation relation [d, ~+] = i strictly throughout. Usually, it is assumed that the operator function changes only its form by these ordering processes, but not its effect when applied to any state belonging to the domain of definition off. This assumption has been investigated for functions depending, because of mathematical simplicity, on the number operator d+d only, but not on d and c~÷ separately, first for the exponential exp (-/a~+d) 0a complex number) [3], and later on for general functionsf(d+d) [4]. It turned out that the assumption is correct for normally ordered functions. However, antinormal ordering of f may produce a new operator function with properties different from those o f f . In ref. [4] general expressions of normally and antinormally ordered forms of operator functions have been presented, along with formulae describing the effect of these expressions on number operator eigenstates. In the present letter alternative relations on this subject are derived and applied to some examples. Let f(x) be a c-number function having a power series expansion
f(x)= ~ b~xv,
(1)
v=O
so that f(d+d) may be defined, at least formally, by replacing x by d+d • f(a+d)
=
* v ~ . b v . ( a * ÷ a) .
(2)
v=0
The coefficients %, dv of the normally and antinormally ordered forms a , f(n)(~+~) = ~ 0 e,,a.+~Av
f(a)(d+~) = ~
=
dvd~d +v,
(3,4)
v=O
are related to the Fourier transform 332
0.375-9601/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 102A, number 8
1
g(~.) = ~
f
PIIYSICS LETTERS
4 June 1984
(5)
f(x)ei?'Xdx
o f f ( x ) by the following expressions oo
1 f g(X)(e - i x - 1)vdk cv=--~,
and
1 f dv =-~.
g(X)ei?'(l - eiX)vdX.
(6,7)
--oo
--oo
Furthermore, if the normally ordered form (3) is applied to an occupation number eigenstate Im)(m = 0, 1, 2 .... ) we always find that
(8)
f(n)(d+d) Im > =f(m) Im), which means that f(n)(d+d) and f(d+d) are merely different forms of the same operator. If, however,f(a)(d+d)acts upon Im)the result
f(a)(~+d)[m)=I~im :
g(X) eihSn,m(iX)dX][m),
(9)
with n +m
(io) v=0
equalsf(m) Irn)under rather restrictive conditions only. Relations (6)-(10) have been proved in ref. [4]. 2. When expression (5) is inserted into (6) and (7) the integration over A can be performed. Interchanging the order of integrations and using the binomial formula we obtain for cv
if
%= 2rw!
f(x)
1
=--v! ~ (--1) k=0
dXeiXX ~ (_l)V_ k v e_iX k dx k=O k
v-k v k
f(x)8(x-k)dx
V!k=o(-l)
v-k v k f(k).
(ll)
-~
This relation has also been derived by another method by Gluck [5]. For the case of antinormal ordering we get in quite the same manner 17
v! k=0 Let us first verify that (8) follows from (3) and (11). From ~lm) = .v/-m Im
1),
d+ Ira)= ~-m-+ 11m + 1),
(13a,b)
we obtain
333
Volume 102A, number 8
PHYSICS LETTERS
f(n)(d÷8)lm)=lm)v~aOV!( m)c v= Ira) ~ ( m] ~v =
V
v=0\
(-1)v-k
4 June 1984
( kV) f(k)__
~'1 k = 0
[ m ) ~m k=0
f(k) v~=k( - 1 ) =
v-k ( m ) ( k ) i
14)
l)
Since (k) = 0 for v < k the inner sum becomes (-1)v-k(mv)(k)
(-l'k
-
~
= ~
v= k
(-l)V(7)v(v-1)...(v-k+l)
v=0
(--l)k[dk ~ (_l)v(mluv 1 = ( - 1 ) k F dk k! dukv=o \Vl _lu= 1 k! Ld---~ ( 1 - u ) m
] u =1
=
(
) [(I mk -
u)m-klu= 1 = 6 k , m
(15)
Hence only the term with k = m contributes to the outer sum of (14) yielding eq. (8). On the other hand, we get for antinormal ordering
90
v
k=0
k
from eqs. (4), (12), (13a,b). The outer sum over v which, contrary to eq. (14), does not terminate, must not be interchanged with the summation over k because in the resulting expression
k=0
=
v
the inner sum would be evidently divergent. Therefore, we start from the Nth partial sum N
v
SN=v~=o(m+v) k~=O(-1)k(k)f(-k-1),
(18)
where it is now allowed to interchange the summations, since they both are finite. N
S~v=
N
(-1)kf(-k
- 1) ~
,,=k
k=0
(m+v~(v~ v ]\g]"
(19)
Substitution of n = v - k in the inner sum yields
m v
=
v=k
m+k+n n=O
=[m+k] I l k
n=0
k +n
)(k+n k
=
(m + 1)(m + k!~. 2) n=O
(
(m+k+n) _ m ;
m k+n = m k~(m+N+l +k ]\m + k + 1 '
m+k n
)
(20)
where use has been made of the relation [6]
~=0
\
l+1
"
When the binomial coefficients are written out and terms are rearranged we get from (20)
334
(21)
Volume 102A, number 8
PHYSICS LETTERS
4 June 1984
IV
~ (m + v'l(v]=(N+m](N]m+ N + l v=k v f \ k f \ m l \ k / m + k + l "
(22)
Since f(a)(d+d)lm)=
Im)
lira N -i, e~
sN,
we obtain from (19) and (22)
.m {,N.m+
N--,,o
k=O
""*
nfYkTl
"/ "
k
(23) • . : N + m
Clearly, whether this limit does exist or not depends u p o n : It is true that (N + m + l)( m ) tends to infinity but the members of the sum over k might cancel as N ~ 0% so the limit possibly exists. Let us now consider some specific functions.
3. Examples. (i) f(x) = x I (1 >10 integer). For the (~+d)I we obtain from eqs. (11), (12)
coefficients
cv, d v of the
normally and antinormally ordered forms of
V
= 1 ~ (-l)V-k(V)kl= cv v! k=0 k
S/v)
(24)
and v+l
(
)k+l "
k=0
=A S
k
-(-1)l+vV+l
)l+k-I
v! k = 1
(_l)V+l_k.+l
( v + l ) ! k= 1
k
(
v
V k .... 1
)
k-i
kl+l
( l)l+vV+l
(
-~ (_l)V+l_ k v+l (v+l)!k= 0 k
)
kl+l=(_l)l+vs(~.;l),
(25)
where S) v) are Stirting numbers of the second kind 16-8]. The result (24) has also been obtained by Katriel [9] using another method. Since S(q) = 0 for q > p the sums (3), (4) terminate, thus yielding 1
[(d+d)l](n)= ~ S(iV)d+vdv
(26)
v=0
and I
[(d+e)l] (a) = ( - 1 ) / ~
(-l)VS(+;1)dvd +v.
(27)
v=O
When the expressions (26), (27) act upon the state Im)we expect that [(d+d)/] (n)Jm) =
[(d+d)t] (a)Im) = mllm),
since only a finite number of interchanges is necessary for ordering well-known relation (loc. cit.)
(28)
(a+d)l. (28)
is readily verified by use of the
l
~-I S(V)x(x -
l ) . . . ( x - v+ l ) = x / .
(29)
v=O
335
Volume 102A, n u m b e r 8
PHYSICS LETTERS
4 June 1984
Thus, when eq. (26) is applied to Im> the result is l
[(a+d)l](n)lm>= Ira> ~ S(V)m(m -
1).., (m - v + 1) =
mtlm).
v=0
Analogously, we obtain from (27) l
[(a+O)l](a)lm)= ( - 1 ) t i m )
(-I)VS]~I)(m + v)... (m + 1)
~ v=0
l+1
-(-1)t+llm>~ t ,~SO~)(rn+ta_l)(m+la_2) m ~=1 ~ - - ~ / I+1
"'"
m.
e(0) _- u n we add the term with/~ = 0, and applying (29) with x = - m and with (I + 1) instead of I we find that Sinc_e ~'t+l (28) holds for antinormal ordering too. (ii) f ( x ) = sin(Trx). Since f(k) = 0 for all integers it follows from (3), (4), (11), and (12) that [sin(Trd+a)] (n) = [sin0r0+a)] (a) = 0.
(30)
Furthermore, sin (nO+a) Im) = sin 0rm) Im) = 0,
(31 )
so that
f(n)(d+d) Ira)= f(a)(o+d) Im) = f(d+a) Im) = 0
(32)
is valid. (iii) f(x)= cos0rx). Noting
f(k)
f(-k -
= cos(Trk) = ( - 1 ) k,
1)= ( - 1 ) k+l
(33a,b)
we obtain v
V! k=O
v
k-
v-----7(-. '
dr=-
v! ~=o
-
(34,35)
~.' '
Clearly, from (8) we must have [cos(Trd+d)] (n)Im) = cos(m rr)Im) = ( - l ) m Im).
(36)
However, (33b) and (23) yield [c°s (Tr0+~)] ( a ) I n ) = - [m)N~lim
{(N+m+l)(N+m)k~= O m
m+k+ll
(N)}=_oolm),k
(37)
evidently divergent, i.e. the antinormal expansion of cos(trY+d) is purely formal and has an empty domain of definition. (iv) f(x)= 0rx)-1 sin0rx). As in examples (ii), (iii), f(x) has a power series expansion convergent everywhere. When k is integer f(k) = 0 except for k = 0 where f(0) = 1. Thus we obtain
cv=(-1)v/v!,
dr=0,
and it follows for all states Im) that
[sinOrd+a)/lr(d+a)]( a ) I m ) 336
= 0.
(38)
Volume 102A, number 8
PHYSICS LETTERS
4 June 1984
[ Iowever, sin(lrm) [ m ) = O, //'Ell = I0),
i f m integer > 0, i f m = 0.
(39)
Thus, antinormal ordering o f sin(rrd+~)/(rtd+~)does not change the effect o f this o p e r a t o r on the basis states Im) except for m = 0. (v) 3"(x) = 1 / F ( x + 1 ) = 1/x!. f(x) has a power series expansion with infinite radius o f convergence [6]. Since f(-k - 1 ) = l / F ( - k) = 0 it follows d v = 0 and therefore ll/r(l
+ d+d)l (a) Im} = 0.
(40)
However, [ I / F ( I + d+d)] Im) = [ I / F ( 1 + m ) ] Im}= In
[I/mP(m)]
general,f(a)(d+d)Ira) 4:f(d+d)Im>i f f ( - k
Im):~ O.
(41)
- 1 ) = 0, but f ( k ) ~: 0 for k = 0, I, 2 .....
Financial support by the " F o n d s der Chemischen lndustrie" is gratefully acknowledged.
References [1 I [21 [31 [41 [51 [6] [7] [81 19]
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