On the completeness of the roth functions

Volume 33A, n u m b e r 6

PH Y S I C S L E T T E R S

[

• MD-64 Q - 1.89 • neutmm Q = 186

F(Q't) = S(Q) - A 2 - ~t 2. + A ~=A "~,t4!4

• q = I.,57

| t" ~.

tC

~

• q = 1.53

f o r s m a l l t i m e s (Qt < 0 . 3 p s ~ - 1 ) . H e r e A 2 = Q 2 k B T / M , k B is Boltzmann's constant, M is the

1.0

a t o m i c m a s s a n d A 4 d e n o t e s a f a c t o r of t h e f o u r t h m o m e n t a s o b t a i n e d f r o m MD. Since for these times there is an excellent a g r e e m e n t i n t h e t i m e d e p e n d e n c e of F ( Q , t ) taking into account that the wavevector and the m e a n t e m p e r a t u r e of e x p e r i m e n t a n d MD a r e n o t precisely the same, it is most likely that also the observed deviations at larger times are m a i n l y d u e to t h e m .

u.

0.5

o,

O.5

. . . .

~:o T

'

t [psl

, o

1o

t [psi

References

Fig. 2. The i n t e r m e d i a t e s c a t t e r i n g function obtained by e x p e r i m e n t and by m o l e c u l a r dynamics calculations for two values of

[1] C. D. A n d r i e s s e , Physica 48 (1970) 61. [2] P. Zandveld, C. D. A n d r i e s s e , J. D. B r e g m a n , A. Hasman and J. J. van Loef, Physica, to be published. [3] C. Bruin, Phys. L e t t e r s 28 (1969) 777. [4] P. G. Mikolaj and C. J. Pings, J. Chem. Phys. 46 (1967) 1401, 1412. [5] D. Levesque and L. Verlet, Phys. Rev. L e t t e r s 20 (1968) 905.

In fig. 2 t h e i n t e r m e d i a t e s c a t t e r i n g f u n c t i o n s , t y p i c a l i n t h i s r a n g e of w a v e v e c t o r s , a r e s h o w n f o r two v a l u e s of t h e l a t t e r . It h a s b e e n v e r i f i e d that the intermediate scattering function for both e x p e r i m e n t a n d MD o b e y s t h e r e l a t i o n

ON

THE

30 November 1970

COMPLETENESS

OF

THE

ROTH

FUNCTIONS

P. K. MISRA

Department of Physics, Utkal University, Bhubaneswar-4, Orissa, India Received 27 October 1970

The Roth r e p r e s e n t a t i o n has been widely used in the problem of Bioch e l e c t r o n s in a magnetic field but the completeness of the Roth functions has not been proved. We prove that the Roth functions a r e complete with r e s p e c t to the wave function of the Hamiltonian of an electron in a periodic potential and an uniform magnetid field. T h e R o t h r e p r e s e n t a t i o n [1] h a s b e e n w i d e l y u s e d in t h e p r o b l e m of B l o c h e l e c t r o n s in a m a g n e t i c f i e l d [ 1 - 3 ] . H o w e v e r , t h e c o m p l e t e n e s s of t h e R o t h f u n c t i o n s h a s n o t b e e n p r o v e d . In a r e c e n t p a p e r [4], we h a v e o b t a i n e d a c l a s s of r e p r e s e n t a t i o n f o r B l o c h e l e c t r o n s in a m a g n e t i c f i e l d b y u s i n g o n l y t h e t r a n s l a t i o n a l p r o p e r t i e s of t h e H a m i l t o n i a n . We h a v e s h o w n t h a t if we p u t in t h e c o n d i t i o n t h a t t h e s e b a s i s f u n c t i o n s r e d u c e to B l o c h f u n c t i o n s f o r z e r o m a g n e t i c f i e l d , we o b t a i n a" s e t of m a g n e t i c B I o c h f u n c t i o n s w h i c h a r e

the R o t h f u n c t i o n s . In t h i s p a p e r , u s i n g t h e r e s u l t s of r e f . [4], we s h a l l p r o v e t h a t t h e R o t h f u n c t i o n s a r e c o m p l e t e w i t h r e s p e c t to the w a v e f u n c t i o n of t h e H a m i l t o n i a n of a n e l e c t r o n i n a periodic potential and an uniform magnetic field. In t h e R o t h r e p r e s e n t a t i o n , t h e b a s i s f u n c t i o n s are

dPnk(r) = u n K . ( r ) exp ( i k ' r )

(1)

where K = k + (e/~c)A(iVk), A is the vector potential, unK.(r) is obtained from Unk(r), the

339

Volume 33A, number 6

PttYSICS

p e r i o d i c p a r t of the B l o c h f u n c t i o n s , by r e p l a c i n g k by the o p e r a t o r K* s y m m e t r i c a l l y and the o t h e r s y m b o l s have t h e i r u s u a l m e a n i n g s . We a s s e r t that the set of f u n c t i o n s ePnk(r) a r e c o m p l e t e with r e s p e c t to the w a v e f u n c t i o n of the H a m i l t o n i a n of an e l e c t r o n in a p e r i o d i c p o t e n t i a l and an u n i f o r m m a g n e t i c field. If it is not t r u e , let the s e t of f u n c t i o n s 6nk(r), Xmk(r) f o r m a c o m p l e t e set w h e r e we want to d e t e r m i n e the p r o p e r t i e s of the unknown f u n c t i o n s X m R ( r ) . T h e n the w a v e f u n c t i o n of the H a m i l t o n i a n of an e l e c t r o n in a p e r i o d i c p o t e n t i a l and an u n i f o r m m a g n e t i c f i e l d can be w r i t t e n as

~ ( r , t ) : ~ 6 n k ( r ) ~ n ( k , t ) -~ ~ X m k ( r ) ~ m ( k , t ) nk mk

LETTERS

30 N o v e m b e r

1970

n e n t i a l , i.e.; s y m m e t r i c a l l y . F u r t h e r , s i n c e Grn(K, r) is p e r i o d i c in r, Gin(k, r ) is a l s o p e r i o d i c in r . Since any p e r i o d i c f u n c t i o n can be e x p a n d e d in t e r m s of the p e r i o d i c p a r t of the B l o c h f u n c t i o n s w h i c h a r e c o m p l e t e with r e s p e c t to p e r i o d i c f u n c t i o n s , Gin(k, r) can be w r i t t e n a s Cm(k. r ) : ~ anm Unk(rl

(7)

w h e r e Unk(r) a r e the p e r i o d i c p a r t of the B l o c h f u n c t i o n s and anm a r e i n d e p e n d e n t of k . T h u s we have

Cm(K*,r)

~anm,nK.(rl

(8)

H

(2)

We can a l w a y s w r i t e Xmk-

Fm e x p ( i k . r )

(3}

w h e r e F m is an o p e r a t o r which o p e r a t e s on e x p ( i k , r) to give Xrnk. F r o m e q s . (1), (2) and (3), by f o l l o w i n g the p r o c e d u r e o u t l i n e d in r e f . [4 [. we obtain

w h e r e urcg.(r) is o b t a i n e d f r o m a n k ( r ) by r e p l a c i n g k by K* s y m m e t r i c a l l y . D e h a v e a l s o p r o v e d in r e f . [4] that f o r any s y m m e t r i c f u n c t i o n of g, (fj-(K))* : f ( g * ) . So f r o m eq. (8) we obtain F m = (G~(g))*

= ~.~ anmUnK.(r)

(9)

H

F r o m e q s . (1), (3) and (9), we obtain

~( r, t) : n~kexp (ik" r)UnK ~n(k, t) Xmk :: )-~ a nm %~k + ~

mk

exp (ik. r)Cm~m(k, t)

w h e r e Gm = ( F ~ ) $ . By u s i n g the t r a n s l a t i o n a l p r o p e r t i e s of the H a m i l t o n i a n and f o l l o w i n g the p r o c e d u r e o u t l i n e d in r e f . [4], it can be e a s i l y shown that G m m u s t be p e r i o d i c in r a s w e l l as a s y m m e t r i c f u n c t i o n of K. To obtain the s y m m e t r i c o p e r a t o r Gm(K,r), we want K to a p p e a r in a s i n g l e exponent. We f o u r i e r a n a l y z e GIn(K, r) Gm(~;,r )

[ d@exp(-iK.@)Gm(@,r)

(5/

We d e f i n e the f u n c t i o n

Gm(k , r) = ./-d~ exp (- i k ' ~ ) ) a m ( ~, r )

(6)

It f o l l o w s that Gm(K , r) is the o p e r a t o r o b t a i n e d f r o m Grn(k, r) by r e p l a c i n g k by K in the e x p o -

340

(10)

1t

(4!

T h u s any function in the set Xmk is f o r m e d f r o m a l i n e a r c o m b i n a t i o n of the Roth f u n c t i o n s . So Xmk a r e not i n d e p e n d e n t f u n c t i o n s . H e n c e we p r o v e that the Roth f u n c t i o n s f o r m a c o m p l e t e s e t with r e s p e c t to the w a v e function of the H a m i l t o n i a n of an e l e c t r o n in a p e r i o d i c p o t e n t i a l and an u n i f o r m m a g n e t i c field.

I~(c~'c12cc5 [1] L.M.Roth, J. Phys. Chem. S,,lids 23 (1962) 433. [2] L. M. Roth, Phys. Rex. 145 (1966) 434, [31 P,K. Misra and L.M. Roth, Phys. Rev. 177 (1969) 1089. [4l P.K. Misra. Phys. Roy, t~,be published,