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8. 225 / ST S . 042, P h y s i c s i n t h e 20t h C e n t u r y Pr of e s s or Da v id K a is e r , 7 O c tobe r 2 0 2 0
1. Q u a n t u m N u m b e rs a n d S p i n
2. H e i s e n b e rg a n d M a t ri x M e c h a n i c s
3. T h e U n c e r t a i n t y Pri n c i p l e
A t o m ic St r u c t u re a n d Sp e c t ra l L in e s
R e c a l l t ha t one of t he g r e a t s u c c e s s e s of B ohr ’ s ( 1 9 1 3 ) mod e l of t he a t om w a s t ha t i t c ou l d a c c ou nt for t he sp ec t r a l l i n es of h y d r og e n,
of h y d r og e n
Em i s s i o n l i n e s o f t h e “ B a l m e r s e r i e s ” s u c h a s t he B a l me r s e r i e s .
a 0
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4 a 0
9 a 0
A t o m ic St r u c t u re a n d Sp e c t ra l L in e s
R e c a l l t ha t one of t he g r e a t s u c c e s s e s of B ohr ’ s ( 1 9 1 3 ) mod e l of t he a t om w a s t ha t i t c ou l d a c c ou nt for t he sp ec t r a l l i n es of h y d r og e n,
of h y d r og e n
Em i s s i o n l i n e s o f t h e “ B a l m e r s e r i e s ” s u c h a s t he B a l me r s e r i e s .
B y t he 1 8 9 0 s , ph y s i c i s t s ha d obs e r v e d s p l i t t i n gs i n s u c h s pe c t r a l
l i ne s , w he n t he g a s w a s pl a c e d i n a n e x t e r n a l m a gn e t i c f i e l d : a s i ngl e Em i s s i o n l i n e , n o e xt e r n a l B fi e l d
s ha r p l i ne ( w he n B = 0 ) w ou l d a ppe a r a s a c l os e l y - s pa c e d tr i p l e t of
l i ne s ( w he n B ≠ 0 ) : t he “Z e e ma n e ffe c t . ”
Em i s s i o n l i n e s i n t h e p r e s e n c e o f a n e xt e r n a l B fi e l d
Ar n o l d S o m m e r f e l d , t he O r d i na r i u s P r ofe s s or of t he or e t i c a l ph y s i c s i n Mu ni c h, w or k e d w i t h hi s g r a d u a t e s t u d e nt s * t o u nd e r s t a nd t he pu z z l e s of “a t omi c s t r u c t u r e a nd s pe c t r a l l i ne s . ” T he y be g a n b y ge n e r a l i z i n g B ohr ’ s mod e l t o el l i p t i c a l or b it s .
1s t e d i t i o n : 1919
* We r n e r H e i s e n b e r g , Wo l f g a n g P a u l i , Ha n s B e t h e , ...
So m m e rf e ld ’ s A p p ro a c h
+
F or s i mpl i c i t y , B ohr ha d c ons i d e r e d c i r c u l a r or bi t s of e l e c t r ons i n s i mpl e a t oms l i k e h y d r og e n. B u t ( j u s t a s i n c e l e s t i a l me c ha ni c s ) , S omme r fe l d a r gu e d t ha t t he mos t g e ne r a l mot i on s hou l d be el l i p ses .
( L : a n g u l a r mo me n t u m)
W he r e a s a c i r c u l a r or bi t ha s onl y on e d e g r e e of fr e e d om ( r ) , a n e l l i pt i c a l or bi t ha s tw o d e g r e e s of fr e e d om ( r a nd ) . S o S omme r fe l d pr opos e d a ge n e r a l i z a t i o n of B ohr ’ s “q u a nt u m c ond i t i on. ” Ea c h d e g r e e of fr e e d om s hou l d be s u bj e c t t o q u a nt i z a t i on:
- 1
(S a m e a s B o h r ’ s e x p r e s s io n , bu t w i t h n → n r + n )
So m m e rf e ld ’ s A p p ro a c h
I f one qua nt iz e s t he or bi t a l a ngu l a r mome nt u m L ( i n u ni t s of ℏ ) , t he n t he pr o je c t i o n
of L a l ong a gi v e n d i r e c t i on z w i l l onl y t a k e d i s c r e t e v a l u e s , m l ℏ = L . z , w i t h
- n ≤ m l ≤ n
F or n = 1 , m l ∈ { - 1 , 0 , + 1 } ; for n = 2 , m l ∈ { - 2, - 1 , 0 , + 1 , + 2 } . F or a ny i nt e g e r n , t he r e w i l l be ( 2 n + 1 ) v a l u e s of m l : alw ay s an o dd n u m be r .
S o no w i n pl a c e of onl y on e q u a nt u m n u mbe r , a s i n B ohr ’ s mod e l , S omme r fe l d a nd hi s s t u d e nt s be g a n c ons i d e r i ng th r e e q u a nt u m n u mbe r s : ( n r , n , m l ) , e a c h of w hi c h c ou l d onl y t a k e on i nt e g e r v a l u e s .
Not e S omme r fe l d ’ s s t r a t e g y : m u c h l i k e ot he r w or k w i t hi n “ol d q u a nt u m t he or y , ” he be g a n w i t h cl a s s i ca l d e s c r i pt i ons of obj e c t s ’ mot i on, a nd t he n a ppe nd e d s pe c i a l “q u a nt u m c ond i t i ons ” t o c ons t r a i n t he a l l o w a bl e mot i on.
So m m e rf e ld ’ s A p p ro a c h
W h y w ou l d a ny one pu r s u e s u c h ba r oq u e Em i s s i o n l i n e , n o e xt e r n a l B fi e l d
c ompl e x i t y ? B e c a u s e S omme r fe l d q u i c k l y fou nd a w a y t o a d d r e s s Z e e ma n ’ s obs e r v e d s pl i t t i ng of
s pe c t r a l l i ne s i nt o t r i pl e t s . Em i s s i o n l i n e s i n t h e p r e s e n c e o f a n e xt e r n a l B fi e l d
A n e l e c t r i c c ha r g e q t ha t i s mo v i ng w i t h s ome a ngu l a r mome nt u m L w i l l ha v e a
m a gn e t i c m o m e n t
I n a n e x t e r na l ma gne t i c fi e l d , t he e ne r g y of t he s y s t e m w i l l d e pe nd on t he r e l a t i v e or i e nt a t i on of a nd B :
If o n e q u a n t i z e d L ( a nd he nc e ) , t he e ne r g y l e v e l s of a n e l e c t r on i n a n e x t e r n a l m a gn e t i c f i e l d w ou l d be sp l i t i nt o m l d i s t i nc t l e v e l s . T he Z e e ma n t r i pl e t s m u s t ha v e c ome fr om e l e c t r ons ma k i ng t r a ns i t i ons fr om a n or bi t w i t h n = 1 ( a nd he nc e m l = - 1 , 0 , + 1 ) t o a n or bi t w i t h n = 0 ( a nd he nc e m l = 0 ) . T he l i ght e mi t t e d fr om t hos e t r a ns i t i ons w ou l d ha v e s l i gh t l y d i f f e r e n t e n e r gi e s , y i e l d i ng t he t hr e e c l os e l y - s pa c e d s pe c t r a l l i ne s .
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“ A n o m a lo u s ” Z e e m a n E f f e c t
S omme r fe l d ’ s a ppr oa c h — t r e a t a n e l e c t r on ’ s mot i on w i t h a l l t he t ool s of c l a s s i c a l me c ha ni c s , t he n i mpos e “q u a nt u m c ond i t i ons ” t o r e s t r i c t v a l u e s of c e r t a i n q u a nt i t i e s — a d d r e s s e d t he “or d i na r y ” Z e e ma n s pl i t t i ng i nt o t r i pl e t s . B u t t he r e w a s a l s o e v i d e nc e of a n “a noma l ou s ” Z e e ma n e ffe c t : do u b le t s !
O ne of S omme r fe l d ’ s s t u d e nt s , W o l f g a n g P a u l i , w or k e d on t he c ha l l e ng e w hi l e a pos t d oc w i t h Ni e l s B ohr i n Cope nha g e n i n t he
Em i s s i o n l i n e , n o e xt e r n a l B fi e l d ea r l y 1 9 2 0 s .
Em i s s i o n l i n e s i n t h e p r e s e n c e o f a n e xt e r n a l B fi e l d
“ A c ol l e a gu e w ho me t me s t r ol l i ng r a t he r a i ml e s s l y i n t he be a u t i fu l s t r e e t s of Cope nha g e n s a i d t o me i n a fr i e nd l y ma nne r , ‘ Y ou l ook v e r y u nha ppy ’ ; w he r e u pon I a ns w e r e d fi e r c e l y , ‘ H o w c a n one l ook ha ppy w he n he i s t hi nk i ng a bou t t he a noma l ou s Z e e ma n e ffe c t ? ’ ” – P a u l i r e c ol l e c t i ons
Co p e n h ag e n c a. 1900
Image is in the public domain.
“ A n o m a lo u s ” Z e e m a n E f f e c t
Gr a d u a t e s t u d e nt s of H e nd r i k L or e nt z i n L e i d e n, Ge o r ge Uh l e n b e c k a nd Sa mu e l G o u d s mi t , w e r e a l s o w or k i ng on t he a noma l ou s Z e e ma n e ffe c t .
T he y r e a s one d t ha t t he Ea r t h ha s tw o k i nd s of a ngu l a r mome nt u m: i t or bi t s t he S u n ( K e pl e r i a n e l l i pt i c a l or bi t ) , an d i t s pi ns on i t s o w n a x i s ( d a y / ni ght ) .
If t he e l e c t r on ha d a n int r ins ic “s pi n” S ( a k i n t o t he E a r t h ’ s r ot a t i on a r ou nd i t s o w n a x i s ) , an d t ha t s pi n we r e qua nt iz e d ,
t he n t he e l e c t r on w ou l d ha v e a n addi t i o n al ma gne t i c mome nt :
“ A n o m a lo u s ” Z e e m a n E f f e c t
Gou d s mi t a nd Uhl e nbe c k t he n a r gu e d a s S omme r fe l d ha d d one : i n a n e x t e r na l B fi e l d , . If t he “s pi n” c ou l d
Em i s s i o n l i n e , n o e xt e r n a l B fi e l d onl y e v e r l i ne u p par alle l or an t i par alle l t o B , t he n t he r e s hou l d be
do u b le t s of v a r i ou s s pe c t r a l l i ne s , w hos e s e pa r a t i on d e pe nd e d on
| S | = ℏ /2 . T he y c a l l e d t hi s “ sp a c e q u a n t i za t i on ”: t he s pi n v e c t or
Em i s s i o n l i n e s i n t h e p r e s e n c e o f a n e xt e r n a l B fi e l d c ou l d onl y poi nt a l ong di s c r e t e d i r e c t i ons i n s pa c e .
B y fi x i ng t he ma gni t u d e | S | = ℏ / 2 , t he y fou nd a c l os e ma t c h t o Z e e ma n ’ s r e s u l t s . B u t w i t h a ( c onc e pt u a l ) pr i c e : w i t h t ha t ma gni t u d e of | S | , a poi nt on t he e l e c t r on ’ s e q u a t or w ou l d be s pi nni ng fas t e r t h a n t h e s p e e d o f l i gh t , a nd i t s ma s s w ou l d d i v e r ge . B y 1 9 2 4 , s u c h a v a l u e of s pi n — i f i ma gi ne d a s a r e a l , ph y s i c a l mot i on — s e e me d a bs u r d ; t he e ffe c t s e e me d i mpos s i bl e t o vi s u ali z e .
T he i r a d v i s or , L or e nt z , c a u t i one d t he m not t o pu bl i s h. B u t t he i r ot h er a d v i s or , Pa u l E h r e n f e s t , ha d a l r e a d y s e nt t he i r pa pe r t o a j ou r na l w i t hou t t e l l i ng t he m! “ Y ou ’ r e s t i l l y ou ng e nou gh t o a ffor d a s t u pi d i t y , ” he e x pl a i ne d . *
*B y t o d a y ’ s s t an d ar d s , E h r e n f e s t ’ s ac t i o n s w e r e t o t al l y un e t h i c al . A d vi s o r s t o d a y w o ul d p ut
th eir o w n n a m e on t he pa pe r a nd th en su b m i t i t t o a j o u r n a l , b e h i n d t h e i r st u d e n t s’ b a c k s . . .
“ A n o m a lo u s ” Z e e m a n E f f e c t
Co p e n h ag e n c a. 1900
Image is in the public domain.
I nd e pe nd e nt of Gou d s mi t a nd Uhl e nbe c k , W ol fg a ng P a u l i i nt r od u c e d hi s “e x c l u s i on pr i nc i pl e ” e a r l y i n 1 9 2 5 .
F ol l o w i ng hi s o w n a d v i s or , A r nol d S omme r fe l d , P a u l i c ons i d e r e d a d d i ng a fo u r t h q u a nt u m n u mbe r , n s , w hi c h ( l i k e t he ot he r n i ) c ou l d onl y t a k e on c e r t a i n d i s c r e t e v a l u e s .
H e a r gu e d t ha t i f t hi s fou r t h q u a nt u m n u mbe r ha d a “ cl a s s i ca l l y ind e s c r ib a b l e d oub l e - va l u e d n e s s ” ( i . e . , c ou l d onl y t a k e on 2 v a l u e s ) , t he n one c ou l d a c c ou nt for t he a noma l ou s Z e e ma n e ffe c t a s w e l l a s ot he r fe a t u r e s of a t omi c s t r u c t u r e .
T hi s be c a me k no w n a s t he Pa u l i e x c l u s i o n p r i n c i p l e : e l e c t r ons i n a n a t om m u s t be d e s c r i be d b y fou r q u a nt u m n u mbe r s ( n r , n , m l , n s ) , a nd no t w o e l e c t r ons c a n ha v e t he s a me s e t of q u a nt u m n u mbe r s
a t t he s a me t i me .
Pa u l i l a t e r c l a i m e d t o h a v e b e e n i n s p i r e d b y t h e p r e c i s i o n o f Ca n - Ca n da n c e r s , wh o al w a y s m an ag e d t o g e t o u t o f e ac h o t h e r ’ s s p o t at t h e l as t m o m e n t .
Q u e s tion s ?
H e is e n b e rg a n d Ma t rix Me c h a n ic s
O ne of P a u l i ’ s c l os e fr i e nd s , We r n e r H e i s e n b e r g — a not he r r e c e nt P hD s t u d e nt of S omme r fe l d ’ s — s ha r e d P a u l i ’ s fr u s t r a t i on w i t h t he a ppr oa c h of t r y i ng t o fi nd v i s u a l i z a bl e , c l a s s i c a l mod e l s a nd t he n a ppe nd i ng a d hoc “q u a nt u m c ond i t i ons . ” I n 1 9 2 4 , H e i s e nbe r g be g a n a pos t d oc pos i t i on w i t h Ni e l s B ohr i n Cope nha g e n.
H e i s e nbe r g s ou ght a ne w “ qua nt um me c h a nic s ”: a fi r s t - pr i nc i pl e s t r e a t me nt of t he a t omi c r e a l m, r a t he r t ha n a k l u d g e . H e s ou ght t o br e a k t he i mpa s s e of t he B ohr - S omme r fe l d a ppr oa c h b y r e t u r ni ng t o E i ns t e i n ’ s ( Ma c hi a n) pos i t i v i s m of 1 9 0 5 : w e c a n ne v e r ob ser v e a n e l e c t r on i n i t s or bi t w i t hi n a n a t om, s o w e s hou l d s t op t r y i ng t o c a l c u l a t e a t omi c pr ope r t i e s on t he ba s i s of qua si - c l a s s i c a l or bi t s .
He i se n b e rg , “ O n t h e q u a n t u m - th e o r e tic a l re i n t e r p re t a t i o n o f k i n e m a t i c a n d me c h a n i c a l r e l a t i o n s h i p s , ” 1 9 2 5
Image is in the public domain.
H e is e n b e rg a n d Ma t rix Me c h a n ic s
He i se n b e rg , “ O n t h e q u a n t u m - th e o r e tic a l re i n t e r p re t a t i o n o f k i n e m a t i c a n d
O ne of P a u l i ’ s c l os e fr i e nd s , We r n e r H e i s e n b e r g — a not he r r e c e nt P hD s t u d e nt of S omme r fe l d ’ s — s ha r e d P a u l i ’ s fr u s t r a t i on w i t h t he a ppr oa c h of t r y i ng t o fi nd v i s u a l i z a bl e , c l a s s i c a l mod e l s a nd t he n a ppe nd i ng a d hoc “q u a nt u m c ond i t i ons . ” I n 1 9 2 4 , H e i s e nbe r g be g a n a pos t d oc pos i t i on w i t h Ni e l s B ohr i n Cope nha g e n.
“I t s e e ms s e ns i bl e t o d i s c a r d a l l hope of obs e r v i ng hi t he r t o u nobs e r v a bl e q u a nt i t i e s [ l i k e a n e l e c t r on ’ s or bi t ] . I ns t e a d i t s e e ms mor e r e a s ona bl e t o t r y t o e s t a bl i s h a t he or e t i c a l q u a nt u m me c ha ni c s , a na l og ou s t o c l a s s i c a l me c ha ni c s , bu t i n w hi c h onl y r e l a t i ons be t w e e n ob ser v a b l e qua nt it ie s a ppe a r . [ . . . P r e v i ou s a ppr oa c he s c ou l d be ] s e r i ou s l y c r i t i c i z e d on t he g r ou nd s t ha t t he y c ont a i n, a s ba s i c e l e me nt s , r e l a t i ons hi ps be t w e e n q u a nt i t i e s t ha t a r e a ppa r e nt l y uno b s e r v a b l e in p rinc ip l e , s u c h a s pos i t i on a nd pe r i od of r e v ol u t i on of t he e l e c t r on. ”
me c h a n i c a l r e l a t i o n s h i p s , ” 1 9 2 5
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E i ns t e i n ’ s r e s pons e : “ A g oo d j ok e s hou l d n ’ t b e r e pe a t e d t o o oft e n. ”
H e is e n b e rg a n d Ma t rix Me c h a n ic s
H e i s e nbe r g a r gu e d t ha t ph y s i c i s t s s hou l d foc u s on em p i r i c a l qua nt it ie s , s u c h a s t he fr e q u e nc i e s of s pe c t r a l l i ne s . I n pa r t i c u l a r , t he
Em i s s i o n l i n e s o f t h e “ B a l m e r s e r i e s ” fr e q u e nc i e s of s pe c t r a l l i ne s obe y e d a law o f addi t i o n .
of h y d r og e n
T hi s r e l a t i ons hi p c ou l d be e x t e nd e d : . I n ot he r w or d s , a n e l e c t r on c ou l d j u mp fr om ( s a y ) m = 6 t o n = 1 a l l a t onc e ( 16 ) , or v i a 6 → 3 , 3 → 1 , or 6 → 3 , 3 → 2 , 2 → 1 ,
a nd s o on.
He i s e n b e r g b e g a n t o c o n s i d e r ar r ay s o f the s e ob s e r v a b l e qua nt i t i e s , nm .
H e is e n b e rg a n d Ma t rix Me c h a n ic s
H e i s e nbe r g a r gu e d t ha t ph y s i c i s t s s hou l d foc u s on em p i r i c a l qua nt it ie s , s u c h a s t he fr e q u e nc i e s of s pe c t r a l l i ne s . I n pa r t i c u l a r , t he
Em i s s i o n l i n e s o f t h e “ B a l m e r s e r i e s ” fr e q u e nc i e s of s pe c t r a l l i ne s obe y e d a law o f addi t i o n .
of h y d r og e n
I n Ma y 1 9 2 5 , i n t he mi d s t of t he s e s t u d i e s , H e i s e nbe r g s u ffe r e d fr om ha y fe v e r a nd t r a v e l e d t o t he i s l a nd of He l i go l a n d i n t he Nor t h S e a . T he r e he c ont i n u e d hi s w or k
Is l a n d o f H el i g o l a n d , o f f t h e co a s t o f D en m a r k for a bou t t w o w e e k s .
Image is in the public domain.
He i s e n b e r g b e g a n t o c o n s i d e r ar r ay s o f the s e ob s e r v a b l e qua nt i t i e s , nm .
H e is e n b e rg a n d Ma t rix Me c h a n ic s
O n t he i s l a nd , H e i s e nbe r g r e a s one d : t he s e fr e q u e nc i e s r e fe r t o s pe c t r a l l i ne s , t ha t i s , t o l i gh t , s o t he y a ppe a r i n t he ex p on en t w he n
Em i s s i o n l i n e s o f t h e “ B a l m e r s e r i e s ” d e s c r i bi ng l i ght w a v e s : fr e q u e n c y
of h y d r og e n
I f t he fr e q u e nc i e s add , t he n t he a mpl i t u d e s A mu s t mul t i p l y :
am p l i t ud e
S o i f , one s hou l d r e q u i r e . But H e i s e nbe r g fou nd :
T h e or d e r o f m u l t ip l ic a t io n c ha ng e d t h e r e s u l t !
H e is e n b e rg a n d Ma t rix Me c h a n ic s
S oon a ft e r fi ni s hi ng hi s pa pe r , H e i s e nbe r g be g a n a ne w pos i t i on i n Göt t i ng e n, w or k i ng c l os e l y w i t h t he ma t he ma t i c a l ph y s i c i s t Ma x B o r n . B or n ’ s r e a c t i on: “ Y ou d u mmk opf ! Y ou ’ r e s t u d y i ng ma t ri c e s !” *
G öt t i ng e n I ns t i t u t e f or T he or e t i c a l P h y s i c s
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T h e fa c t t ha t A x B ≠ B x A i s a ge n e r a l f e a t u r e o f ma t r i x m u l t i pl i c a t i on : ma t ri c e s d o no t c o mmut e .
*a r o ug h p a r a p h r a s e . . .
H e is e n b e rg a n d Ma t rix Me c h a n ic s
S oon a ft e r fi ni s hi ng hi s pa pe r , H e i s e nbe r g be g a n a ne w pos i t i on i n Göt t i ng e n, w or k i ng c l os e l y w i t h t he ma t he ma t i c a l ph y s i c i s t Ma x B o r n . B or n ’ s r e a c t i on: “ Y ou d u mmk opf ! Y ou ’ r e s t u d y i ng ma t ri c e s !” *
G öt t i ng e n I ns t i t u t e f or T he or e t i c a l P h y s i c s
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T h e fa c t t ha t A x B ≠ B x A i s a ge n e r a l f e a t u r e o f ma t r i x m u l t i pl i c a t i on : ma t ri c e s d o no t c o mmut e .
*a r o ug h p a r a p h r a s e . . .
Q u e s tion s ?
T h e Un c e r t a in t y P rin c ip le
I n H e ise nbe rg ’ s for m ul a t i on, ph y sic a l qu a n t itie s a re re pr e se n t e d b y m a tr i c e s ; he n c e t he o u tc o m e of t ra ns for ma t i ons d e pe nd s on t he o r d e r of ope ra t i ons .
I n t he s pri ng of 1927 , H e ise nbe rg r et u r ne d t o B ohr ’ s I n st it ut e i n Cope nha g e n. He a i me d t o w ork o ut a p h y s i ca l in te r p r e t a ti o n of w ha t non - c omm ut i ng ma t r ic e s mi ght me a n for t he qu a n t u m re a l m. He i ma gi ne d a
N ie ls B oh r ’ s I n s titu te , C op e nh a g e n
g a mm a - r a y mi c r os c o p e .
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s c a t t e r e d
g a mm a r a y e l e c tr o n
g a mm a r a y ( h ig h - e n e r g y li gh t )
I f w e w a nt t o me a su re t he p os iti o n of a n e le c t ron, w e c a n bou n c e li ght off of it a nd c o lle c t t he sc a t t e re d li ght . But t he e le ct ron is s m a l l , s o w e n e e d li ght w it h a s m a ll w a v e l e n gth ( l a rg e fr e que n c y) t o g e t g ood r e s o lut i on.
A ny li ght sc a t t e re d w it hi n a n a ng le w ill e n t e r t he a pe r t u re of t he m ic ro sc ope . R e s o l v in g p o w e r :
T h e Un c e r ta in t y P rin c ip le
I n H e ise nbe rg ’ s for m ul a t i on, ph y sic a l qu a n t itie s a re re pr e se n t e d b y m a tr i c e s ; he n c e t he o u tc o m e of t ra ns for ma t i ons d e pe nd s on t he o r d e r of ope ra t i ons .
I n t he s pri ng of 1927 , H e ise nbe rg r et u r ne d t o B ohr ’ s I n st it ut e i n Cope nha g e n. He a i me d t o w ork o ut a p h y s i ca l in te r p r e t a ti o n of w ha t non - c omm ut i ng ma t r ic e s mi ght me a n for t he qu a n t u m re a l m. He i ma gi ne d a
N ie ls B oh r ’ s I n s titu te , C op e nh a g e n
g a mm a - r a y mi c r os c o p e .
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T he a pe r t u re w ill c o lle c t sc a t t e re d li ght w it h mome nt a p pho t o n w it hi n a c one of a ng ul a r size : t he
g a mm a r a y
s c a t t e r e d sc a t t e re d phot ons c o ul d ha v e a n y c ompone nt p x w it hi n
e l e c tr o n p x
= | p pho t o n
| si n .
p pho t o n
p x
F o ll o w i ng t he sc a t t e ri ng, t he e l e c t r o n w ill a c qui re
x pho t o n
g a mm a r a y s ome mome n t u m w it hi n p = | p | si n .
( h ig h - e n e r g y li gh t )
But | p pho t o n | = h / .
T h e Un c e r ta in t y P rin c ip le
N ie ls B oh r ’ s I n s titu te , C op e nh a g e n
I n H e ise nbe rg ’ s for m ul a t i on, ph y sic a l qu a n t itie s a re re pr e se n t e d b y m a tr i c e s ; he n c e t he o u tc o m e of t ra ns for ma t i ons d e pe nd s on t he o r d e r of ope ra t i ons .
I n t he s pri ng of 1927 , H e ise nbe rg r et u r ne d t o B ohr ’ s I n st it ut e i n Cope nha g e n. He a i me d t o w ork o ut a p h y s i ca l in te r p r e t a ti o n of w ha t non - c omm ut i ng ma t r ic e s mi ght me a n for t he qu a n t u m re a l m. He i ma gi ne d a g a mm a - r a y mi c r os c o p e .
information, see https://ocw.mit.edu/help/faq-fair- use/
s c a t t e r e d
g a mm a r a y e l e c tr o n
g a mm a r a y ( h ig h - e n e r g y li gh t )
One c a nnot ma k e b o t h x a n d p x a rb it ra r ily s ma ll a t t he s a me t i me !
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Combi ne :
T h e Un c e r ta in t y P rin c ip le
N ie ls B oh r ’ s I n s titu te , C op e nh a g e n
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H e ise nbe rg i nt re pr e te d t h is r e sult a s a d is t ur ba n c e : w e a re c lu ms y , a nd w e c a n ’ t h e l p b ut d ist u rb t i ny t hi ngs li k e e le c t rons w he n w e t r y t o me a su re t h e i r prope r t ie s .
B ohr st rong ly d is a g r e e d . Du ri ng i n t e n se — s om e t i m e s t e a r - st r e a k e d
— d isc ussi ons t hrou gho ut t he s pri ng a nd su mme r of 1927 , H e ise nbe rg a nd B ohr a rg ue d o v e r ho w t o ma k e se n se of t he ne w un ce r ta in ty p r i n i cp l e . *
B ohr ’ s i n t e r pr et a t i on: x p x ~ h is n o t a r e sult of ou r c lu m si n e s s , b ut a fa c t a bo ut qu a n t um o bj e ct s th e m s e l v e s : t h e y si mp ly d o not a nd c a nnot ha v e si m ult ane o usly s ha r p v a lues for c e r t a i n pa i rs of pr ope r t ie s . T o B ohr , t he e le c t ron di d n o t ha v e x p x = 0 , e v e n b e f o r e it w a s s ma c k e d b y t he phot on.
* S e e M e g a n S hi e l d s F or m a to r e a d i ng: B oh r w as a l w a y s w or k i ng i n d i a l ogu e w i t h ot he r pe opl e , m os t of t e n hi s w i f e M a r g r e t h e B oh r (w ho r a r e l y r e c e i v e d a ny c r e d it) . S om e t i m e s t he s e d i a l ogu e s b ec a m e qu it e e m o t i ona l !
T h e Un c e r ta in t y P rin c ip le
Ni e l s B o h r ’ s I n s t i t u t e , C o p e n h a g e n
I f t he u nc e r t a i nt y pr i nc i pl e he l d for q u a nt u m obj e c t s t he ms e l v e s ( a nd not j u s t a s a c ons e q u e nc e of ou r i nt e r a c t i ons w i t h t he m) , t he n t he r e c ou l d be no t ra j e c t o rie s for q u a nt u m obj e c t s : a ft e r a l l , a t r a j e c t or y r e q u i r e s k no w i ng wh e r e an o bje c t i s a nd w h e r e i t i s go i n g at e a c h mome nt i n t i me .
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T o B ohr , a t l e a s t , t he u nc e r t a i nt y pr i nc i pl e s e e me d t o i mpl y t ha t gi v e n x ( t 0 ) , one ca n n o t k no w x ( t 1 ) w i t h c e r t a i nt y . T hi s s u g g e s t e d t he fall o f de t e r m i n i s m : gi v e n t he pr e s e nt s t a t e of a s y s t e m a nd k no w l e d g e of t he for c e s a c t i ng on i t w a s n o l o n ge r su f f i c i en t t o pr e d i c t w i t h c e r t a i nt y w ha t w ou l d ha ppe n i n t he fu t u r e .
B ohr e v e nt u a l l y c on v i nc e d He i s e nbe r g of t hi s br oa d e r c onc e pt i on; B ohr c a me t o c a l l i t “t he g e ne r a l e pi s t e mol ogi c a l l e s s on of t he q u a nt u m. ”
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STS.042J / 8.225J Einstein, Oppenheimer, Feynman: Physics in the 20th Century
Fall 202 0
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