W a v e s a n d P r o b a b i l i t i e s

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8 .2 2 5 / S T S .0 4 2 , P h y s ic s in t h e 2 0 t h C e n t u r y P r o f e s s o r D a v id K a is e r , 1 3 O c t o b e r 2 0 2 0

1. S c h r ö di n g e r a n d W a v e M e c h a n i c s

2. B o r n ’ s I n t e r p r e t a t i o n a n d t h e Do u b l e - Sl i t E x peri m ent *

3. S l i t D e t e c t o r ?*

* S e e o p t i o n a l Le c t u r e N o t e s on t he d ou bl e - sl i t e xp e r i m e n t

Ma t r i x Me c h a n i c s R e c a p

In spr i ng 1 9 2 5 , We r n e r H e i s e n b e r g in t r o d u c e d a n e w , f ir s t - pr i nc i pl e s ap p r o ac h t o q u an t u m t h e o r y . R at h e r t h an b e g i n n i n g w i t h c l as s i c al co n ce p t s a n d ex p r es s i o n s — su c h a s f o r t h e o r b i t o f a n e l e c t r o n i n a n at o m — He i s e n b e r g a r g u e d t h a t o n e s h o u l d f o c u s o n l y o n ob s e r v a b l e fe a t u r e s , s u c h a s t h e p r o p e r t i e s o f s p e c t r a l l i n e s .

He r e a s o n e d t h a t s i n c e t h e fr e q u e n c i e s of s pe c t r a l l i ne s obe y e d a la w o f addi ti o n , t h e am pl i tu de s of t he c or r e s pond i ng s pe c t r a l l i ne s sh o u l d mu l t i p l y . B u t t h e n h e f o u n d t h a t t h e ou t c om e de p en ded o n th e or d e r of m u l t i p l i c a t i on . M a x B o r n c l a r i f i e d : t h e s e a r e ma t r i c e s !

He i s e n b e r g , “ O n t h e q u a n t u m - th e o r e ti c 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

mec h a n i ca l r el a t i o n s h i p s , ” 1 9 2 5

Images are in the public domain.

In spr i ng 1 9 2 7 , a m i d e m ot i ona l d e ba t e s w i t h N i e l s B ohr , He i s e n b e r g d e r i v e d a c o n s e q u e n c e o f h i s n o n - co m m u t i n g m a t r i ces : th e unc e r t a i nt y p r i nc i p l e .

Sc h r ö di n g er a n d W a v e M ec h a n i cs

Du r i n g t h e w i n t e r a n d s p r i n g o f 1 9 2 6 — be g i nni ng j u s t s i x mo n t h s a f t e r He i s e n b e r g ’ s f i r s t p a p e r o n ma t r i x me c h a n i c s — Er w i n S c h r ö d i n g e r in t r o d u c e d a di f f e r e n t fi r s t - pr i nc i pl e s a ppr oa c h t o qua n t um t h e o r y . W h e r e a s H e i s e n b e r g h a d b ui l t up o n i d e a s a b o ut di s c r e te n e s s fr o m am o n g t h e h o d g e - pod g e of “ ol d q u a nt u m t he or y , ” Sc h r ö d i n g e r p i c k e d up o n t h e i d e a o f d e B r o g l i e ’ s ma t t e r w a v e s .

Sc h r ö d i n g e r — or i g i na l l y f r om V i e nna a nd a t t he t i m e t e a c hi ng in Z ü r ic h — wa s a g e n e r a t i o n o l d e r t h a n H e i s e n b e r g a n d P a u l i ; h e ha d a l r e a d y pu bl i s he d ma n y re s e a rc h a r t i c l e s . H i s a p p ro a c h w a s t o re t a i n as m u c h o f t h e “ l o o k an d fe e l ” o f t i m e - te s te d p h y s i c s a s h e co u l d.

Sc hr öd i ng e r , “Qu a nt i z a t i on a s a n

Ei g e n v a l u e P r obl e m, ” 1 9 2 6 co n s t r u ct i v e i n t er f er en ce de s tr u c ti v e i n te r fe r e n c e

Sc h r ö di n g er a n d W a v e M ec h a n i cs

Sc h r ö d i n g e r b e g a n w i t h t h e us ua l e x p r e s s i o n f o r e n e r g y :

De Br o g l i e h a d s u g g e s t e d t h a t ma t t e r w a v e s we r e c r u c i a l . S o Sc h r ö d i n g e r c o n s i d e r e d t h e us ua l e qua t i o n f o r a s t a n d i n g w a v e :

De Br o g l i e h a d i d e n t i f i e d a wa v e l e n g t h fo r m at t e r :

Sc hr öd i ng e r , “Qu a nt i z a t i on a s a n Ei g e n v a l u e P r obl e m, ” 1 9 2 6

new “ q u a nt u m oper a t or ” :

ℏ bu i l t i n f r o m t h e s t ar t .

Sc h r ö di n g er a n d W a v e M ec h a n i cs

Sc h r ö d i n g e r w a v e eq u a t i o n :

Al l c o n t i n u u m ! No “ s p e c i a l s t a t e s ” o r o r b i t s w i t h d i s c r e t e pos i t i ons ( a s i n B ohr ’ s m od e l ) . P l u s , t he m a t he m a t i c s w a s fa m i li a r to p h y s i c i s ts : d i f f e r e n ti a l e q u a ti o n s r a th e r th a n a b s tr a c t m a tr i c e s .

Fi r s t b i g t e s t : s u b s t i t u t e . T h e n s o l u t i o n s t o Sc h r ö d i n g e r ’ s e qua t i o n c o r r e s p o n d t o

n = 1 , 2 , 3 , . . .

Sc hr öd i ng e r , “Qu a nt i z a t i on a s a n Ei g e n v a l u e P r obl e m, ” 1 9 2 6

Ex a ct l y t h e s a m e s p ect r u m a s t h e B o h r m o d el , s o i t r e p r o d u ces th e B a l m e r s p e c tr u m !

Sc h r ö di n g er a n d W a v e M ec h a n i cs

An a l o g o u s t o o n e - di m en s i o n a l s t a n di n g w a v es o n a s t r i n g :

L

di s c r e t e al l o w ab l e w a v e n u m b e r s ,

or “ ei g en v a l u es ”

appl y bo u n dar y c o n di t i o n s

Sc hr öd i ng e r , “Qu a nt i z a t i on a s a n Ei g e n v a l u e P r obl e m, ” 1 9 2 6

Fo r t h e H y d r o g e n at o m , ap p l y t h e bou nd a r y cond i t i ons :

Sc h r ö di n g er a n d W a v e M ec h a n i cs

So l ut i o n s  to S c h r ö d i n g e r ’ s w a v e e q u a ti o n o b e y e d su p er p o si t i o n : if  1 ( t , x ) we r e a s o l u t i o n a n d  2 ( t , x ) we r e a s o l u t i o n , t h e n

 3 ( t , x ) =  1 ( t , x ) +  2 ( t , x ) wa s al s o a s o l u t i o n .

Th a t m e a n t t h a t S c h r ö d i n g e r ’ s w a v e f u n c t i o n s  co u l d u n der g o

co n s t r u ct i v e an d de s tr u c ti v e in t e r f e r e n c e .

So  ha d d e f i ni t e w a v e l i k e pr ope r t i e s . B u t wh a t wa s  ? A

co n s t r u ct i v e

2 in t e r f e r e n c e

1

1 2 3 4 5 6 7

wa v e o f wh a t ?

Ev en m o r e b i z a r r e: i f o n e co n s i d er ed a s ys t em w i t h tw o el ect r o n s , V → V ( x 1 , y 1 , z 1 , x 2 , y 2 , z 2 ) . S o t h e n  wo u l d b e a wa v e i n s o m e si x - di m e n s i o n al s pac e ! Cl e a r l y  wa s n o t j u s t l i k e a

wa t e r wa v e. . .

- 1 de s tr u c ti v e

in t e r f e r e n c e

- 2

A N e w Q u a n t u m M e c h a n i c s

Sc h r ö d i n g e r a n d H e i s e n b e r g h a d s t r o n g f e e l i n g s a b o ut e a c h o t h e r ’ s w o r k .

So i t w a s a l l t h e m o r e s ur p r i s i n g w h e n s e v e r a l p h y s i c i s t s ( Sc hr ö d i n g e r hi m s e l f , Pa s c u a l J o r d a n , a n d Pa u l D i r a c ) d e m o n s tr a te d l a te r i n 1 9 2 6 th a t H e i s e n b e r g ’ s ma t r i x me c h a n i c s a n d S c h r ö d i n g e r ’ s w a v e me c h a n i c s w e r e ma t h e ma t i c a l l y eq u i v a l en t .

By t h e e n d o f 1 9 2 6 , p h y s i c i s t s b e g a n t o r e f e r t o a n e w “ qua nt um m e c h a ni c s .”

Qu e s t i o n s ?

Th e do u b l e - sl i t ex p er i m en t is o n e o f t h e m o s t c e le b r a t e d e x p e r im e n t s in mo d e r n p h y s i c s .

He i s e n b e r g le c t u r e d o n it in 1 9 2 9 ; Sc hr ö d i n g e r in 1 9 3 6 .

Ni e l s B o h r fe at u r e d i t i n h i s d i s c u s s i o n o f h i s l o n g d e b at e w i t h Al b e r t Ei n s t e i n ov e r q u a n t u m t h e o r y ( 1 9 4 9 ) .

Ri c h a r d F e y n ma n decl a r ed t h a t i t “ h a s i n i t t h e he a r t of q u a nt u m me c h a n i c s ” ( 1 9 6 2 ) .

Re a d e r s o f Ph y si c s W o r l d ma g a z i n e v o t e d i t t h e # 1 a l l - ti m e s i n gl e m o s t be a u t i f u l e x pe r i m e nt i n a l l of ph y s i c s ( 2 0 0 2 ) !

Ni el s B o h r , “Di s cu s s i o n s w i t h Ei n s t ei n o n Epi s t e mol ogi c a l P r obl e ms i n A t omi c Ph ys i cs ” ( 1 9 4 9 )

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* S e e o p t i o n a l Le c t u r e N o t e s on t he d ou bl e - sl i t e xp e r i m e n t

Ca s e 1 : C l a s s i c a l P a r t i c l e s

Fi r e bu l l e t s one a t a t i m e a t a bu l l e t - pr oof w a l l t ha t c ont a i ns t w o na r r o w s l i t s , A an d B .

Th e n c o u n t t h e n u m b e r o f b u l l e t s t h a t a r r i v e a t t h e b a c k - st o p a s a fu n c t i o n o f p o si t i o n a l o n g th e b a c k - st o p .

sl i t A open, s l i t B cl os ed

sl i t A cl os ed , s l i t B open

Ca s e 1 : C l a s s i c a l P a r t i c l e s

sl i t s A an d B open

Th e c o u n t s o f b u l l e t s p e r l o c a t i o n y i e l d pr o babi l i ty di s tr i bu ti o n s . F o r c l a s s i c a l p a r t i c l e s , t h e p r o b a b i l i t i e s f o r in d e p e n d e n t e v e n t s add .

Ca s e 2 : C l a s s i c a l W a v e s

wa t e r wa v e

pr oj e c t i on of s l i t A

A

B

pr oj e c t i on of s l i t B

re e f sh o r e

Courtesy of Elsevier, Inc., https://www.sciencedirect.com. Used with permission.

A w a t e r w a v e a p p r o a c h i n g t h e s h o r e e n c o u n t e r s a r e e f w i t h t w o n a r r o w s l i t s i n i t , A

an d B . M e a s u r e t h e i n t e n s i t y o f t h e w a v e a l o n g t h e s h o r e a s a f u n c t i o n o f p o s i t i o n .

sl i t A open, s l i t B cl os ed

sl i t A cl os ed , s l i t B open

Ca s e 2 : C l a s s i c a l W a v e s

pea k i nt ens i t y

occu r s be t w e e n

A an d B

sl i t s A an d B open

Th e i n t e n s i t y o f t h e w a v e g o e s a s t h e abs o l u te s q u ar e of th e a m p l i tu d e . B e c a u s e o f su p er p o si t i o n , t h e r e s u l t i n g wa v e s h o w s a c h a r a c t e r i s t i c in t e r f e r e n c e p a t t e r n .

|  A +  B | 2 ≠ |  A | 2 + |  B | 2

Ca s e 3 : Q u a n t a

ar r a y o f e l e c t r o n de t e c t o r s

el ect r on s ou r ce

Di s t a n c e b e t w e e n s l i t s A an d B is 1 0 , 0 0 0 t im e s Courtesy of Elsevier, Inc., https://www.sciencedirect.com. Used with permission.

la r g e r t h a n t h e e le c t r o n s ’ d e B r o g lie w a v e le n g t h

Re l e a s e o n e e l e c t r o n a t a t i m e t o w a r d t h e b a r r i e r ; w a i t o n e h o u r b e t w e e n r e l e a s e s . Re p e a t t h i s pr oc e d u r e 1 0 , 0 0 0 t i m e s ( i . e . , g e t a pa t i e nt g r a d u a t e s t u d e nt ) , a nd t he n pl ot t he n u m be r of el ect ro n s d et ect ed at a g i v en p o s i t i o n al o n g t h e ar ra y o f d et ect o rs .

sl i t A open, s l i t B cl os ed

sl i t A cl os ed , s l i t B open

W h e n o n ly o n e s lit is o p e n , t h e e le c t r o n r e s u lt s lo o k a lo t lik e t h e b u lle t r e s u lt s , c le a r ly c lu m p e d b e h in d t h e o p e n s lit . Y o u m ig h t r e a d ily d is m is s t h e sm a l l w i g g l es o n ei t h er si d e o f t h e cen t r a l p ea k a s s p u r io u s e x p e r im e n t a l e r r o r ; t h e y n e v e r r is e a b o v e

~ 1 % o f t h e c e n t r a l p e a k . “ A h a , ” y o u s a y : “e l e c t r ons a r e l i k e bu l l e t s a f t e r a l l … ”

sl i t A open, s l i t B cl os ed

sl i t s A an d B open

Wh e n b o th s l i ts a r e o p e n , th e p a tte r n — aft e r 1 0 , 0 0 0 i n d i v i d u al el ect r o n s h a v e b een s h o t a t t h e s l i t s , o n e a t a t i m e , a n h o u r a p a r t ! — is e x a c t ly lik e t h a t o f in t e r f e r in g w a t e r w a v e s . E v e r y in d i v id u a l el ect r o n w a s det ect ed a s a l o ca l i z ed p a r t i cl e . I t w a s n ’ t t h e el ect r o n s th a t s p r e a d o u t l i k e a w a v e a n d i n te r f e r e d w i th e a c h o th e r ; i t w a s

sl i t A cl os ed , s l i t B open th e i r pr o babi l i ty am pl i tu de ,  . Ma x B o r n (1 9 2 6 ): Pr o b a b i l i t y = |  | 2 .

t = 1 / 3 0 s t = 1 s t = 1 0 0 s

Th r e e s n a p s h o t s o f t h e d e t e c t i o n o f i n d i v i d u a l p h o t o n s a f t e r t h e y h a v e p a s s e d t h r o u g h a

ba r r i e r w i t h s l i t s . E a c h q u a nt u m i s d e t e c t e d a s a l oc a l i ze d pa r t i c l e , y e t t he pa t t e r n t ha t bu i l d s u p ov e r t i m e r e v e a l s w a v e l i k e i n t e r f e r e n c e . Co u r t e s y Ro b e r t A u s t i n a n d L y ma n P a g e , P r i n c e t o n

© Robert Austin and Lyman Page. All rights reserved. This

content is excluded from our Creative Commons license. For more information, see https://ocw.mit.edu/help/faq-fair- use/

Qu e s t i o n s ?

te s t

pa r t i cl es

el ect r ons

S

d

We c o u l d m o d i f y o u r a p p a r a t u s t o t r y t o d e t e r m i n e th r o u g h w h i c h s l i t an el ect r o n re a l l y p a s s e d : pl a c e te s t p a r ti c l e s be hi nd s l i t A . I f t h o s e pa r t i c l e s ge t s c a t t e r e d , t h e n w e k n o w t h a t t h e el ect r o n pa s s e d t hr ou g h s l i t A en r o u t e t o t h e s cr een .

* S e e o p t i o n a l Le c t u r e N o t e s on t he d ou bl e - sl i t e xp e r i m e n t

Re s u l t s w h e n b o t h s l i t s a r e o p e n , bu t we Re s u l t s w h e n b o t h s l i t s a r e o p e n , an d we d o no t

me a s u r e t h r o u g h w h i c h s l i t e a c h e l e c t r o n p a s s e d . me a s u r e t h r o u g h w h i c h s l i t e a c h e l e c t r o n p a s s e d .

As k a “ p a r t i c l e - lik e ” q u e s tio n — “t h r o ugh w h i c h s l i t d i d t h e e l e c t r o n p as s ? ” — an d w e w i l l ge t a p a r t i c l e - lik e a n s w e r ( eith er sl i t A or sl i t B , w i t h p a r t i c l e - lik e s ta tis tic s ). A s k a “ w a v e - lik e ” que s t i o n — “h o w d o e s  be ha v e be t w e e n t he s l i t s a nd t he d e t e c t or ? ” — an d w e w i l l g et a wa v e - lik e a n s w e r (in te r f e r e n c e p a tte r n w ith w a v e - lik e s ta tis tic s ).