Rational Bureau of Standards

Library, N.W. Bldg

MAY 2 1966 NBS

^Gcknic&l <yiote ^0.328

SOME PROPERTIES of the EIGENFUNCTIONS

and EIGENVALUES INTRODUCED by BETHE

for the LINEAR CHAIN of ATOMS

BY HENRY UNRUH, JR.

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
THE NATIONAL BUREAU OF STANDARDS

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Croup, Joint Institute for Laboratory Astrophysics at the University of Colorado,

olorado.
NATIONAL BUREAU OF STANDARDS

technical ^ote 32S

ISSUED April 19, 1966

SOME PROPERTIES OF THE EIGENFUNCTIONS AND EIGENVALUES

INTRODUCED BY BETHE FOR THE LINEAR CHAIN OF ATOMS

BY HENRY UNRUH, JR.

Radio Standards Laboratory

Institute for Basic Standards

National Bureau of Standards

Boulder, Colorado

NBS Technical Notes are designed to supplement the Bureau's regular publications program. They provide a means for making available scientific data that are of transient or limited interest. Technical Notes may be

listed or referred to in the open literature.

sale by the Superintendent of Documents, U.S. Government Pri Washington D.C. 20402

Price: 25<f

Contents

Page

Abstract 1

1. Introduction 2 2. Calculations 5

3. Summary .. 20

4. References .. 21

Some Properties of the Eigenfunctions and Eigenvalues

Introduced by Bethe for the Linear Chain of Atoms.

Henry Unruh, Jr.

2 ± The operations S \|r and S i|r, in which \|/ is an

exact eigenfunction obtained by Bethe for the linear

chain of localized spins, are discussed. The operations

2 ±

S and S are the usual square of the total angular momentum and the raising and lowering operators respectively. The results, valid for an arbitrary number of spin inversions, show, as expected, that all possible

states of the system can be achieved by omitting all excitations associated with zero wave number but requir^

ing each state to take all possible orientations in space.

Key Words: Spin waves, linear chain, ferromagnetism.

^Permanent address: Dept. of Physics, Wichita State University, Wichita, Kansas.
1, Introduction

The exact energy eigenstates of a linear chain of N

localized spins, each of spin 1/2, and subject to the isotropic

Hamiltonian,

^--|>*s* 1 + sjsj+1

+ s;s;

+1 >

1* are well known . In Jf the subscript j locates the position of a

spin in the lattice, and the spin operator for the spin in that location

S . = i sx+ i sy + k s

z

.

Introducing

N N 2 x N \2 , N \2

s

2

=

j=i j ^ v j=i j ^

v- j =i j ^

and

s*-S(i

=1 - J J

* The notation we are using, and will use for the most part, is identical to that of Bethe (reference 1).
have at once the following commutation relationships:

s

2

, S* I = 0, (1)

3C r

S (2)

S

+

, S" = 2S

Z

(3)

According to reference 1 the eigenstates of the system are given by

^(N,q)= Z a(m .m

2 ,...m ) "^ - ~ Pm ••• P, <m>q i c m, m q

with the restriction that

1 < m < m < . . • < m < N J

and

a(m

1 ,m2

, * *' m

q

} =pt

6XP

[* ^+ ^+

' * ' + ^l + 1/2 J/^^ ^^

To obtain a numerical value for a(m , . . . , m ) one first solves for the f s and cb

1 s in the following equations:
Nf = 2Trn 1+

i?i*"'

Nf^= 2TMv + k

rnk L

T ^^

Nf = 27rn + /,<))-

q q „_i q^ i=i

(6a)

tan- *W C °S 2

fk

+f

i V'i

cos ~

k i

+11 = ° kk *

for k£i

(6b)

in which n = 0.1. 2. . . , N-l, and cb ., . After the f's are obtained

k

Y ik

they may be written down (together with the associated <|>'s) in some

arbitrary order. Such a sequence is one of the possible q ! sequences

and is characterized by a particular value of the index P (which is written as a super scipt on the f's and cj)' s). Thus from equation (5)

it is seen that a(m , . . . ,m ) includes all possible permutations of the f's for an associated specification of an n set. At this time we should also note that (6b) can be written

in this form

eXp(i*ki ) exptMfj+f^)] + 1 - 2exp(ifk

) expLMfj+f^J + 1 - 2exp(if

i

) (7)
2. Calculations

We now evaluate S i|;(N, q) for the special case when none

of the f's are zero. At this point it is convenient to write l|f(N, q) for this special case in the form i|f (N, r) in which the r has re- placed the q to make the notation consistent with that in reference

2. Now

S\ <N,r)=£a(m ... mJ?a* • . . p . . . Pm

. . . Pm . • • <fc <m> 1 l r

<m> i=l 1 l r

Z b(n ... n )«« ... p Pn

... aN . (8) <n> 1 r-1

where

n.-l

r l (n^.-. n

r _ 1 )= £ I a(n

l) n

2 ...ni _ 1) m. ) n....n

r _ 1

) ( (9) i=l m. = n. , + 1 l i-l

in which it is understood that when i = 1, rn runs from 1 to n _]^ and

when i = r, m runs from n . + 1 to N.

' r r-1 Combining (5) and (9), there results

b(nlf ... n

r _ 1

) =

Tj Tj T

P=l i=l m.= n. , + 1 l i-l

exp / P P P P P P ,, n p\i ffn. + ffn. + ... + f. .n. . + f.m. + f.,.n. + ... + f n . + Vz > 4>_ .

* V 1 I 2 2 i-l i-l 11 l+l i r r-1 ," ki,
The third sum can be evaluated, and if the second is written out

explicitly we have

b(n . . . n ) =

1 r-1

P=l L

P Pi

expti^ ] - expfi^ i^J- p

1 -expLif j

] exp[i(f

2 Y»-. +fr Vl+1/2 ^*ki )] k<4

+ exp[i(f ini )] exp[ n +l] -exp[if n„ l-exp[if ] pexp[i(f

3 n2+

k<£

+ . . .

+exp[i(f^n

1+. .,+f.^n.^)]

P P

exp[if. (n. +l)]-exp[if. n.] l-exp[if. ] jeXp[i(£.^n.H

k<e

+ exp[i(f

P

n.+ . . .+fP

n.)l ii li

exp[ if

,-j.i (n,+l)]-exp[if. n.,_]

i+1 i i+1- 1-exptif^]

exp[i(f

P

2

n. +1+. <vi+ *2£" k<£

+ exPLi(f

1P

n

1 +...+f

P

_ 1 n

r _ 1 )] exptiff^+ Dj-expLif^N+l)] ]^^^^^^!

, r.J, J k«! W J [iff]
or b(n . . . n^ )

ri rr

r; exp[pLii,f J 1 p p p

^llLl-exp[ifP

] J 1-expfi^ ] 2 X V V l ^

exp[i(f

1 n^]

1-expCi^ ] l-exp[if

2

] ^-]exp[i(^n1+ ...<nr. 1+ y2 Z^)]

xn if ^<t

exp[i(f

P

n

1+...+f

P

n

i )]|"

exp[if.

+1 ] -i p

:p; "

:

ti^t J exp[i(f

i+i n

i+ -

• Ll-exptif. ] l-exp[if.

+1 ] +<Vi+ *2£»

.exp[if (N+l)]

wicfv..-<i-*-i"[ggr] expU(1/i^ (10)

Now consider one term of (10),

T P

= exp[i(ffn+...+ffn)]r1 x l

L.1-,

exp[ if.i+1- P P

e^[if. ] l-exp[if

i+1 ] Jexp[i(f

P

1ni+..

r r-1 ^ kl
For any sequence of the f' s that determines T. one always has present

p another sequence, P', that does nothing more than interchange f. and

P

i

f. ,. Therefore

i + 1

exp[if.

+1 ] l-exp[if. ] l-exp[if

i+1 ] J exp[ i

/

k<0

l-exp[if ] l-exp[if. ]

exp[if

j

] Wi^^l

Now let

# 2 +w = $+1/^

k<£ M i, i+1

so that *££-•+*<,,-•-*

k<£

i, i+1 and after some reduction

^exp[if

P

] exptif^] + 1-Zexptif^]

•rP . f P

T L eXp[if ] exp[iff~ ] + l-2exp[if. ] exp[i<>

P

]

Substituting exp[i<j,^

i+i ] aS given by (7), it is clear that T^TP'= -1 r^P P'

T. + T. = 0.

i l
Therefore all the terms in (9) go to zero except possibly for the first and lost one, so that p

exptifj

<v- vi> = £ {—TFT h

p[i(f2 V-+f

r vi**

«n

1 r p=lU-l-exp[if, llU-expti^ ]J k<£ fP

n ^M,)]

exp[i(f

1 n1+ ...f

r+1n

r _ 1

)] exp[if (N+l)] _

l-exp[if ] exp[lHE<JMlla ) k<£

Since all possible permutations of the f's appear in (Ha), it is clear

that for any sequence, P,

P P P

f12, f , . . . f r there is another sequence P' such that P' P P' P P' P

f 1=223 = f , f = f , . . . f = f . r 1 Now adding the first term on the right-hand side of (11a) in the permutation P to the second term on the right-hand side of (11a) in the per- mutation P' there results expti^ . l-exp[il.

exp[i(ffn+...+ffn +% £ <>*)] 2 1 r r-i

k<£ k*

Texptif^ (N+l)]

3 «rp[itqn

1+...+£

r n

r _ 1

)] exptifj ]P~ 1-expfi^ ]

exp[ i % 2 4>fj

- exp[ if N ] exp[ i% £ <}> ] (Hb)
Now

and

E

k<0 -ki

= *12 +

*13 + *14 + "' + *l,r-l

+

*lr

+ *23 + *24 + "- + *2,r-l

+ *2r

+ *34 + -" + *3,r-l

+ *3r

r-2, r-1 r-2, r

+ 4> r-1, r

k<£ M +U + *i3 +

*i4

+ --- +

+i,r.l

+

*i r

+ *23 + *24 + -" + *2,r-l

+ *2r

+ *34 + --- + +3,r-l

+ *3r

+ *;.2,r.l

+ *Ur

r-1, r

10
' 2 3 + *24

+ *25 + --- + *2r

+

+2]

+ +34 ++35 +— ++3r

++31

+ *45 + --- + *4r

+

+41

+ + r-1,, r ++11 r-1,

, 12- <t> 13-',, 14' <,> 15 'l,r-l " v lr

+ + 12 ++13 ++14 ++15 + + *l,r-l

+ *lr

+ +23 + *24 + *25 + + *2,r-l

+ *2r

+ + 34 ++35 + + +3,r-l

+ *3r

+ Vl,r-+l,r-l

lr

1 1
and finally

j = 2

X

J k<f k<0

Therefore the last bracketed term in (11a) is

|~exp [i%l «£ ] - exp[ iNff] exp[ i/a £ «£'] =

L k<£

l k<0 J

exp[ i % £

^] -exp[ iNfj ] exp[ i# £ <j£ ] exp[ -i J cblj

]

k<« k<0 k=2

exp[ tf £ «£ ] f

1 - exp[ i (Nf

xP- £ ^. ) ] k<0 j=2

s zero. But from the first of equation (6a) it is clear that the result i (It is understood that terms like ^= in (6b)). Therefore the terms

on the right hand side of (11a) drop out in pairs, leaving us with the result that b(n

2

. .. n ) = 0.

When this result is substituted into (8) we have what we set out to accomplish,

S%

o

(N, r) = 0, r<~ . (12)

1 2
It should be emphasized that (12) has been proved only for the case when all r of the f' s are non zero, since otherwise zero denominators would have occurred, thus requiring a more careful analysis

than was done. It should also be emphasized that (12) has been proved

N

only for the case r<~ . This is because we have made use of the

f's supplied by Bethe's theory, and he has supplied them only for the N

case r< —— which, however, is all that is necessary. Indeed it can

be shown that S%q

(N, r) * if r>-y. (13]

This may be done as follows: We realize first of all that the eigenfunctions have always been written so that a(m's) coefficients contain

indices that locate the position of the inverted spins but are not dependent on the location of the individual up spins. Mathematically, there

is no justification for this; one could just as well make the a(m's)

dependent on the location of the up spins rather than the inverted ones and there would be a one to one correspondence of all eigenstates obtained under these conditions with the states expressed by

(4) which have been used exclusively so far. Specifically, if we

consider ty (N, r) with r<— we may write a $ as just described which

has the a(m's) depending on the location of the or

1 s and these are N-r

in number. We write the subscript zero on $ because each dis- tribution of the 8's determines uniquely a distribution of the or's, which is to say that the a(m's) in <1? are dependent on all the indices

locating the ar's. Now there is nothing in the system to distinguish

up from down. Such a space inversion is equivalent to exchanging

S and S and exchanging the <y*s and (3's. Therefore when we have

1 3
N

r> —— to start with, we wind up with an eigenfunction with

N

r'> —— so that S% (N, r) =

is equivalent to

S i|, (N, N-r) =

S> (N,r')= 0, r'>^-. (14) o- ' ' ' 2 At this point it is convenient to introduce

s

2 = s-s

+ + s

z + (s z

) 2 (15) sV -s

z + (s z

) 2 (16)

Thus, using (12), (15), and (16)

S 4> Q (N,r)= 1/2 (N-2r)[ 1/2 (N-2r)+ 1]^ ( N, r)

= S S>

Q

(N, r)+ 1/2(N-2r)[ 1/2 (N^r)-l]qJ (N, r) (17) N

y . Similarly we find, using (14), (15) and (16), for r<

1 4
s\o

(N, r)= 1/2(N-2r)[ 1/2 (N-2r)-l]LlJo (N, r), (17b)

N 2 for r> — . Thus i\j (N, r) is an eigenstate of S , which is expected

since [S , $?] = . But we now see from (17) that for r< —

S

+ S> (N,r)= (N-2r)^ (N,r), (18)

that is, S> (N,r)* r<^-. (19) T o 2 The corresponding statement when a space inversion is made is

(13), so its correctness is established.

Now since [S",3C ] = 0, S~\\i (N, r) must be an eigenfunction of 3C , that is, S> (N, r) = Ki)j (N, r)

f

= K> (N, r+1)

since the total number of inversions certainly increases by one. The

subscript on if; represents the number of inversions having associated

f values of zero. We may exclude K'iWN, r+1) as a possibility since

then S \\j (N, r+1) would be zero [according to (12)] so we would have a contradiction in (18). Therefore

S>q

(N, r) = K^(Nf r]

1 5
If we assume all the state functions normalized, the constant may be

evaluated as follows:

(S ^

o (N,r), S ^(N, r) )= K*K

(v|j (N, r), S

+S> (N, r) ) = K* K

o o '

so that, using (18), K*K = (N-2r),

and finally,

Now (18) becomes

S>o (N,r)= (N-2r) /2

l|j1 (N,r) . (20)

S^N, r) = (N-2r)

1/2 ^(N, r) . (21

)

We now assume for some particular h, where h<N-2r, it is true

that S>h

(N, r) = {(h+l)(N-2r-h)} \+

j(N, r)

, (22)

and

S

+ ^h+1

(N, r) = {(h+l)(N-2r-h)}% (N, r) , (23)

1 6
where all the state functions are orthonormal. "We now establish the correctness of (22) and (23) by induction. Operating on (23) with S~, and with the help of (4), there results

S

" S\+1 (N' r) =

{(h+1 ><N> 2r - h » ^"^(N, r)

f

(S+ S'-2S Z

)q; h+i

(N, r) = (h+l)(N-2r-h)^

h+1

(N, r)

S

+ S>h+1

(N, r)= (h+2)(n-2r-h-lH

h+1

(N, r) . (24) Now

S ^+1 (N,r) =K^ (N.r)

= K' l|Jh+1 (N,r+l).

Choosing the latter, we may substitute back into (24), giving

K'S%

h+1

(N, r+1) = (h+2)(N-2r-h-lH

h+1

(N, r)

and, using (23) ^ K

, {(h+l)(N-2r)} \(N, r+1) = (h+2)(N-2r-h-l)ib (N, r),

which is impossible since ik (N, r+1) is orthogonal to ik (N, r). If choose the former, we may determine K according to

1 7
(s ~WN,r) ' s ~WN>r)

> = K* K

K*K = \+1

(N, r), S

+ S>

h+1

(N, r) )

= (h+2)(H-2r-h-l)

when using (24). Then

and

S>h+1

(N, r) = {(h+2)(N-2r-h-l)} \+2

(N, r)

S

+

4; h+2

(N, r) = {(h+2)(N-2r-h-l)} ^ \+1

(N, r)|

i. e. , if (22) and (23) are true for a particular h, then they must also be true for h+1. Since (22) and (23) are true for h = 0, we immediately

have

(S~) ip (N, r) = Ai|j, (N, r), h<N-2r o n

where

= 0, h> N-2r, (25)

2 h

A = n (j+l)(N-2r-j).

j=0

1 8
So far we have proved that each of the h inversions introduced

by the S operator has an f value of zero. Further, it is apparent by- looking at the explicit specification of \\j following (4) that the S operation introduces inversions whose location indices do not appear

in the a(m) coefficients. Now since there is only one state of the system, ik(N,r), for each specified set of f values and since such a state can be generated from \\j (N, r) for the same r specified f values,

it follows that the states, ijj (N, r), and those generated from them be means of S will include all possible states of the system. We

therefore have established, that only the h inversions introduced by

the S operator have an f value of zero. Further, the inversions in- troduced by the S operator, and only those, have location indices

that do not appear in the a(m) coefficients. Therefore, we have proved

the following statement: Those inversions, and only those, introduced

by the S operator, i.e., all those inversions whose location indices

do not appear in the a(m) coefficients, individually have an associated

f value of zero. This statement, together with the result (25), is most

important for use in reference (2). It should be noted that in reference

(2) the f's are replaced by k's and non-normalized state functions are used.

1 9
3. Summary

To summarize what has been accomplished we note that we

have used the exact eigenstates of a linear chain that were derived by

Bethe. These states are complete and are characterized by collections of f's, which can be real or complex as well as zero or nonzero and which become the usual wave numbers in the simple spin wave theory. We then focused our attention on all the f's collectively

that are zero and found that they may be given a physical interpretation that is distinct from the non-zero f's. This interpretation is that excitation of the f = modes simply corresponds to tipping the

total angular momentum vector with respect to the vertical and does

not lead to a diminution of the magnitude of the vector, while

excitation of f ^ modes corresponds to a diminution of the vector

but does not change its orientation. We may make this interpretation

since from (1), (17b) and (25) it can be seen that the eigenvalue of S is independent of h as long as h<N-2r. These results are exact and permit an arbitrary number of inversions in the system and so

are not limited to the usual spin wave theory in which the temperature

must be low compared to the Curie temperature to insure that the

relative number of inversions be small.

2
4. References

1. H.A. Bethe, Z. Physik, 7l_, 205, (1931)

2. H. Unruh, Jr., NBS Tech. Note 32 7.

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