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Rahul SiddharthanDepartm ent of PhysicsInd ian Inst i tu le o f Sc ienceBangalore 560 012, india.

The Uncert ain!y Principle for D u m m i e sI couldn't resist the choice of title, but in fact I am not supposingthat the readers are dummies, I assume no prior knowledge ofquantum mechanics, but a basic knowledge of vectors and ofmatrices would be useful. Readers who are unfamiliar withthese can try ploughing ahead anyway, and they can look up adefinition of matrix multiplicat ion and play around the exampleswhen they get that far.The uncertainty principle of Heisenberg is one of the mostfamous statements in science this century, but it causes a lot ofconfusion among students and even among teachers. Forinstance, this periodical has been receiving requests to clarifyissues like 'Is the uncertainty principle compatible withSchr/Sdinger's equation? ' 'Where does the uncertainty principlecome from?' and 'Must any solution of Schr6dinger's equationobey the uncertainty principle?'In fact the uncerta inty principle has nothi ng to do withSchr6dinger's equation. It comes from the basic assumptions ofquantum mechanics: it must automatically be obeyed by a n ystate, wh ether or no t i t obeys the Schr 6d inge r eq ua t ion . In addition,of course, any state given at an instant as a function of spatial co-ordinates must evolve in time according to Schr6dinger's timedependent equation.The confusion perhaps stems from the usual statement of theuncertainty principle in terms of position and momentum, andone's confusion with the classical versions of these objectswhich always have exact values. The student can't see why theydon't (and cannot) have exact values in quantum mechanics (atleast not simultaneously). The reason is that in quantum mecha-nics, these objects are not numbers, but 'operators' which act ona state, and may either leave it unchanged but multiplied by aconstant (so that they' re effectively like numbers) or change itentirely to something else (so that t hey' re not like numbers atall). Our classical idea of a position, or a momentum, is then

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s o m e s o rt o f a v e r a g i n g o f t h e r e a l e f fe c ts o f t h e s e o p e r a to r s .A l l t h i s t e n d s t o b e a b i t t o o c o n f u s i n g fo r t h e b e g i n n i n g s t u d e n ts o w e ' ll t r y t o c l a r i fy it w i t h t h e e x a m p l e o f s p i n s, w h i c h h a v et h e i r o w n a n g u l a r - m o m e n t u m u n c e r t a i n t y r e la t io n s , a n d w h i c ha r e a ls o n o t a c o m f o r t a b l e t o p ic f o r b e g i n n e r s . A f t e r h o p e f u l l ym a k i n g t h i s a b i t c l ea r , w e ' l l m o v e o n t o t h e c a s e o f p o s i t i o n a n dm o m e n t u m . W e w o n ' t a c t u a ll y d e r iv e t h e u n c e r t a i n t y p r in c i p le ,b u t m e r e l y m a k e i t p la u s i b l e.B a s i c P o s t u la t es o f Q u a n t u m M e c h a n i c sT h e f o ll o w i ng a r e t h e b a s i c id e a s o f q u a n t u m m e c h a n i c s . T h e s ea r e g e n e r a l l y i n t i m i d a t i n g t o b e g i n n e r s , so I ' ll t r y t o e x p l a int h e m u s i n g t h e s p e c i f ic e x a m p l e o f a sp i n s y s t e m . I f t h e r e a d e rg e t s c o m f o r t a b l e w i t h t h e i d e a o f a n o b s e r v a b l e ( li k e p o s it i o n o rm o m e n t u m o r s p in ) b e i n g a n o p e r a to r , r a t h e r t h a n a n u m b e r ( or ,m o r e l ik e l y , g i v e s u p t r y i n g t o u n d e r s t a n d a n d d e c i d e s to t a k e ito n f a i t h ) , t h e n t h e r e s t i s n o t d i f f ic u l t .E a c h s t a te o f a s y s t e m i s r e p r e s e n t e d b y a v e c t o r o f u n i t l e n g t h i na H i l b e r t s p a ce .W e w o n ' t g e t i n t o t h e d e f i n i t i o n o f a H i l b e r t s p a c e h e r e ; b u t o u ro r d i n a r y t h r e e d i m e n s i o n a l s p ac e , w i t h a v e c t o r b e i n g t h e t r i a do f c o o r d i n a t e s o f a p o i n t , i s a n e x a m p l e ( th o u g h a v e r y s i m p l eo n e ) . M o r e g e n e r a l l y , s o is a n N d i m e n s i o n a l v e c t o r s p a c es i m i l a r to o u r t h r e e d i m e n s i o n a l o ne . I n q u a n t u m m e c h a n i c s ,t h e H i l b e r t s p a ce s w e n e e d a r e u s u a l ly i n f in i t e d i m e n s i o n a l , a n da l w a y s c o m p l e x - t h a t i s , t h e c o m p o n e n t s o f v e c t o r s a r e c o m p l e xn u m b e r s , a n d w e c a n m u l t i p l y t h e m b y c o m p l e x n u m b e r s .F o r i n s t a n c e , i f t h e s y s t e m is a p a r t ic l e w h o s e s p a ti a l c o o r d i n a t e sw e a r e i g n o r i n g , a n d w h o s e i n t r i n s i c a n g u l a r m o m e n t u m ( o r' s p i n ' ) c a n p o i n t i n e i t h e r t h e + z a n d - z d i r e c t i o n , it s s ta t es c a nb e ( 1 , 0 ) ( s p i n u p ) , ( 0, I ) ( s p i n d o w n ) , o r a n y o t h e r l i n e a r c o m b i -n a t i o n o f t h e s e w i t h c o m p l e x c o e f f ic i e n ts w h o s e s q u a r e d s u m i su n i t y : t h a t i s, a g e n e r a l t w o c o m p o n e n t c o m p l e x v e c t o r w i t hu n i t m a g n i t u d e .

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A n e s s e n t ia l d i f f e r e n c e o f t h e s e s t a t e s f r o m c l a ss i c a l s ta t e s i s t h ei d e a o f ' s u p e r p o s i t i o n s o f s t a t e s ' - c l as s ic a ll y , a sp i n m u s t p o i n ti n o n e d i r e c ti o n o r a n o t h e r, b u t q u a n t u m m e c h a n i c a l l y b o t hp o s s i b i l i ti e s c a n e x i s t, w h i c h i s w h y w e n e e d a t w o c o m p o n e n tv e c t o r t o d e s c r i b e t h e s t a t es r a th e r t h a n a n o r d i n a r y n u m b e r .I f t h e s y s t e m i s a s i n g l e p a r t i c le i n o n e d i m e n s i o n , t h e ' v e c t o r 'c o u l d b e a c o m p l e x - v a l u e d f u n c t io n o f t h e c o o r d i n a t e . I t c o u l da ls o b e a c o m p l e x - v a l u e d f u n c t io n o f t h e m o m e n t u m , o r a n yo t h e r o b s e r v a b l e . T h e c h o i c e i s o u r s , b u t i n m o s t e l e m e n t a r yp r o b l e m s o n e c h o o s e s t h e c o o r d in a t e . Y o u c a n t h i n k o f t h ew a v e f u n c t i o n , ~ x ) , a s a ' v e c to r ' w h o s e i n d e x i s c o n t i n u o u s ( xv a r y i n g i n a n i n t e r v a l o n t h e r e a l a x i s ) , r a t h e r t h a n d i s c r e t e a s i nthe u sua l v ec to r s ~gn w he re n = 1 , 2 , 3 f o r i n s t an ce .A l l o b s e r v a b l e s o f a s y s t e m a r e r e p r e s e n t e d b y l i n ea r H e r m i t i a no p e r a t o r s a c t in g o n t h e H i l b e r t s p a c e , a n d t h e i r a l l o w e d v a l u e sa r e t h e e i g e n v a l u e s o f t h e s e o p e r a t o r s . F o r e a c h o b s e r v a b l e , th ec o r r e s p o n d i n g e i g e n v e c to r s f o rm a c o m p l e t e s e t w h i c h s p a n s t h eH i l b e r t s p a c e .A n o p e r a t o r is j u s t s o m e t h i n g w h i c h t r a n s f o r m s a v e c t o r i n t oa n o t h e r v e c to r . L i n e a r m e a n s th a t w h e n i t a ct s o n t h e s u m o ft w o v e c t o r s t h e r e s u l t i s t h e s a m e a s i f i t a c t e d o n e a c h i n d i v i d u a l l ya n d t h e n y o u a d d e d t h e t r a n s f o r m e d v e c t o r s. A n e i g e n v e c t o r o ft h e o p e r a t o r i s a v e c t o r w h i c h i s u n c h a n g e d a p a r t f r o m am u l t i p l i c a t i v e f a c t o r w h e n t h e o p e r a t o r a c t s o n it . A n e i g e n v a l u ei s t h e c o r r e s p o n d i n g m u l t ip l i c a ti v e f a c to r . A n N d i m e n s i o n a ll i n e a r o p e r a t o r h a s N e i g e n v a l u e s. F o r a H e r m i t i a n o p e r a t o r a llt h e e i g e n v a l u e s a r e r e a l .F o r i n s ta n c e , i n t h e t w o - c o m p o n e n t s p i n c a se a bo v e , t h e zc o m p o n e n t o f t h e s p i n c o u l d b e r e p r e s e n t e d b y a 2 2 m a t r i x ,a n d s o c o u l d t h e x a n d y c o m p o n e n t s . T h e s e a r e li n e ar o p e r a t o r s ,a n d t h e i r o p e r a t i o n o n t h e s t a te i s a m a t r i x m u l t i p l i c a t i o n i n t ot h e t w o - c o m p o n e n t v e c t o r w h i c h r e p r e s e n ts th e st a te . A ne i g e n v e c to r o f t h e o p e r a t o r is a v e c t o r w h i c h , w h e n a c t e d o n b yt h e o p e ra t o r, m e r e l y g e t s m u l t i p l i e d b y a n u m b e r . T h a t n u m b e ri s c a l l e d a n e i g e n v a l u e .

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T o t a k e a v e r y s i m p l e e x a m p l e , i f t h e z c o m p o n e n t o f t h e s p in , S ,i s t a k e n t o b e

t h e e i g e n v e c t o r s a r eC X 0 / a n a Iw i t h e i g e n v a l u e s 1 /2 a n d - 1 / 2 , r e s p e c t i v e ly .I f a s t a t e i s a n e i g e n s t a t e o f a n o p e r a t o r , a n y m e a s u r e m e n t o f t h ec o r r e s p o n d i n g o b s e r v a b l e w i ll a l w a ys g iv e t h e c o r r e s p o n d i n ge i g e n v a l u e o f t h a t o p e r a t o r . I f a s t a te is a s u p e r p o s i t i o n o fe i g e n s t a t e s , t h e o b s e r v a t i o n w i l l p i c k o n e o f t h e e i g e n v a l u e sa c c o r d i n g t o t h e a b s o l u t e s q u a r e o f t h e w e i g h t o f t h e c o r r e s -p o n d i n g e i g e n s t a t e i n t h e i n i t i a l s ta t e o f t h e s y s t e m , a n d t h es y s t e m w i l l 'c o l l a p s e ' t o t h a t e i g e n s t at e .F o r i n s t a n c e , i f t h e s t a t e o f t h e a b o v e t w o - c o m p o n e n t s p i n i s (~0),a m e a s u r e m e n t o f t h e z c o m p o n e n t o f t h e s p i n w i l l a lw a y s y i e ldt h e v a l u e + 1 / 2 , w h e r e a s i f t h e i n i t i a l s t a t e i s ~ 1 , t h em e a s u r e m e n t w i l l y i e l d + 1 / 2 o r - 1 / 2 w i t h e q u a l p r o b a b i l it y .G en e ra l l y , i f N1, N2, g3 , "'" a r e t he e ige ns t a t e s o f t h e o pe ra to r , andth e s ys t em i s i n a s t a t e ~g = a I N1 + a2N2 + "", w he rea ~ , a 2, ... ar e c o m p l e x n u m b e r s w h o s e a b s o l u t e s q u a r e s s u m t o 1 ,t h e n t h e p r o b a b i l i t y t h a t a m e a s u r e m e n t w i ll f in d t h e s y s t em i nth e st at e ~gl is [ a 1 J2 a n d s o o n .T h e s e a r e t h e p o s tu l a t e s o f q u a n t u m m e c h a n i c s , a n d o n l y t h ed y n a m i c s r e m a i n s t o b e d e s c r i b e d : t h a t ' s w h e r e S c h r 6 d i n g e r' se q u a t i o n c o m e s i n . T h a t e q u a t i o n d e s c r i b e s h o w a s t a te e v o l v e sw i t h t i m e . B u t w e w o n ' t g e t in t o t h at , b u t r a t h e r d e s c r ib e h o wt h e a b o v e p o s t u l a t e s , w i t h o u t S c h r 6 d i n g e r ' s e q u a t i o n , l e a d t ot h e u n c e r t a i n t y p r i n c i p l e .SpinsT h o u g h t h e a b o v e i s n o t a s h a rd a s i t s e e m s a t fi r st s i g h t, i t

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r e q u i re s a n e w w a y o f t h i n k i n g a b o u t th e s y s t e m . T o g e t t h er e a d e r u s e d t o th i s , le t u s d i s c u s s t h e i n t r i n s i c a n g u l a r m o m e n t a ,o r s pi n s, o f q u a n t u m m e c h a n i c a l p a r t i c l e s .

A n g u l a r m o m e n t u m , l ik e l in e a r m o m e n t u m a n d p o si t io n , d oe sn o t m e a n q u i t e t h e sa m e t h i n g i n q u a n t u m m e c h a n i c s a s i t d o e si n c la s s ic a l m e c h a n i c s . I t b e c o m e s t h e s a m e t h i n g i n t h e l i m i t o fl a rg e q u a n t u m n u m b e r s - b u t w h i l e th e ' o r b i t a l ' a n g u l a rm o m e n t u m c a n b e c o m e a s l a rg e a s y o u li k e , t h e i n t r i n s i c ' s p i n 'i s f i x e d , a n d t h e r e f o r e t h e r e i s n o c l a s s i c a l a n a l o g u e o f i t. T h es p i n o f a n e l e c t r o n i s n o t i t s r o t a t i o n a b o u t a n a x i s . I t is a t h i n gi n i t s e l f , w i t h o u t a c l a s s i c a l c o u n t e r p a r t .B a s ic a ll y , h o w e v e r , q u a n t u m m e c h a n i c a l a n g u l a r m o m e n t u mh a s t h r e e c o m p o n e n t s , ea c h r e p r e s e n t e d b y a n o p e r a t o r . I na d d i ti o n , t h e s q u a re o f t h e t o ta l an g u la r m o m e n t u m is t h e s u m o ft h e s q u a r e s o f t h e s e o p e r a t o r s. T h e s e o p e r a t o r s o b e y c e r t a i n' c o m m u t a t i o n r e l a t i o n s ' w h i c h w e ' l l c o m e t o s o o n , a n d W h i c hi m p l y t h a t t h e s q u a r e d t o ta l a n g u la r m o m e n t u m , j 2 , c a n o n l yh a v e e i g e n v a l u e s o f t h e f o r m h j (1 + I ) w h e r e j is an i n te g e r o rh a l f an o d d i n t e g e r; a n d t h e z c o m p o n e n t o f t h e a n g u l a rm o m e n t u m , J z , c a n o n l y ha v e v a lu e s m = j , j - 1 , . .. , - j . T h e r e ' sn o t h i n g s p e c i a l a b o u t J : t h i s is t r u e O f J x a n d J y t o o , a n d o n e c a np i c k a x e s i n a n y d i r e c t i o n o n e c h o o s e s , a n d t h i s i s s t i ll tr u e . I ts o u n d s c o u n t e r i n t u i t iv e i f o n e t h i n k s o f a n a n g u l a r m o m e n t u ma s a c la s si ca l v e c t o r w h o s e z c o m p o n e n t m u s t v a r y c o n t i n u o u s l ya s o n e t u r n s t h e a x es . B u t th i s a n g u l a r m o m e n t u m i s n o t ac l as s ic a l v e c t o r . I t is a q u a n t u m m e c h a n i c a l v e c t o r o p e r a to r , a n dn o m a t t er h o w y o u p i c k y o u r a x es , t h e z e i g e n v a l u e m u s t b e a ni n t e g e r o r h a l f a n o d d i n t e g e r ( w e c a l l t h e l a t t e r a h a lf - i n te g e r f o rs h o r t) . T h e t o ta l s q u a r e d a n g u l a r m o m e n t u m m u s t b e h j ( j + I )w h e r e j i s t h e m a x i m u m a l lo w e d v a lu e o f t h e z c o m p o n e n t . F o rs i m p l i c i t y , i n t h i s s e c t i o n w e w i l l p u t h = 1 .

L e t u s t a k e t h e s m a l l e st v a l u e a ll o w e d f o r j , n a m e l y 1/2 . T h ea l l o w e d v a l u e s o f S z a r e + 1 /2 , a n d t h e t o t a l s p i n i s S 2 = (1 /2) (1 /2 + 1 ) = 3 /4 . W e c a n r e p r e s e n t a s t a t e o f t h e s y s t e m b y

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denoting respectively an up spin or a down spin; a state can alsobe a linear comb ina tion o f these states

where a and fl are complex numbers and la J2 + [fl j z = 1.The most general linear operator on these states would be a2 2 matrix which multiplies these and transforms them intosomethin g else. If we want the states representing up and downspins to be eigenstates of S , S must be a diagonal 2 2 matrix;it is easy to see tha t

Th e eigenvalues of thi s matr ix are ___1/2, corresponding to theup a nd down states above.Now what are the matrices representing S and S ? It can beshown that the three matrices representing S , S and S shouldsatisfy the commu tatio n relations

S S = i Sa y - y Xs s - s s y = i S x ,s s x - s s = i s

and t he total spin S z should commut e with all the componen ts(S 2S - S x S 2 = O, etc). Th is is satisfied if we take

1 1

The re is nothi ng special about choosing eigenstates of S as ourbasis: we could equally choose

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= 1 an d - 1

w h i c h a r e e i g e n s t a t e s o f S w i t h e i g e n v a l u e s 1 /2 a n d - 1 / 2 ,r e s p e c t i v e l y ; o r

9< = i a n d - i

w h i c h a r e e ig e n s ta t e s o f S . O r w e c o u ld s i m p l y c h o o s e d if f e re n tm a t r i c e s ( p e r h a p s p e r m u t a t i o n s o f t h e a b o v e ) fo r S , S y, Sk e e p i n g t h e i r c o m m u t a t i o n r e la t io n s a n d e i g e n v a l u e s t h e s a m e .T h e a b o v e c ho i c e is t h e c o n v e n t i o n a l o n e , a n d t h e s e m a t r ic e s a rec a l le d t h e P a u l i s p i n m a t r i c e s .T h e p o i n t t o b e n o t e d i s t h i s : T h e e i g e n s t a t e s o f S a r e n o t th ee i g e n s ta t e s o f S , b u t a l in e a r c o m b i n a t i o n o f t h e m . S o a s t a tew h i c h h a s a d e f i n i t e v a l u e o f S d o e s n o t h a v e a d e f i n i t e v a l u e o fS a n d v ic e v e r sa . T h i s i s a n u n c e r t a i n t y p r i n c i p l e f o r t h e c o m -p o n e n t s o f s p i n (a n g u l ar m o m e n t u m ) .M o s t g e n e r a l l y , a n u n c e r t a i n t y p r i n c i p l e a r i s e s w h e n e v e r t h eo p e r a t o rs c o r r e sp o n d i n g t o t w o o b s e rv a b l e s d o n ' t c o m m u t e . I ft w o o p e r a t o rs c o m m u t e , o n e c a n a l w a y s f i n d s i m u l t a n e o u se i g e n f u n c t i o n s f o r t h e m , s o o n e c ~ n h a v e s t a t e s w i t h d e f i n i t ev a l u e s f o r b o t h o b s e r v a b le s . B u t i f t h e y d o n o t c o m m u t e , o n ec a n n o t i n g e n e r a l f in d s t a te s w h i c h a r e s i m u l t a n e o u s e i g e n -f u n c t i o n s o f b o t h , a n d w e m u s t h a v e u n c e r t a i n t y .S u p p o s i n g w e h a d a h y p o t h e t i c a l s t a te g / w h i c h w a s a n e i g e n s ta t eo f S , w i t h e i g e n v a l u e s a n d s u p p o s e th e s a m e s t at e w e r e a ne i g e n s t a t e o f s x w i t h e i g e n v a l u e s . T h e n c o n s i d e r t h e o p e r a t io no f t h e c o m m u t a t o r o f S a n d S x o n ~ :

( s s - s s ) v , = s ( s v,) - s ( s v ,) = ~ s ~ , - ~ s v , =s s 9 ' - s sx 9 ' = 0

s i n c e s a n d s a re o r d i n a r y n u m b e r s w h i c h c o m m u t e w i t h t h eo p e r a t o r s S x a n d S . B u t w e a ls o k n o w t h a t

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SSx - SS = i S.

S o u n l e s s t h e e i g e n v a l u e o f S f o r t h i s s t a t e o f z e r o , s u c h a s t a t ec a n n o t e x i st . I n t h e s p i n h a l f ca s e w e w e r e d i s c u s s i n g a b o v e , t h ee i g e n v a l u e s o f S w e r e _ 1 /2 , n o t z e r o ; s o n o s t a t e c a ns i m u l t a n e o u s l y h a v e e x a c t v a l u e s f o r S a n d S .T h u s , t h e u n c e r t a i n t y p r i n c i p l e r e a ll y a r is e s n o t f r o m i m p e r f e c tm e a s u r i n g d e v i c e s o r l i m i t a t i o n s i n o u r u n d e r s t a n d i n g , n o rf r o m t h e d y n a m i c s a s d e s c r i b e d b y S c h r 6 d i n g e r ' s e q u a t i o n : i ta r i se s f r o m t h e v e r y s t r u c t u r e o f q u a n t u m m e c h a n i c s i t se lf .W h e n w e st a te t h a t o b s e r v a b l e s a re n o t p u r e n u m b e r s b u to p e r a t o r s , a n d t h e o b s e r v e d v a l u e s o f t h e s e o b s e r v a b l e s a re t h ee i g e n v a l u e s o f t h e s e o p e r a t o r s , t h a t a l o n e is s u f f i c i e n t to e n s u r et h a t t w o o p e r a t o r s w h i c h d o n o t c o m m u t e m u s t b e r e l a te d b y a nu n c e r t a i n t y p r i n c i p l e.T h e r e is m u c h m o r e to th e s p i n m a t r ic e s a n d t h e c o m m u t a t i o nr e l a t i o n s t h a n h a s b e e n d e s c r i b e d h e r e . I n f a c t t h e a l lo w e de i g e n v a lu e s o f t h e a n g u l a r m o m e n t u m c a n b e d e d u c e d e n t i re l yf r o m t h e c o m m u t a t i o n r e la t io n s . M o r e o v e r , t h e r e a r e s i m p l ew a y s t o w o r k o u t t h e a n a l o g u e s o f t h e P a u l i m a t r i c e s f o r s y s t e m sw i t h h i g h e r s p i n . T h e s e t o p i c s a r e d i s c u s s e d i n s e v e ra l t e x t b o o k s ,s o w e d o n ' t p u r s u e t h e m f u r t h e r h e re .P o s i t i o n a n d M o m e n t u m

T h e s a m e t h in g h a p p e n s w i t h p o s i ti o n a n d m o m e n t u m : t h eo p e r a t o r s X a n d P ( t h e p o s i t i o n an d t h e m o m e n t u m ) d o no tc o m m u t e . T h e i r c o m m u t a t o r is g iv e n b y

X P - P X = i h

T h i s i s a f u n d a m e n t a l p o s t u l a t e o f q u a n t u m m e c h a n i c s ; i t is no tp r o v a b l e . ( In a s e n se , t h e a n g u l ar m o m e n t u m c o m m u t a t i o nr e l a t i o n s a r e a ls o p o s t u l a t e s : i n t h e c a s e o f o r b i t a l a n g u l a rm o m e n t u m , t h e y c a n b e d e r i ve d f r o m t h e a b o v e p o s it io n -m o m e n t u m u n c e r t a in t y r e l at io n , b u t s p in a n g u la r m o m e n t u m isa th i n g i n i t s e l f a n d n o t d e r i v a b l e f r o m t h e a b o v e , s o w e m u s t

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t a k e t h e c o m m u t a t i o n r e l a t i o n t h e r e a s a n a s s u m p t i o n . )N o t e t h a t h e r e b o t h X a n d P a re o p e r at o rs . W h a t t h e s e o p e r a to r sl o o k li ke d e p e n d s o n h o w w e a r e d e s c r i b i n g o u r s y s t e m ; m o s tc o m m o n l y , w e s i m p l y d e s c r i b e it b y a w a v e f u n c t io n w h i c h i s af u n c t io n o f t h e p o s i t i o n x . T h e n w e c a n ta k e t h e p o s i t io no p e r a t o r X t o b e ju s t t h e p o s i t i o n x ( a p u r e n u m b e r ) a n d t h em o m e n t u m o p e r a to r to be - i t t d / d x , w h i c h i s a d i f f e r e n ti a l o p e -r a to r a c t in g o n t h e w a v e fu n c t i o n , n o t a n u m b e r . T h e r e a d e r c a nv e r i f y t h a t t h i s c h o i c e s a ti s fi e s t h e c o m m u t a t i o n r e l a ti o n .N o w w h e r e d o e s th e u r ~ c er ta i nt y p r i n c i p l e c o m e i n ? I t a r i se sf r o m t h e f a ct t h a t P a n d X d o n ' t c o m m u t e . S o w e c a ni m m e d i a t e l y s e e , a s a b o v e , t h a t a s t a t e w i t h d e f i n i t e v a l u e s o f P xa n d X c a n n o t e x is t. I f i t d id , t h e v a l u e o f ( X P - P X ) a c t in g o nt h i s s t a t e w o u l d b e z e r o : b u t i t c a n n o t b e z e r o ; i t m u s t b e i ttt i m e s t h e o r i g i n a l s t a t e (n o m a t t e r w h a t s t a t e i t a c ts o n ).T h e r e p r e s e n t a t io n o f t h e p o s i t io n a n d m o m e n t u m o p e r a t o rsa b o v e i s u s e f u l i n u n d e r s t a n d i n g w h a t ' s g o i n g o n . I n t h i s re p r e -s e n t a t i o n , a p o s i t i o n e i g e n s t a t e is a w a v e f u n c t i o n ~ ( x ) w h i c h i ss h a r p l y p e a k e d a t o n e p o i n t a n d v a n i s h e s e v e r y w h e r e e l se , b u tw h o s e i n t e g r a l o v e r a ll s p a c e i s 1 . T h i s i s c a l l e d t h e D i r a c d e l t af u n c t i o n , a n d r e q u i r e s s o m e s o p h i s t i c a t e d a r g u m e n t s t o s t r i c t l yj u s t i f y - t h e j u s t i f ic a t i o n w a s n o t m a d e t il l t h e 1 9 60 s b u t p h y s i c i s t sf r e e l y u s e d i t e v e r y w h e r e a f t e r D i r a c i n t r o d u c e d i t i n t h e 1 9 3 0 s.T h i s i s o b v i o u s l y a n e i g e n s t a t e o f t h e ' o p e r a t o r ' X = x b e c a u s ex ~ ( x ) = x o ~ ( x ) , x 0 b e i n g t h e o n l y p o i n t w h e r e ~ (x ) i s n o n z e r o .T o a p p l y P o n i t is a b i t t r i c k y , an d t h e b e s t w a y i s t o t a k e ~ ( x )a s a l i m i t o f a h i g h , n a r r o w w a v e p a c k e t . B u t o n e c a n c o n v i n c eo n e s e l f q u i t e e a s i l y t h a t ~ ( x) is n o t a n e i g e n s t a t e o f P .P b e i n g a d i f f e r e n t i a t i o n o p e r a t o r , i t i s c l e a r t h a t i t s e i g e n s t a t e s( f u n c t i o n s t h a t s t a y t h e s a m e a p a r t f ro m a m u l t i p l i c a t iv e c o n s t a n tw h e n a c te d o n b y P ) w i ll b e e x p o n e n t ia l s . I f P is H e r m i t i a n ( h a sr e a l e i g e n v a l u e s ) t h e e x p o n e n t w i l l b e c o m p l e x , s o t h e e i g e n s t a t ew o u l d l o o k li k e e x p ( i k x ) ( = c o s k x + i s i n k x ) . T h i s i s a n o s c i ll a t -i n g f u n c t i o n w i t h w a v e l e n g t h 2 ~ k , b u t i t ' s o s c i l l a t i n g n o t i n

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a m p l i t u d e b u t i n t h e c o m p l e x 'p h a s e ' : i t s a b s o l u t e v a lu e is t h es a m e e v e r y w h e r e . S o t h e m o m e n t u m e i g e n s ta t e is c o m p l e t e l yu n c e r t a i n i n p o s i t i o n , a n d t h e p o s i t i o n e i g e n s t a t e ( i t c a n b es h o w n ) is co m p l e te l y u n c e r t a i n in m o m e n t u m . T h e m o m e n t u me i g e n v a l u e i s h k .T o r e a l l y f o ll o w t h e s e i d e a s t o t h e f u l le s t e x t e n t , i t w o u l d b e g o o df o r t h e r e a d e r t o b e f a m i l i a r w i t h F o u r i e r s e r ie s a n d F o u r i e rt r a n sf o r m s . T h e ' m o m e n t u m s p a c e' r e p r e s e n ta t i o n , a n d t h eu n c e r t a i n t y p r i n c i p l e i n t h i s c a s e i s r e a l ly a n e x a m p l e o f a w e l l-k n o w n u n c e r t a i n t y re l a ti o n i n w a v e p h y s ic s b e t w e e n t h e ' sp r ea d 'o f a w a v e p a c k e t a n d t h e ' s p r e a d ' o f i ts c o n s t i t u e n t w a v e l e n g t h s( F o u r i e r t ra n s f o r m ) . S o m e o f t h i s g r o u n d , a n d r e l a te d' u n c e r t a i n t y p r i n c i p l e s ' i n m a t h e m a t i c s , a r e d i s c u s s e d el s e w h e r ei n t h i s i s su e .E v e n i f t h e r e a d e r is n o t f a m i l i a r w i t h F o u r i e r t r a n s f o r m s , o n ec a n c o n v i n c e o n e s e l f t h a t a l a rg e n u m b e r o f w a v e s w i t h n e a r l ye q u a l w a v e l e n g t h s c a n f o r m a ' w a v e p a c k e t ' 9 t h e r e w i l l t h e n b ea p l a c e w h e r e ( b y d e s i g n , o r c o i n c i d e n c e ) t h e y a ll a d d u pc o n s t r u c t i v e l y to g i v e a f i n i t e v a l u e , b u t a s w e m o v e a w a y f r o mt h a t p o i n t , t h e i r a m p l i t u d e s b e c o m e m o r e a n d m o r e u n c o r r e l a te ds i n c e t h e i r w a v e l e n g t h s a r e a ll d i f fe r e n t , a n d s o t h e t o t a l a m p l i t u d ef al ls . T o g e t a n a r r o w e r w a v e p a c k e t , w e n e e d a b i g g e r s p r e a d i nw a v e l e n g t h s f o r m o r e e f f e c t i v e c a n c e l l a t i o n a w a y f r o m t h em a x i m u m .C o n c l u s i o n

A b o v e , w e h a v e t r e a t e d t w o s p e c i a l c a se s o f t h e u n c e r t a i n t yp r i n c ip l e . T h e p r i n c i p l e is m u c h m o r e w i d e l y a p p l ic a b le t h a nt h a t , b u t t h e s e t w o e x a m p l e s s e r v e to c o n v e y th e f l a v o u r o f w h a ti s g o i n g o n .H i s t o ri c a ll y , t h e d e v e l o p m e n t o f q u a n t u m m e c h a n i c s w as n o tq u i t e s o s i m p l e a s a b o v e , a n d t h e p o s t u la t e s w h i c h w e m a d e i nt h e s e c o n d s e c t i o n d e v e l o p e d q u i t e g r a d u a l l y . T h e f ir s t v e r s i o no f q u a n t u m m e c h a n i c s w a s H e i s e n b e r g ' s ' m a t r i x m e c h a n i c s ' ,w h e r e h e t r e a t e d s o m e s p e c i f i c p r o b l e m s u s i n g m a t r i c e s fo r

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momentum and position rather than numbers. Schr6dinger'sequation followed soon after, and was a partial differentialequation whose solutions yielded the same energy levels asHeisenberg's matrices. At that time this seemed strange, butSchr6dinger soon set his method into the more general frameworkof vector spaces and showed that his method and Heisenberg'swere really the same. Born, Dirac, and many others contributedto this step. So while the uncertainty principle relating toposition and moment um was first suggested by Heisenberg andis known by his name, in a more general context such a principlefollows from the work of several other people, includingSchr6dinger, whom this issue commemorates. The final structureof quantum mechanics emerged from the collective efforts ofvarious people including Heisenberg and Schr6dinger, and fromthis structure various generalizations of he uncerta inty principlecan be derived.

D i n o I u r e n d r a n1 2, L a n g h a m R o a d , U ni v er s i tyo f Z i m b a b w e , P . O , o x . M P 1 6 7,H a r a r e , Z i m b a b w e .

One of the most famous problems in the world is the fou r colourproblem. This merely states a fact that any six-year old armedwith crayons has long suspected - it is possible to colour anymap in the family atlas with only four colours so that no twoneighbouring regions have the same colour.Mathematicians have this habit of being precise, and theydefine a map to be a partition of some finite area into finitelymany contiguous regions. The contiguous bit merely stressesthe point that each region must be connected, so countries likeAngola which are composed of two different parts (the main partof Angola and the Cabinda enclave) are considered as twodifferent regions. Another technical point to note is that twocountries which only touch at a point (like Zimbabwe andNamibia) are not considered to be neighbouring.Another problem with mathematicians (in the last centuryanyway) is that they tend to wait for someone to state a problemformally before they have a go at it. With the four colour

7 6 RESONANCE ) February 1999