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Branching diagram for “elementary particles” with arbitrary spin
CHEMICAL PHYSICS LETI-ERS
Volume 46, number 2
BRANCHlNG DIAGRAM FOR “ELEMENTARY
PARTICLES’
1 hfarcit 1977
WITH ARBlTRARY
SPIN
Ruben PAUNCZ and Jacob KATKIEL Department of Chemistry, Haifa 32000, Israel
Technion - Israel Institute of Technology,
Received 2 November 1976 A brancbmg diagram IS constructed for B system of identical particles with .m arbitrary spin, a. A formula is derived tar the number of state5 belonging to a resultant S quantum number. ?be case (I = 1, relevant to the Serber-
type constructlon 1sconsIdered In dctad.
1. Introduction The spin degeneracy prablcm has been the SUbJect of several invcstlgations [ 1] . Recently Carrington and Doggett [2] considered the case when the spin clgenfunction is built up from units having u = 1. In calcuIations connected with NMR and ESR experiments one has to deal with composite systems whose umts can have spin 3/2,2, etc. It seems, therefore, worth-
while to give a general treatment for the dctennmation of the number of spm states built up from “elementary particles” having an arbitrary spin.
The cigcnflmction% of S2 are linear combinations of the primitive spin-functions; the latter are products of one-particle spin-functions and they are eigenfunctlons of the Sz operator; their number in equal to (2~ + 1)“. Let us denote the number of primitive spin-functions with a given S, cigcnvalue, M by Fo(rr. M). From these functions we can form spin eigenfurictions with the spin quantum number S = M, M-I, .._, tn. From the primitive spin-functions with the S, cigcnvalue, M+l we can form spin eigenfunctions which belong to S = Af+l ,M+2, . ...w. Therefore go@, S) is related to F,(n, M) in the following way: g,(n, S) = I$(& S) -- E&
2. The number of spiu states for an arbitrary spin
s,=g_u, g,(n - 1,a.
(2)
WChave the additional relations:
Given n Identical elementary particles, each with spin u WCwish to find out the number of spin states with resultant spin S. Let us denote thus number by g&z, 5’). For u = l/2 particles the solution is given by the well known branching diagram [3 ] . We can construct similar branching diagrams for “elementary particles” with an arbitrary spm. The essence of the branching diagram is that one constructs the n-partick spin eigcnfunctions from the 01 - 1) particle eigenfunctions with the use of the appropriate Clcbsch-Cordan coefficients_ From the addition theorem of angular momenta it is evident that g&z, S) is related to g&z - 1,S’) in the following way: s+o
g,h S) =
s+l).
(1)
E
F(n,M)=(2fJ+
M=-no
5
(2s
S
l)“,
(3)
= (20 + ly’ .
(4)
o
+
I)g,(n,S)
The functions F,(n, M) satisfy the recurrence relation:
which can easily be shown to follow from (I) and (2) and the symmetry relation E& -M) = F,(n. M). The crucial step will be to show that the coefficients in the power series expansion Q. #)
= (x” +x O-l + ... +x-o )n
(6) 319
CHEMICAL PHYSICS LElTERS
Volume 46, number 2
1977
Formulas ( 13)-( 15) together with (2) give the number of spin states for an arbitrary spin.
satisfy h recurrence relation identical with (5). As @&)
1 March
= E CY(n,i)xi i=__na a a-l + . . . +x-a) = @a,n- 1 (x) (x” + x h-l)0
a
rra
C
=
3. The number of spin states for (I = 1 As an illustration of formula (13) consider the case ofu= l/2. F&z,
a
C
a,(12 - 1,k - j)xk,
M) = (n; f n+S, f n-S)
=
;
(16)
(7)
k=-naJ=-a
hcncc
wcgct
01,2(n~9=(ny)-(y:
,)9
k+a a,(n-W.
(8)
We can check that we have the same initial condrtions for cra(n, k) and F,(n, M) and puttmg x = 1 in (6) WC have relation (3) satrsficd; so we have established the isomorphism be twcen o~,(n, k) and Fa(n. M). Usmg the multinomral expansion WCget Qa,,(x) = f=
(n;fa)xn(Ja),
(9
with v = f N - S, and this is a well known result. For u = 1 particles Ft(n,M)=
C(n;il,io,i_t),
i, + i. + i_, = n, i, - i_,
=M,
I (n+M)/2 1 =
c
i=M
(n;i,n+M-2i,
i-M)
a WhMC
(17)
a
f, ‘(i,,i,_1,...,
C
i -a ),
j=-a
i.=n, J
(10)
a
7?tfa)=
c Ii,,
(11)
J'-a
and
(n;ia)=n!/(i a’1i a-l is
! . .. i_al)
(12)
the muitinomial coefficient. Hence,
Fa(n. M) = aa@. M) = f;
(n;f,).
In figs. I and 2 we give the branching diagrams for the functions F1(n, M) and gt(n, s), respectively. The case u = I is cspccially interesting as spin eigerifunctions burlt up from triplet functions play an important role in the construction of the Serbcr-type [4] functions (Carrington and Doggett [2]). One can establish an interesting relation between g&z, s) and gl(n, s). Let us use the following notations:
sl12h S) E f(n. s) =
(“y)-(v-_nl). v=in-S,
(13)
a
f(n, s) is the number of spin states for an n-electron
with the relations IY
C i.=n J
+a
(14)
and a
c
j=-0
320
j$=M.
(15)
system with resultant spin S. For the sake of simplicity consider the case of n being an even number n = 2t. According to the construction of Carrington and Doggett, one can obtain all the spin eigenfunctions (in the Serber-scheme) by interposing the required number of singlet coupled pairs in all possible positions in the all-triplet coupled spin-functions. There,fl2t, S) is given
CHEMICAL PHYSICS LETTERS
Volume 46. number 2
1 March! 1977
f(2.s) =h(t,S) f
0
; h(r-1,s)
f
r
0
kft--2,s)
2
f .** i-
?
f(2t-2,s)
2
= h(t-
1,s) +
I MC -I
f(S,S) = h(S, S).
-2
We can easily soIve the-system of (rS+I ~cqu~tio~s (20) unknown &(t, S),lr(t- I$), ..,Fr(S. S) by using Cramcrs rule. The determinant of the coefficients is equal to unity and the detcrmmants in the numerators can be expanded &ding to the f~~~ow~ng simple expressions:
tW
for the (r-Si-1)
-II _ -4
I
2
3
4
n dqrdm for the number of primitive spinfunctions for sli-trtplet spin c~Eenfun~tIons_
I’ig. 1. Brawling
*.-
*
I as we can interpose
the k singict pair-functions
In (i) different ways. Let us use eq. (19) for t, t-1, _._etc
(‘T2)
The validity of (2 I) can be simply proved by induction. Using (2 1) we have the relation (22) Comparing (22) and (19) we see that f(Zn.S) and h(n, S) are related to each other by invene relations (Rlordan [S] ). Substituting the expression forl”(2rr.S) and usulg (2) one obtains a new relation for Fl(a,M):
(23)
n
This paper is part of a research project supported by the US-Israel Binational Science Foundation.
Fig. 2. Branching diagram for the number of spin states for ~11.triFlet spm eig~nfu~ctions.
321
Volume 46, number 2
CHEMICAL PHYSICS LETTERS
Note added in proof The special case o = 1 has recently been considered by P.W. Atkins and T.P. Lambert, Mol. Phys. 32 (1976) I 151, using a combinatorial argument.
References II] P-0 Lijwdin and 0. Goscinski, Intern. J. Quantum Chcm. 3s (1970) 535, and rcfcrcnccs thercm.
322
1 March 1977
PI P.J. Carrington and G. Uoggett. Mol. Phys. 30 f 1975) 49. I31 E.M. Carson. Perturbation methods in the quantum mechanics of the n-electron systems (BCckic and Son, Glasgow, 1951) pp. 189,214. [41 R. Serbcr, Phys. Rev. 45 (1934) 461; K. Ruedenberg and W.I. Salmon, J. Chem. Phys. 57 (1972) 2776,2787; K. Ruedenberg, W.1. Salmon and L.M. Cheung. J. Chem. Phys. 57 (1972) 2791. I5J J. Rlordan, Combinatorial Jdcntities (Wiley, New York, 1968) p. 43.
Volume 46, number 2
BRANCHlNG DIAGRAM FOR “ELEMENTARY
PARTICLES’
1 hfarcit 1977
WITH ARBlTRARY
SPIN
Ruben PAUNCZ and Jacob KATKIEL Department of Chemistry, Haifa 32000, Israel
Technion - Israel Institute of Technology,
Received 2 November 1976 A brancbmg diagram IS constructed for B system of identical particles with .m arbitrary spin, a. A formula is derived tar the number of state5 belonging to a resultant S quantum number. ?be case (I = 1, relevant to the Serber-
type constructlon 1sconsIdered In dctad.
1. Introduction The spin degeneracy prablcm has been the SUbJect of several invcstlgations [ 1] . Recently Carrington and Doggett [2] considered the case when the spin clgenfunction is built up from units having u = 1. In calcuIations connected with NMR and ESR experiments one has to deal with composite systems whose umts can have spin 3/2,2, etc. It seems, therefore, worth-
while to give a general treatment for the dctennmation of the number of spm states built up from “elementary particles” having an arbitrary spin.
The cigcnflmction% of S2 are linear combinations of the primitive spin-functions; the latter are products of one-particle spin-functions and they are eigenfunctlons of the Sz operator; their number in equal to (2~ + 1)“. Let us denote the number of primitive spin-functions with a given S, cigcnvalue, M by Fo(rr. M). From these functions we can form spin eigenfurictions with the spin quantum number S = M, M-I, .._, tn. From the primitive spin-functions with the S, cigcnvalue, M+l we can form spin eigenfunctions which belong to S = Af+l ,M+2, . ...w. Therefore go@, S) is related to F,(n, M) in the following way: g,(n, S) = I$(& S) -- E&
2. The number of spiu states for an arbitrary spin
s,=g_u, g,(n - 1,a.
(2)
WChave the additional relations:
Given n Identical elementary particles, each with spin u WCwish to find out the number of spin states with resultant spin S. Let us denote thus number by g&z, 5’). For u = l/2 particles the solution is given by the well known branching diagram [3 ] . We can construct similar branching diagrams for “elementary particles” with an arbitrary spm. The essence of the branching diagram is that one constructs the n-partick spin eigcnfunctions from the 01 - 1) particle eigenfunctions with the use of the appropriate Clcbsch-Cordan coefficients_ From the addition theorem of angular momenta it is evident that g&z, S) is related to g&z - 1,S’) in the following way: s+o
g,h S) =
s+l).
(1)
E
F(n,M)=(2fJ+
M=-no
5
(2s
S
l)“,
(3)
= (20 + ly’ .
(4)
o
+
I)g,(n,S)
The functions F,(n, M) satisfy the recurrence relation:
which can easily be shown to follow from (I) and (2) and the symmetry relation E& -M) = F,(n. M). The crucial step will be to show that the coefficients in the power series expansion Q. #)
= (x” +x O-l + ... +x-o )n
(6) 319
CHEMICAL PHYSICS LElTERS
Volume 46, number 2
1977
Formulas ( 13)-( 15) together with (2) give the number of spin states for an arbitrary spin.
satisfy h recurrence relation identical with (5). As @&)
1 March
= E CY(n,i)xi i=__na a a-l + . . . +x-a) = @a,n- 1 (x) (x” + x h-l)0
a
rra
C
=
3. The number of spin states for (I = 1 As an illustration of formula (13) consider the case ofu= l/2. F&z,
a
C
a,(12 - 1,k - j)xk,
M) = (n; f n+S, f n-S)
=
;
(16)
(7)
k=-naJ=-a
hcncc
wcgct
01,2(n~9=(ny)-(y:
,)9
k+a a,(n-W.
(8)
We can check that we have the same initial condrtions for cra(n, k) and F,(n, M) and puttmg x = 1 in (6) WC have relation (3) satrsficd; so we have established the isomorphism be twcen o~,(n, k) and Fa(n. M). Usmg the multinomral expansion WCget Qa,,(x) = f=
(n;fa)xn(Ja),
(9
with v = f N - S, and this is a well known result. For u = 1 particles Ft(n,M)=
C(n;il,io,i_t),
i, + i. + i_, = n, i, - i_,
=M,
I (n+M)/2 1 =
c
i=M
(n;i,n+M-2i,
i-M)
a WhMC
(17)
a
f, ‘(i,,i,_1,...,
C
i -a ),
j=-a
i.=n, J
(10)
a
7?tfa)=
c Ii,,
(11)
J'-a
and
(n;ia)=n!/(i a’1i a-l is
! . .. i_al)
(12)
the muitinomial coefficient. Hence,
Fa(n. M) = aa@. M) = f;
(n;f,).
In figs. I and 2 we give the branching diagrams for the functions F1(n, M) and gt(n, s), respectively. The case u = I is cspccially interesting as spin eigerifunctions burlt up from triplet functions play an important role in the construction of the Serbcr-type [4] functions (Carrington and Doggett [2]). One can establish an interesting relation between g&z, s) and gl(n, s). Let us use the following notations:
sl12h S) E f(n. s) =
(“y)-(v-_nl). v=in-S,
(13)
a
f(n, s) is the number of spin states for an n-electron
with the relations IY
C i.=n J
+a
(14)
and a
c
j=-0
320
j$=M.
(15)
system with resultant spin S. For the sake of simplicity consider the case of n being an even number n = 2t. According to the construction of Carrington and Doggett, one can obtain all the spin eigenfunctions (in the Serber-scheme) by interposing the required number of singlet coupled pairs in all possible positions in the all-triplet coupled spin-functions. There,fl2t, S) is given
CHEMICAL PHYSICS LETTERS
Volume 46. number 2
1 March! 1977
f(2.s) =h(t,S) f
0
; h(r-1,s)
f
r
0
kft--2,s)
2
f .** i-
?
f(2t-2,s)
2
= h(t-
1,s) +
I MC -I
f(S,S) = h(S, S).
-2
We can easily soIve the-system of (rS+I ~cqu~tio~s (20) unknown &(t, S),lr(t- I$), ..,Fr(S. S) by using Cramcrs rule. The determinant of the coefficients is equal to unity and the detcrmmants in the numerators can be expanded &ding to the f~~~ow~ng simple expressions:
tW
for the (r-Si-1)
-II _ -4
I
2
3
4
n dqrdm for the number of primitive spinfunctions for sli-trtplet spin c~Eenfun~tIons_
I’ig. 1. Brawling
*.-
*
I as we can interpose
the k singict pair-functions
In (i) different ways. Let us use eq. (19) for t, t-1, _._etc
(‘T2)
The validity of (2 I) can be simply proved by induction. Using (2 1) we have the relation (22) Comparing (22) and (19) we see that f(Zn.S) and h(n, S) are related to each other by invene relations (Rlordan [S] ). Substituting the expression forl”(2rr.S) and usulg (2) one obtains a new relation for Fl(a,M):
(23)
n
This paper is part of a research project supported by the US-Israel Binational Science Foundation.
Fig. 2. Branching diagram for the number of spin states for ~11.triFlet spm eig~nfu~ctions.
321
Volume 46, number 2
CHEMICAL PHYSICS LETTERS
Note added in proof The special case o = 1 has recently been considered by P.W. Atkins and T.P. Lambert, Mol. Phys. 32 (1976) I 151, using a combinatorial argument.
References II] P-0 Lijwdin and 0. Goscinski, Intern. J. Quantum Chcm. 3s (1970) 535, and rcfcrcnccs thercm.
322
1 March 1977
PI P.J. Carrington and G. Uoggett. Mol. Phys. 30 f 1975) 49. I31 E.M. Carson. Perturbation methods in the quantum mechanics of the n-electron systems (BCckic and Son, Glasgow, 1951) pp. 189,214. [41 R. Serbcr, Phys. Rev. 45 (1934) 461; K. Ruedenberg and W.I. Salmon, J. Chem. Phys. 57 (1972) 2776,2787; K. Ruedenberg, W.1. Salmon and L.M. Cheung. J. Chem. Phys. 57 (1972) 2791. I5J J. Rlordan, Combinatorial Jdcntities (Wiley, New York, 1968) p. 43.