Randell L. Mills- The Hydrogen Atom Revisited

8/3/2019 Randell L. Mills- The Hydrogen Atom Revisited

http://slidepdf.com/reader/full/randell-l-mills-the-hydrogen-atom-revisited 1/39

© 20 00 by BlackLight Power , In c . Al l r ight s r e se rved .

5 5 9

THE HYDROGEN ATOM REVISITED

R a n d e l l L. M il l s

BlackLight Power, Inc.4 9 3 O ld T r e n t o n Ro a d

Cr a n b u r y , NJ 0 8 5 1 2




5 6 0

A b s t r a c t

Se v e r a l m y t h s a b o u t q u a n t u m m e c h a n ic s e xis t d u e t o a l o ss o f a wa r e n e s s o f it s de t a ils s in c e it s in c e p t i on i n t h e be g i nn in g o f t he l a s tc e n t u r y o r b a s e d o n r e c e n t e xp e r i m e n t a l e vid e n c e. It is ta u g h t int e xt b o o k s t h a t a t o m ic h y d r o g e n c a n n o t go b e lo w t h e gr o u n d s t a t e.

At o m ic h y d r o g e n h a v in g a n e xp e r i m e n t a l g r o u n d s ta t e o f 1 3 .6 e V ca no n ly e x is t in a v a c u u m o r i n i so la t io n , a n d a t o m ic h y d r o g e n c a n n o t g obe l ow t h i s g r ou n d s t a t e in i s o la t i on . Howe ve r , t h e r e is n o kn ownc o m p o s it io n o f m a t t e r co n t a i n i n g h y d r o g e n i n t h e g r o u n d s t a t e o f 1 3 .6e V. It is a m y t h t h a t h y d r o g e n h a s a t h e o r e t ic a l g r o u n d s t a t e b a s ed o nf ir s t p r i nc ip le s . H is t o r i ca l ly t h e r e we r e m a n y d ir e c t ion s in whi c h t op r oc e e d t o s o l ve a wa ve e qu a t i on f o r hyd r oge n . The Sc hr o d in ge re q u a t io n g iv e s th e o b s e r v e d s p o n t a n e o u s ly r a d ia t iv e s ta t e s a n d t h en on r a d ia t i ve e ne r gy le ve l o f a t om ic hyd r oge n . On t h is ba s is a lon e , it is jus t i fi ed d espi t e i t s in con s is ten cy with p h ys ica l laws as we ll a s wi th m an y

e xpe r i m e n t s . A s o lu t ion c om p a t i b le w it h f ir s t p r in c ip l es a n d h a v in g fir s tp r i nc i p l es a s t h e ba s is o f qu a n t iz a t ion wa s ne ve r f oun d . Sc a t t e r i ngr e s u lt s r e q u ir e d t h e s o lu t i o n t o b e in t e r p r e t e d a s p r o b a b ilit y w a v e s t h a tg ive r is e t o t h e un c e r t a in t y p r i nc ip l e whi ch i n t u r n f o r m s t he ba s i s o f t h ewa ve p a r t i c le d ua lit y . The co r r e s pon d e n c e p r in c ip a l p r e d ic t s t h a tq u a n t u m p r e d i ct io n s m u s t a p p r o a c h c la s sic a l p r e d i ct io n s o n a la r g es ca le . H o we v e r , r e c e n t d a t a h a s s h o w n t h a t t h e H e is e n b e r g u n c e r t a i n t yp r i nc ip le a s t he b a s is o f t h e wa ve pa r t ic le d u a lit y a n d t h ec o r r e sp o n d e n c e p r i n c ip l e t a u g h t i n t e x t b o o k s a r e e x p e r im e n t a l lyi n c o r r e c t .

Re c e n t l y , a r e c ons i de r a t ion o f t h e pos t u la t e s o f qu a n t u mm e c ha n i cs , h a s g ive n r i se t o a c los e d f o r m s o lu t i on o f a Sc hr od in ge r -li kewa ve e qu a t i on ba s e d on fir s t p r i nc i p l e s [ 1 ] . Hydr oge n a t p r e d i c t e dlowe r e ne r gy le ve ls ha s be e n id e n t ifie d i n p r e v iou s da t a . The t r a n s it iono f h y d r o g e n t o f r a ct io n a l q u a n t u m e n e r g y l ev e ls is r e p o r t e d b y t h ea s s ignm e n t o f s o ft X–r a y e m is sion s fr om t h e i n t e r s t e l la r m e d i u mo b s e r v e d b y La b o v a n d Bo w y e r [2 ] a n d b y t h e a s sig n m e n t o f t h e s o u r c eo f 50 eV a n o m a lo u s t h e r m a l b r o a d e n i n g o f t h e Ba lm e r lin e s o b se r v e d b yKur a ic a a n d Kon je v ic [3 ] d u r in g a g low d is ch a r ge o f hyd r oge n -a r gonm ix tu r e s w h i ch w a s n o t o b s er v e d w it h n e o n - h y d r o g e n m ixt u r e s o r p u r e

h y d r o g e n ir r e s p e c t iv e o f ca t h o d e m a t e r ia l.Cer ta in ino rgan ic ions p red icted b y Mil ls [1] se rve as " t r an s it ionc a t a ly s t s" w h ic h r e s o n a n t ly a c ce p t e n e r g y fr o m h y d r o g e n a t o m s a n dr e l e a se t he e ne r gy t o t h e su r r ou n d in gs . Ar gon i on i s a ca t a l ys t t h a t wa sp r e s e n t i n t h e h yd r oge n g low d is ch a r ge o f Ku r a i ca a n d Kon je v ic [3 ] .Lowe r -e n e r gy hy d r oge n a t om s , h ydr ino s , c a n c a u s e a n a u t oc a t a l y t ica c c e le r a t i on o f t h i s " t r a n s it ion r e a c t i on" wh i ch i s t h e m e c ha n i smc or r e s p on d in g t o t he in t e r s t e lla r l in e s .




5 6 1

T h e d e t e ct io n o f t h e t r a n s it io n o f a t o m ic h y d r o g e n f r o m t h et r a d it io n a l "g r o u n d " s t a t e ( n = 1) t o t h e f r a ct io n a l q u a n t u m e n e r g y l ev e ln = 1/ 2 be l ow t he t r a d it ion a l "g r ou n d " st a t e —h yd r i nos — is fu r t h e rr e p o r t e d b y t h e a s sig n m e n t o f t h e a n o m a lo u s 3 1 e V b a c k w a r d p e a kob s e r ve d b y Ru d d , e t a l . [ 4 ] in t h e e le c t r on s pe c t r u m f r om c o ll is ion s o f

7 0 k e V p r o t o n s w it h h y d r o g e n a t o m s . T h e t r a n s it io n o c c u r s b y a" r e s on a n t c o llis ion " me c h a n i sm p r e d ic t e d b y Mill s [ 1 , 5 ] . Pr o t o n s e ffe c tt h is t r a n s it i on o f hyd r oge n by a r e s ona n t in e la s t ic co l lis ion r e a c t i on . Int h is ca s e , a b a c k wa r d 4 0 .8 e V e le c tr o n is p r o d u c e d w h i ch u n d e r g o e sFr a n c k-He r t z sc a t t e r i ng [ 6 ] t o g ive r i se p r e d om in a n t l y t o a 30 .6 e Vb a c k w a r d p e a k , a 2 7 .2 e V b a c k w a r d p e a k , a n d a 2 0 . 4 e V b a c k w a r d p e a k .Dis con t i nu i t ie s in t h e ba c k s ca t t e r i n g sp e c t r u m a t t h e s e en e r g i e s we r eob s e r ve d b y Ru d d , e t a l . [ 4 ] in t h e e le c t r on s pe c t r u m f r om c o ll is ion s o f 7 0 k e V p r o t o n s wit h h y d r o g e n a t o m s . T h e m a x im u m in t e n s it y o f b a c ksca t t e r ing i s p red icted to b e 16 5° fa ll in g to ze ro a t 90 ° wh ich i s in

a g r e e m e n t w it h t h e o b s er v e d m a x im u m a t 1 6 0 ° w h i ch d e c r e a s es w it hs m a l le r a n g le s t o t he a b s e nc e o f t h e ba c kwa r d s ca t t e r i n g a t 90 °.

N ew e vid e n c e m a n d a t e s t h a t o l d t h e o r ie s b e r e v is ed o r a b a n d o n e d .Re c e n t l y lin e s pe c t r a o f f r a c t ion a l qua n t u m e n e r gy l e vels o f a t om ich y d r o g e n h a v e b e e n m e a su r e d b y a 4 ° g r a zin g i n c id e n c e ext r e m eu l t r avio le t spec t r om ete r a t INP Gre i fswa ld , Germ an y [7] . In ten se EUVe m i ss ion wa s obs e r ve d by Mill s e t a l . [ 7 -12 ] a t low t e m p e r a t u r e s ( e .g .< 10

3 K ) fr o m a t o m i c h y d r o g en a n d c er t a in a t o m i ze d p u r e e l em e n t s o r

c e r t a i n ga s e ous i ons whi c h i on i ze a t in t e ge r m u lt ip le s o f t h e po t e n t ia le n e r g y o f a t o m ic h y d r o g e n . T h e s e a t o m iz e d p u r e e le m e n t s o r ga s e o u s

io n s c o m p r i se h y d r o g e n c a t a ly s t s t o fo r m lo w e r e n e r g y h y d r o g e n . N ewc o m p o s it io n s o f m a t t e r c o n t a i n i n g h y d r o g e n a t p r e d i ct e d l o we r e n e r g yle ve ls ha ve r e c e n t ly be e n obs e r ve d in t h e la bo r a t o r y , wh ic h e n e r gy l eve lsa r e a c h i e ve d u s in g t he nove l ca t a l ys t s. The Sc hr od i n g er w a v e e q u a t io ns o lu t i o n s c a n n o t e x p la in t h e s e r e s u lt s ; t h u s , t h e t h e o r y m u s t b em o d i f i e d .




5 6 2

I n t r o d u c t i o n

J. J. Ba lm e r s h o w e d , in 1 8 8 5 , th a t t h e f r e q u e n c ie s fo r s o m e o f t h elin e s o b s e r v e d i n t h e e m i s sio n s p e c tr u m o f a t o m ic h y d r o g e n c o u l d b ee xpr e s s e d wit h a com p l et e ly e mp ir i ca l r e l a t ion s h i p . Th i s a p p r oa c h wa sl a t e r e xt e n d e d by J. R. Ryd be r g , who s howe d t h a t a l l o f t he s p e c t r a l lin e s

o f a t o m ic h y d r o g e n w e r e g iv e n b y t h e e q u a t io n :

= R1

n f 2 −

1

ni2

( 3 8 . 1 )

w h e r e R = 109,677 cm−1 , n f = 1,2,3,... , ni = 2,3,4,... , a n d ni > n f . Niels Boh r , in

1 9 1 3 , d e v e lo p e d a t h e o r y fo r a t o m ic h y d r o g e n b a s e d o n a nu n p r e c ed e n t e d p o s t u la t e o f s t a b le cir c u l a r o r b i t s t h a t d o n o t r a d ia t e .Alt h ou gh n o e xp l a n a t i on wa s o f fe r e d f o r t h e e x is t e nc e o f s t a b i lit y f o rt h e s e o r b i t s, t h e r e s u l t s ga ve e n e r gy l eve ls in a g r e e m e n t w it h Ryd be r g ' se q u a t i o n .

E n = − e2

n28 o a H

= −13.598 eV n2 ( 3 8 . 2 )

n = 1,2,3,... ( 3 8 . 3 )w h e r e a H is t h e Bo h r r a d iu s fo r t h e h y d r o g e n a t o m ( 52.947 pm ) , e i s t h em a g n i t u d e o f t h e c h a r g e o f t h e e l ec t r o n , a n d o is t h e v a c u u mp e r m it t i vit y . Boh r ' s t h e or y wa s a st r a i gh t f o r wa r d a p p li ca t i on o f Ne wt on ' s l a ws o f m ot i on a n d Cou l omb ' s la w o f e le c t r i c f o r c e - bo t hp i lla r s o f c la s s ic a l p h ys ic s a nd is in a c c or d wit h t h e e xpe r i m e n t a lob s e r va t i on t h a t a t om s a r e s t a b l e . Howe ve r , i t i s n o t in a c c or d wi t he le c t r om a gn e t ic t h e or y - a no t h e r p il la r o f c la s s ic a l ph ys ic s whi ch

p r e d i ct s t h a t a c ce le r a t e d c h a r g e s r a d i a t e en e r g y in t h e f o r m o f e le c t r om a gn e t ic wa ve s . An e le c t r on p ur s u i ng a c u r ve d p a t h i sa c c e le r a t e d a n d t h e r e f o r e sh ou ld c on t in u ou s ly los e e n e r gy , s p i r a l in g i n t ot h e n u c le u s in a f r a c t io n o f a s e co n d . T h e p r e d ic t io n s o f e le c tr o m a g n e t ic t h e o r y h a v e a lw a y s a gr e e d w it h e xp e r i m e n t , y e t a t o m sd o no t c o lla p s e . To t h e e a r l y 20 t h c e n t u r y t he o r e t i ci a ns , t h i sc on t r a d ic t ion c ou l d m e a n o n ly on e t h i ng : The l a ws o f p h ys ic s t h a t a r ev a lid i n t h e m a c r o w o r ld d o n o t h o l d t r u e in t h e m ic r o wo r l d o f t h e a t o m .In 1 9 2 3 , d e Br o g lie s u g g e st e d t h a t t h e m o t io n o f a n e l e ct r o n h a s a w a v e

a s pe c t — =h

p

. Th i s c onc e p t se e m e d u n like ly a c cor d in g t o t he f a mi li a r

p r o p e r t ie s o f e le c t r o n s s u c h a s c h a r g e , m a s s a n d a d h e r e n c e t o t h e la w sof p a r t i c le me c h a n i cs . Bu t , t h e wa ve na t u r e o f t h e e le c t r on wa sc o n f ir m e d b y D a vis so n a n d Ge r m e r i n 1 9 2 7 b y o b s e r v in g d iffr a c t io ne f fe c t s wh e n e l e c t r on s we r e r e f le c t e d f r om m e t a l s . Sc hr od i n g e rr e a s o n e d t h a t if ele c tr o n s h a v e w a v e p r o p e r t ie s, t h e r e m u s t b e a w a v ee q u a t io n t h a t g o v er n s t h e i r m o t io n . An d in 1 9 2 6 , h e p r o p o s e d t h e




5 6 3

Sch r o d inger equ a t ion , H Ψ = E Ψ , whe r e Ψ is t he wa ve f un c t ion , H is t h ew a v e o p e r a t o r , a n d E is t h e e ne r gy o f t h e wa ve. Th i s e qu a t i on , a n d i t sa s s oc ia t e d p os t u la t e s , i s n ow t h e ba s is o f quantum mechan ics , a nd it i st h e ba s i s f o r t he wor l d v i e w t h a t t h e a t om i c r e a l m i nc l ud i ng t he e l e ct r ona n d p h o t o n c a n n o t b e d e s cr ib e d in t e r m s o f " p u r e " wa v e a n d " p u r e "

p a r t i c le bu t in t e r m s o f a wa ve -p a r t i c le d u a l it y . The wa ve -p a r t i c led u a lit y b a s e d o n t h e f u n d a m e n t a l p r in c ip l e t h a t p h y s ic s o n a n a t o m i cs c a le i s ve r y d if fe r e n t f r om p h ys ic s on a m a c r os c op ic s ca l e is c e n t r a l t op r e se n t d a y a t o m ic th e o r y [1 ] .

D e v e lo p m e n t o f At o m i c T h e o r y

Bo h r T h e o r y

In 1 9 1 1 , Ru t h e r fo r d p r o p o s e d a p la n e t a r y m o d e l fo r t h e a t o mw h e r e t h e e le ct r o n s r e v o lv e d a b o u t t h e n u c le u s ( w h ic h c o n t a i n e d t h e

p r o t o n s ) in v a r i o u s o r b it s t o e x p l a in t h e s p e c t r a l lin e s o f a t o m i ch y d r o g e n . T h e r e w a s, h o w e ve r , a fu n d a m e n t a l co n flic t w it h t h is m o d e lan d th e pr eva i ling c la ss ica l p h ys ics . Accord ing to cla ssica le l ec t r om a gn e t i c t h e or y , a n a c c e le r a t e d p a r t i c le r a d ia t e s e ne r gy ( a se l ec t r om a gn e t ic wa ve s ) . Thu s , a n e l e ct r on i n a Ru t h e r f o r d o r b i t ,c ir c u l a t in g a t con s t a n t s p e e d b u t w it h a c on t i nu a l ly c ha n g in g d i r e c t ionof it s ve loc it y ve c t o r is be i ng a c ce le r a t e d ; t h u s , t h e e l e ct r on s ho u l dc ons t a n t ly los e e n e r gy by r a d ia t i ng a nd s p i r a l in t o t he n u c le us .

An e xp l a n a t io n w a s p r o v i d e d b y Bo h r in 1 9 1 3 , wh e n h e a ss u m e dt h a t t h e e n e r g y le v e ls w er e q u a n t iz ed a n d t h e e l e ct r o n w a s co n s t r a i n e d

t o m ove in on ly on e o f a nu m be r o f a llowe d s t a t e s . N ie ls Boh r ' s t he or yfo r a t o m ic h y d r o g e n w a s b a s e d o n a n u n p r e ce d e n t e d p o s t u la t e o f s t a b lec ir c u l a r o r b it s t h a t do n o t r a d ia t e . Alt h ou gh n o e xp la n a t i on wa s o f fe r e df o r t h e e x is t e nc e o f s t a b i li t y fo r t h e s e o r b it s , t h e r e s u l t s ga ve e n e r gyl eve ls in a g r e e m e n t wit h Rydb e r g ' s e qu a t i on . Boh r ' s t he or y wa s as t r a ig h t f o r w a r d a p p l ic a t io n o f Ne w t o n ' s la w s o f m o t i o n a n d Co u l o m b ' sl a w o f e le c t r i c f o r c e . Ac c or d i ng t o Boh r ' s m od e l , t h e po i n t p a r t i c lee le c t r on wa s he l d t o a c ir c u l a r o r b i t a bou t t h e r e l a t ive ly m a s s ive po i n tp a r t i cle n u c l eu s b y t h e b a la n c e b e t we e n t h e c o u l o m b ic fo r c e o f a t t r a c t io n b e t w ee n t h e p r o t o n a n d t h e e le ct r o n a n d c e n t r ifu g a l fo r c e o f

t h e e le c t r o n . e2

4 0r 2=

mev2

r ( 3 8 . 4 )

Boh r p os t u la t e d t h e e x is t e nc e o f st a b l e o r b i t s in d e f ia n c e o f c la s s ic a lphys ics (Maxwel l ' s Equa t ions ) , but he appl ied c la ss ica l phys ics accordingt o Eq . (38 .4 ) . The n Boh r r e a l iz e d t h a t t he e n e r gy fo r m u la Eqs . ( 38 .2 -38 .3 ) wa s g ive n b y p os t u l a t in g non r a d ia t i ve s t a t e s w it h a n gu l a r




5 6 4

m o m e n t u m

L z = mevr = nh n = 1,2,3... ( 3 8 . 5 )

an d b y solv ing the en e rgy equ a t ion c la ss ica lly . The Boh r r ad ius is g ivenby s ub s t it u t in g t he s o l u t i on o f Eq . ( 38 .5 ) f o r v i n t o Eq . ( 38 .4 ) .

r =4 0h

2n2

mee2 = n

2

a0 n =1, 2,3... ( 3 8 . 6 )T h e t o t a l en e r g y i s th e s u m o f t h e p o t e n t ia l en e r g y a n d t h e k i n e t ice n e r gy . In t h e p r e s e n t ca s e o f a n i nve r s e s qu a r e d c e n t r a l f ie ld , t h e t o t a le n e r gy ( whi ch is t he n e ga t ive o f t h e b i nd in g e ne r gy) i s on e h a l f t h ep o t e n t ia l e n e r gy [ 14] . The po t e n t ia l e ne r gy , r( ) , is given b y Poisson 'se q u a t i o n

r( ) = −r '( )dv'

4 0 r − r'

V '∫ ( 3 8 . 7 )

Fo r a p o i n t c h a r g e a t a d i st a n c e r fr o m t h e n u c le u s t h e p o t e n t i a l is

r ( ) =− e2

4 0r ( 3 8 . 8 )

Thu s , t h e t o t a l e ne r gy i s give n b y

E = −Z 2e2

8 or ( 3 8 . 9 )

w h e r e Z = 1. Su bs t i t u t i on o f Eq . ( 38 .6 ) in t o Eq . ( 38 .9 ) w it h t h er e p la c e m e n t o f th e e le c t r o n m a s s b y t h e r e d u c e d e le c t r o n m a s s g iv e sEqs . ( 38 .2 -38 .3 ) .

Bo h r ’s m o d e l w a s in a g r e e m e n t w it h t h e o b s e r v e d h y d r o g e ns p e c t r u m , b u t it fa ile d w it h t h e h e l iu m s p e c tr u m , a n d it c o u ld n o ta c c oun t fo r c he m ic a l bo n d s in m ol e cu l es . The p r e va il in g wis dom wa st h a t t h e Bo h r m o d e l fa ile d b e c a u s e it w a s b a s e d o n t h e a p p lic a t io n o f Ne wt on ia n m e c ha n i cs fo r d is cr e t e p a r t i c le s . An d , it s lim it e da p p lic a b ilit y w a s a t t r ib u t e d t o t h e u n w a r r a n t e d a s su m p t i o n t h a t t h ee n e r gy le ve ls a r e qu a n t i ze d .

Bo h r ' s t h e o r y m a y a l so b e a n a l y z ed a c co r d in g t o t h ec o r r e sp o n d i n g e n e r g y e q u a t i o n . N ew t o n ' s d i ffe r e n t i a l e q u a t io n s o f m ot i on i n t h e c a se o f t h e c e n t r a l fie ld s uc h a s a g r a v it a t i ona l o re lec t ros ta t i c f i e ld a re

m(˙r − r ˙2 ) = f (r ) ( 3 8 . 1 0 )

m(2 r + r ) = 0 ( 3 8 . 1 1 )w h e r e f (r ) i s t h e ce n t r a l f o r c e . The s e con d o r t r a n s ve r s e e qu a t i on , Eq .( 3 8 . 1 1 ) , g iv e s t h e r e s u l t t h a t t h e a n g u la r m o m e n t u m is co n s t a n t .

r 2 ˙ = constant = L / m ( 3 8 . 1 2 )w h e r e L is t h e a n g u l a r m o m e n t u m . T h e ce n t r a l fo r c e e q u a t io n s c a n b e

t r a n s fo r m e d i n t o a n o r b it a l e q u a t io n b y t h e s u b s t it u t i o n , u =1

r . The




5 6 5

d iffe r e n t i a l e q u a t io n o f t h e o r b it o f a p a r t ic le m o v in g u n d e r a c e n t r a lf o r c e i s

2u2 + u =

−1mL2u2

m2

f (u−1) ( 3 8 . 1 3 )

Be c a u s e t h e a n g u l a r m o m e n t u m is co n s t a n t , m o t io n i n o n ly o n e p l a n en e e d b e c ons id e r e d ; t h u s , t h e o r b i t a l e qu a t i on i s g ive n i n p o l a rc oor d i na t e s . The s o lu t i on o f Eq . ( 38 .13 ) fo r a n i nve r s e s qu a r e fo r c e

f (r ) = −k

r 2( 3 8 . 1 4 )

is

r = r 01+ e

1+ e cos( 3 8 . 1 5 )

e = Am

L2

m2

k ( 3 8 . 1 6 )

r 0 =m

L2

m 2

k (1+ e)( 3 8 . 1 7 )

w h e r e e is t h e e c c e n t r i c it y a n d A is a con s t a n t . The e qu a t i on o f m ot i ond u e t o a c e n t r a l fo r c e c a n a l so b e e x p r e s se d in t e r m s o f t h e e n e r g ie s o f t h e o r b it . T h e s q u a r e o f t h e s p e e d i n p o la r c o o r d i n a t e s is

v2 = ( r 2 + r 2 ˙2 ) ( 3 8 . 1 8 )Si nc e a c e n t r a l f o r c e is c ons e r va t i ve , t he t o t a l e ne r gy , E , is e qu a l t o t hes u m o f t h e k i n e t ic , T , a n d t h e p o t e n t ia l, V , a n d is c ons t a n t . The t o t a l

e n e r gy is1

2m(r 2 + r 2 ˙2 ) + V (r ) = E = constant ( 3 8 . 1 9 )

Su bs t i t u t i on o f t h e va r i a b l e u =1

r a n d Eq . ( 38 .12 ) in t o Eq . ( 38 .19 ) g ive s

t h e o r b it a l e n e r g y e q u a t io n .1

2m

L2

m2[(

2u2)+ u2

]+ V (u-1) = E ( 3 8 . 2 0 )

Be c a us e t h e po t e n t ia l e ne r gy f un c t ion V (r ) fo r a n in v e r s e sq u a r e f o r c efield is

V (r ) =- k r =- ku ( 3 8 . 2 1 )

t h e e ne r gy e qu a t i on o f t he o r b i t , Eq . ( 38 .20 ) , is1

2m

L2

m2 [(2u2 )+ u2

]− ku = E ( 3 8 . 2 2 )

2u2 + u2

−2m

L2 E + ku[ ] = 0 ( 3 8 . 2 3 )






5 6 7

Sc hr o d in ge r e qu a t i on s o lu t ion s a r e t h r e e d i m e n s ion a l i n s pa c e a n d fou rd im e n s io n a l in s p a c e tim e

∇2− 1

v2

2

t 2

Ψ(r , , ,t )= 0 ( 3 8 . 2 9 )

w h e r e Ψ(r , , ,t ) a c co r d in g t o q u a n t u m t h e o r y i s t h e p r o b a b ilit y d e n s it yfu n c t io n o f t h e e le c tr o n a s d e s c r ib e d b e lo w . W h e n t h e t im e h a r m o n icf un c t ion is e lim in a t e d [ 15] ,

−h2

2

1

r 2 r r 2

Ψr

+

1

r 2 sinsin

Ψ

r ,

+1

r 2 sin22Ψ

2

r ,

+ V r ( )Ψ r , ,( ) = E Ψ r , ,( )

( 3 8 . 3 0 )w h e r e t h e p o t e n t ia l en e r g y V r ( ) in CGS u n its is

V r ( ) = −e2

r ( 3 8 . 3 1 )

The Sc hr od in g e r eq u a t io n ( Eq . ( 3 8 . 3 0 ) ) ca n b e t r a n s fo r m e d i n t o a su m

c o m p r is in g a p a r t t h a t d e p e n d s on ly o n t h e r a d i u s a n d a p a r t t h a t is af un c t ion o f a ng le on l y ob t a i ne d by s e pa r a t i on o f va r ia b l e s a n d l in e a rs u p e r p o s it io n in s p h e r ic a l c o o r d i n a t e s . T h e g e n e r a l fo r m o f t h es o lu t i ons f o r r , ,( ) a r e

r , ,( ) = Rnlm r ( )l,m∑ Y lm ,( ) ( 3 8 . 3 2 )

w h e r e l a n d m a r e se p a r a t io n c o n s t a n t s . T h e a zim u t h a l ( t h e t a ) p a r t o f Eq . ( 38 .30 ) is t he ge n e r a l iz e d Le ge nd r e e q ua t ion whi c h i s de r i ve d f r omt h e La p la c e e qu a t i on b y Ja c ks on ( Eq . ( 3 .9 ) o f J a cks on [ 16 ] ). Thes o lu t i ons fo r t h e fu ll a n gu l a r p a r t o f Eq . ( 38 .30 ) , Y lm ,( ), a r e t h e

s p h e r ic a l h a r m o n ic s

Y lm ,( ) =2l +1( ) l − m( )!

4 l + m( )!Pl

mcos( )e im ( 3 8 . 3 3 )

By s u b s tit u t i o n o f t h e e i ge n v a l u e s co r r e s p o n d i n g t o t h e a n g u la r p a r t[ 17] , t he Sc hr od i n g e r e q u a t io n b e co m e s th e r a d i a l e q u a t io n , R r ( ) , givenb y

−h2

2mr 2d

dr r 2

dR

dr

+

h2l l +1( )

2mr 2+ V r ( )

R r ( ) = ER r ( ) ( 3 8 . 3 4 )

T h e t im e in d e p e n d e n t Sc h r od i nge r e qu a t i on i s s im il a r t o Eq . ( 38 .20)

e xce p t t h a t t h e s o lu t i on i s fo r t h e d is t r ibu t i on o f a s pa t i a l wa ve f un c t ionin t h r e e d i m e n s io n s r a t h e r t h a n t h e d y n a m ic a l m o t io n o f a p o i n t p a r t ic leo f m a s s m a l ong a on e d i me n s ion a l t r a j e ct o r y . Ele c t r on m ot i on isi mp lic it i n t h e Sc hr od i n g e r eq u a t io n . Fo r w a ve p r o p a g a t io n in t h r e ed im e n s io n s , t h e fu l l t im e d e p e n d e n t Sc h r od i n g e r e q u a t io n is r e q u i r ed ;whe r e a s , t h e c la s s ic a l c a s e con t a i ns t im e d e r i va t ive s . The k i ne t i c e ne r gyof r o t a t i on i s K rot i s given c lass ica l ly b y




5 6 8

K rot =1

2mr 2 2 ( 3 8 . 3 5 )

w h e r e m is t h e m a s s o f t h e e le ct r o n . In t h e t im e in d e p e n d e n tSch r o d inger eq u a t ion , th e k in e t i c en e rgy of ro ta t ion K rot i s given b y

K rot =

l l +1( )h2

2mr 2 ( 3 8 . 3 6 )w h e r e

L = l l+ 1( )h

2 ( 3 8 . 3 7 )is th e v a lu e o f t h e e le c tr o n a n g u l a r m o m e n t u m L fo r t h e s t a t e Y lm ,( ).

In t h e c a s e o f t h e g r o u n d s t a t e o f h y d r o g e n , t h e Sc h r od i n g e re qu a t i on s o l u t i on i s t r iv ia l fo r a n im p lic it c ir c u l a r b ou n d o r b i t wh ic hd e t e r m in e s t ha t t h e e c ce n t r i cit y i s z e r o , a n d wi t h t h e s pe c if ic a t ion t h a tt h e e le c tr o n a n g u la r m o m e n t u m is Pla n c k 's co n s t a n t b a r . W it h k = e2 , Eq .( 38 .25 ) in CGS u n it s be c om e s

E = − 1

2 me

4

h2 = − e

2

2a0( 3 8 . 3 8 )

w h i ch c o r r e sp o n d s t o n = 1 in Eq . (38 .27 ) . Ma ny p r ob l em s in c l a ss ic a lp h y s ic s g iv e t h r e e q u a n t u m n u m b e r s w h e n t h r e e sp a t i a l d i m e n s io n s a r ec o n s id e r e d . In o r d e r t o o b t a i n t h r e e q u a n t u m n u m b e r s , t h e Sc h r o d i n g e re q u a t io n r e q u ir e s t h a t t h e s o lu t i o n is fo r t h e d is t r ib u t io n o f a s p a t ia lw a v efu n c t io n in t h r e e d i m e n s io n s w it h im p l ic it m o t io n r a t h e r t h a n a o n ed im e n s ion a l t r a j e c t o r y o f a po in t pa r t i cl e a s sh own b e low. Howe ve r , t h isa p p r o a c h g iv e s r is e t o p r e d ic tio n s a b o u t t h e a n g u la r m o m e n t u m a n da n g u l a r e n e r g y w h ic h a r e n o t c o n s is t en t w it h e xp e r i m e n t a l o b s er v a t io n s

a s w ell a s a h o s t o f o t h e r p r o b le m s wh ic h a r e s u m m a r i ze d i n t h eDiscuss ion Sec t ion .T h e r a d ia l e q u a t io n m a y b e w r i tt e n a s

d

dr r 2

dR

dr

+

2mr 2

h2 E − V r ( ) −

l l +1( )h2

2mr 2

R r ( ) = 0 ( 3 8 . 3 9 )

Let U r ( ) = rR r ( ), t h e n t h e r a d ia l e q u a t io n r e d u c es t o

′′U +2m

h2 E −V r ( ) −

l l +1( )h2

2mr 2

U = 0 ( 3 8 . 4 0 )

w h e r e

=

1

r U lm r ( )Y lm ,( ) ( 3 8 . 4 1 )Su bs t i t u t i on o f t he p o t e n t ia l en e r gy g ive n b y Eq . ( 38 .31) i n t o Eq . ( 38 .40)gives for suff ic ient ly la rge r

′′U ∞ −2

2

U = 0 ( 3 8 . 4 2 )

p r o v id e d w e d e fin e




5 6 9

2

2

=−2mE

h2 ( 3 8 . 4 3 )

whe r e i s t h e e ige nva l ue o f t h e e ige nf un c t ion s o lu t ion o f t h eSch r o d inger e q u at ion given i n f ra h a v in g un it s o f r e c ip r oc a l le ng t h a n d E is t h e e ne r gy l eve ls o f t h e hyd r oge n a t om . To a r r i ve a t t he s o lu t i onw h i ch r e p r e s e n t s t h e e le ct r o n , a s u it a b l e b o u n d a r y c o n d it io n m u s t b ei m p o s e d . S c h r od i n g er p o st u la t e d a b o u n d a r y c on d it io n : Ψ → 0 a s r → ∞ ,w h i ch l e a d s t o a p u r e ly m a t h e m a t ic a l m o d e l o f t h e e l ec t r o n . T h i se qu a t i on i s n o t ba s e d on f ir s t p r i nc ip le s , h a s no va lid it y a s s uc h , a n ds h o u ld n o t b e r e p r e s e n t e d a s s o . T h e r ig h t h a n d s id e o f Eq . (3 8 .4 3 ) m u s tb e pos tu la ted in o r d e r t h a t t h e Ry d b e r g e q u a t i o n is o b t a in e d a s s h o w nbe l ow. The p os t u l a t e is im p l ic it s in c e Eq . ( 38 .43 ) a r i se s fr om t h eSch r o d inger wh ich i s p os tu la ted . It cou ld be de f ined arbi trari ly , bu t is jus t i fi ed becau se i t g ives th e Ryd be rg form u la . Tha t Sch r od i n g e r gu e s s edt h e a c c ep t e d a p p r o a c h i s n o t su r p r is in g sin c e m a n y a p p r o a c h e s we r e

c o n t e m p l a t e d a t t h is t im e [1 8 ], a n d s in c e n o n e o f t h e s e a p p r o a c h e s w e r es u p e r i o r , Sc h r o d i n g er ' s a p p r o a ch p r e v a ile d .

The s o lu t ion o f Eq . ( 38 .42) t h a t is con s is t e n t w it h t h e bou n d a r yc ond it ion is

U ∞ = c1e / 2( ) r + c2e

− / 2( )r ( 3 8 . 4 4 )In t he ca s e t h a t is r e a l , t h e e n e r gy o f t h e p a r t i c le is n e ga t ive . In t h i scase U ∞ will n o t h a ve a n in t e g r a b l e s qua r e if c1 fa i ls t o va n is h wh e r e i n t h er a d i a l i n t e g r a l ha s t h e fo r m

R2r 2

0

∞

∫ dr = U ∞2∫ dr ( 3 8 . 4 5 )

It is sh own be l ow t ha t t h e s o lu t i on o f t h e Sc hr od i n g e r c o r r es p o n d s t ot h e c a se whe r e in c1 f a il s to van i sh . Th u s , th e so lu t ion s with su f fi cient lyla rge r a r e i n f in it e . The s a m e p r ob le m a r i s e s in t h e ca s e o f a fr e ee le c t r on t h a t i s ion iz e d fr om hy d r oge n . If is im a g in a r y , wh i ch m e a n st h a t E is pos i t ive , Eq . ( 38 .42 ) is t he e q u a t i on o f a l in e a r h a r m on icosc i l l a tor [19] . U ∞ s hows s in u s o id a l be h a v io r ; t h u s , t h e wa ve fu n c t ion fo rt h e f r e e e le ct r o n c a n n o t b e n o r m a liz ed a n d is in fin i t e . In a d d i t io n , t h ea n gu l a r m om e n t u m of t he f r e e e le c t r on is in f in it e s in c e it is g ive n b y

l l+1( )h2 ( Eq . ( 3 8 .3 7 ) ) w h e r e l → ∞ .

In o r d e r t o s o lv e t h e b o u n d e le c t r o n s t a t e s , le tE = −W ( 3 8 . 4 6 )s o th a t W i s p os i t ive . In Eq . (3 8 .39 ) , le t r = x / wh ere is g iven by Eq.( 3 8 . 4 3 ) .

xd 2 R

dx2 + 2dR

dx+

2me2

h2 −

x

4−

l l +1( )

x

R = 0 ( 3 8 . 4 7 )

Eq . ( 38 .47 ) i s t h e d if fe r e n t i a l e qu a t i on f o r a s soc ia t e d La gu e r r e f u n c t ion s




5 7 0

g ive n i n ge n e r a l fo r m by

x ′′ y + 2 ′ y + n *−k −12

−x

4−

k 2 −14 x

y = 0 ( 3 8 . 4 8 )

whi c h h a s a so l u t i on p os s e ss in g a n i n t e g r a b l e squ a r e o f t he f o r m

y = e− x / 2

xk −1( ) / 2 d k

dxk Ln* x( ) ( 3 8 . 4 9 )p r o v id e d t h a t n * a n d k a r e pos i t ive i n t e ge r s . Howe ve r , n * d o e s n o t h a v et o b e a n in t e g e r , it m a y b e a n y arbi t rary c o n s t a n t . T h e n t h ec o r r e sp o n d i n g s o lu t i o n is [2 0 ]

y = e− x / 2 x k −1( ) / 2 d k

dxk L x( ) ( 3 8 . 5 0 )

In t h e c a se t h a t n * is ch o s en t o b e a n in t e g er i n o r d e r t o o b t a i n t h eRy d b e r g fo r m u l a , n *−k ≥ 0 s in c e o t h e r wi se Ln*

k x( ) of Eq . ( 38 .49 ) wou l dva n is h . By com p a r i ng Eq . ( 38 .47 ) a n d Eq . ( 38 .48 ) ,

k 2

−14 = l l +1( ) ( 3 8 . 5 1 )

T h u s , k = 2l +1 ( 3 8 . 5 2 )

a n d

n *−k −12

= n *−l =me2

h 2

−1

( 3 8 . 5 3 )

Su bs t i t u t i on o f t h e va l ue o f a n d s o lv in g f o r W gives

W =1

2

me4

n *−l( )2h

2 ( 3 8 . 5 4 )

Be c a u s e o f t h e c ond i t ion s on n * a n d k , th e q u a n t it y n *−l c an n o t b ez e r o . It is u s ua l ly de n o t e d b y n a n d c a lle d t h e p r i n c ip a l q u a n t u mn u m b e r . T h e e n e r g y st a t e s o f t h e h y d r o g e n a t o m a r e

W n =− E n =1

2

me4

n2h2 ( 3 8 . 5 5 )

a n d t h e c o r r e s p o n d in g e i ge n f u n c t io n s f r o m Eq . ( 3 8 .4 9 ) a r e

Rn,l = cn ,le

− x /2 xl Ln+ l

2l+1 x( ) ( 3 8 . 5 6 )w h e r e t h e v a r ia b l e x is de f in e d by

x = r =8mW

hr =

2me2

nh2 r ( 3 8 . 5 7 )

In t h e Bo h r t h e o r y o f t h e h y d r o g e n a t o m , th e fir s t o r b i t a l h a s a r a d iu s i nCGS u n i ts given b y

a0 =h2

me2 = 0.53 X 10−8 cm ( 3 8 . 5 8 )

T h u s , = 2/ na0 a n d

x =2

n

r

a0

( 3 8 . 5 9 )




5 7 1

The e n e r gy s t a t e s o f t he h yd r oge n a t om in CGS u n it s in t e r m s o f t h e Boh rr a d i us a r e give n b y Eq . (38 .27 ) . Fr om Eq . (38 .56 ) , Rn ,l fo r t h e h y d r o g e na t o m g r o u n d s ta t e is

R1,0 = c1,0e− r / a0 L1

1 = 2a0

−3 / 2e−r / a0 ( 3 8 . 6 0 )For th i s s t a te

Y 00 = cons tan t = 4( )−1 /2 ( 3 8 . 6 1 )w h e n t h e f u n c t io n i s n o r m a liz e d . T h u s , t h e g r o u n d s t a te fu n c t io n i s

0 = a0

3( )−1 / 2

e− r / a0 ( 3 8 . 6 2 )

Im m e d ia t e l y fu r t h e r p r ob le m s a r i s e . Sin c e l m u s t e q u a l ze r o in t h eg r o u n d s t a t e , t h e p r e d i ct e d a n g u la r e n e r g y a n d a n g u la r m o m e n t u m g iv e nby Eq . ( 38 .36 ) a n d Eq . ( 38 .37 ) , r e s p e c t ive ly , a r e z e r o whi c h a r ee xpe r i m e n t a l ly in c or r e c t . In a d d it ion , d i ff er e n t in t e ge r va lue s o f l existi n t h e c a s e o f e xci t e d e l e ct r o n s t a t e s . In t h e s e ca s e s , t h e Sc hr od in g e re qu a t i on s o l u t i on s , Eq . ( 38 .36 ) a n d Eq . ( 38 .37 ) , p r e d i ct t h a t t h e e xc it e d

s t a t e r o t a t io n a l e n e r g y l ev e ls a r e n o n d e g e n e r a t e a s a f u n c t io n o f t h e lq u a n t u m n u m b e r e v e n in t h e a b s en c e o f a n a p p l ie d m a g n e t ic fie ld .Con s id e r t h e c a s e o f t he e xc it e d s t a t e w it h n = 2; l = 1 co m p a r ed t o t h ee xp e r i m e n t a lly d e g e n e r a t e s t a t e n = 2; l = 0 . Ac c or d i ng t o Eq . ( 38 .37 ) t h ed iff er e n c e in a n gu l a r e n e r gy o f t he s e t wo s t a t e s is 3.4 eV w h e r e t h ee xp e c t a t io n r a d iu s , 4a0 , i s g ive n by t h e s qu a r e d in t e g r a l o f Eq . ( 38 .70 )o v e r sp a c e. T h u s , t h e p r e d i ct e d r o t a t i o n a l en e r g y in t h e a b s e n c e o f am a gn e t ic fie ld i s ove r s ix o r de r s o f m a gn it u d e o f t h e obs e r ve dn o n d e g e n e r a t e en e r g y (10−7 −10−6

eV ) in t h e p r e s e n c e o f a m a g n e t i c fie ld .Sch r o d inger r ea l ized th a t h i s eq u a t ion wa s l im i ted . It is n ot

Lore n tz ian inva r ian t ; th u s , i t v io la te s spec ia l r e la t iv ity . It a l so d oes no tc om p ly wi t h Ma xwe ll' s e qu a t i ons a n d o t h e r f ir s t p r i nc i p l e la ws .Sc hr o d in ge r s ou gh t a r e s o lu t i on o f t h e i nc om p a t i b il it y w i t h s pe c ia lr e l a t iv it y f o r t he r e s t o f h i s lif e. He wa s de e p ly t r ou b l e d b y t h e ph ys ic a lc o n s e q u e n c e s o f h is e q u a t i o n a n d i t s s o lu t i o n s . H is h o p e w a s t h a t t h er e s o l u t i on wou ld m a ke h is e qu a t i on f u l ly c om p a t i b le w it h c la s s ic a lp h ys ic s a n d t h e qu a n t iz a t ion wou l d a r i se fr om f ir s t p r i n c ip le s .

Q u a n t u m m e c h a n ic s fa ile d t o p r e d ic t t h e r e s u l t s o f t h e St e r n -Ge r la c h e xp e r i m e n t w h ic h i n d ic a t ed t h e n e e d f o r a n a d d i t io n a l q u a n t u mn u m b e r . Qu a n t u m e le ct r o d y n a m i cs wa s p r o p o se d b y D ir a c in 1 9 2 6 t o

p r o v id e a g e n e r a l iz a t io n o f q u a n t u m m e ch a n i cs fo r h ig h e n e r g ie s inc onf o r m it y wit h t h e t he o r y o f sp e c ia l r e l a t iv it y a n d t o p r ov i de ac o n s is t en t t r e a t m e n t o f t h e in t e r a c t io n o f m a t t e r w it h r a d ia t io n . Fr o mW e is sk o p f [2 1 ], "D ir a c ' s q u a n t u m e le c t r o d y n a m ic s ga v e a m o r ec o n s is t e n t d e r iv a t io n o f t h e r e s u l t s o f t h e c o r r e s p o n d e n c e p r i n c ip l e , b u ti t a l so b r ou gh t a b ou t a nu m be r o f ne w a n d s e r i ous d i ff ic u l t ie s ."Q u a n t u m e le ct r o d y n a m ic s; 1 . ) d o e s n o t e xp l a in n o n r a d ia t io n o f b o u n de le c t r on s ; 2 . ) con t a i ns a n in t e r n a l in c ons i st e n c y wit h s pe c ia l r e la t i v it y




5 7 2

r e ga r d in g t he c la s s ic a l e l ec t r on r a d iu s - t h e e le c t r on m a s s c o r r e s p on d in gto i t s e lec t r i c en e rgy i s inf in i t e ; 3 . ) it ad m i t s so lu t ion s of n ega t ive re s tm a s s a n d n e ga t ive k in e t i c e n e r gy ; 4 . ) t h e in t e r a c t i on o f t h e e le c t r onwit h t h e p r e d ic t e d z e r o - po i n t f ie ld flu c t ua t i on s le a d s t o in f in it e k i ne t i ce n e r gy a nd i n f in it e ele c t r on m a s s ; 5 . ) D ir a c us e d t he un a c c e p t a b le

s t a t e s o f n e ga t ive m a s s fo r t h e d e s cr i p t i on o f t he va c u u m ; ye t , in f in it ie ss t i ll a r ise .

A p h ys ic a l in t e r p r e t a t i on o f Eq . ( 38 .28) wa s sou gh t . Sc hr od i n g e ri n t e r p r e t e d eΨ *( x)Ψ( x) a s t h e c h a r g e -d e n s it y o r t h e a m o u n t o f c h a r g ebe t we e n x a n d x + dx ( Ψ * is t h e c o m p le x co n ju g a t e o f Ψ ). P r e s uma bl y ,t h e n , h e p ic t u r e d t h e e l e ct r o n t o b e s p r e a d o v e r la r g e r e g io n s o f s p a c e .Thr e e ye a r s a f t e r Sc hr o d in ge r ’s in t e r p r e t a t ion , Ma x Bor n , who wa sw o r k in g w i th s ca t t e r in g t h e o r y , fo u n d t h a t t h i s in t e r p r e t a t i o n l e d t ologica l d i ff icul t i e s, an d h e rep laced th e Sch r o d inger in te rp re t a t ion witht h e p r oba b i lit y o f f in d in g t he e l e ct r on be t we e n r , , a n d r + dr , + d , + d

a sΨ(r , , )Ψ *(r , , )drd d ∫ ( 3 8 . 6 3 )

Bor n ’s in t e r p r e t a t i on i s ge ne r a l ly a c c e p t e d . None t h e le s s, in t e r p r e t a t i onof t he wa ve fu n c t ion is a ne ve r - e nd i ng s our c e o f c onf u s ion a n d c on f li ct .Ma ny s c ie n t is t s h a ve so l ve d t h is p r ob le m b y c onve n ie n t l y a dop t in g t heSc hr o d in ge r in t e r p r e t a t i on f o r s om e p r ob l e m s a nd t h e Bor ni n t e r p r e t a t i on f o r o t he r s . Th i s d u a l it y a ll ows t h e e le c t r on t o bee ve r ywh e r e a t one t i me —ye t ha ve n o vo lum e . Alt e r n a t i ve ly , t h e e le c t r onc a n b e v ie w ed a s a d i sc r e t e p a r t i cle t h a t m o v e s h e r e a n d t h e r e ( fr o mr =0 t o r

= ∞) , a n d

ΨΨ* g ive s t he t i me a ve r a ge o f t h is mo t ion .

Sc hr o d in ge r wa s a l so t r o u b l e d b y t h e p h ilos oph ic a l c ons e qu e n c e so f h is t h e o r y s in c e q u a n t u m m e c h a n ic s le a d s t o c e r t a in p h ilo s o p h ic a lin t e r p r e t a t io n s [2 2 ] w h i ch a r e n o t s e n s ib l e . So m e co n ju r e u p m u l t it u d e sof un ive r s e s in c lu d i ng "m in d " un i ve r s e s ; o t h e r s r e q u ir e b e lie f in a l og ict h a t a llo w s t w o co n t r a d ic t o r y st a t e m e n t s t o b e t r u e . T h e q u e s tio na d d r e s se d i s wh e t h e r t h e u n iv e r se is d e t e r m i n e d o r i n flu e n c ed b y t h ep os s ib i lit y o f ou r be i ng c ons c iou s o f it . The m e a n in g o f qu a n t u mm e ch a n ic s is d e b a t e d , b u t t h e C o p e n h a g e n i n t e r p r e t a t i o n isp r e d om i na n t . It s a ss e r t s t h a t "wha t we obs e r ve is a l l we ca n kn ow; a n ys p e c u l a t io n a b o u t w h a t a p h o t o n , a n a t o m , o r e v e n a SQ UID

( Su p e r c on d u c t in g Qua n t u m In t e r f e r e nc e De v ic e ) r e a l ly i s o r wh a t i t isd o i ng whe n we a r e no t l ook in g is ju s t t h a t s pe c u l a t ion " [ 22] . As s hownby Pla t t [ 23 ] i n t h e c a s e o f t h e St e r n -Ge r l a c h e xpe r i m e n t , " t he p os t u l a t eo f q u a n t u m m e a s u r e m e n t [ wh ic h ] a ss e r t s t h a t t h e p r o c e s s o f m e a s u r i n ga n o b s e r v a b l e fo r c e s t h e s t a t e v e ct o r o f t h e s y st e m in t o a n e ig e n v e c t o ro f t h a t obs e r va b le , a n d t h e va lu e m e a s ur e d will be t h e e ige nva l ue o f t ha te ige nve c t o r " . Ac cor d i ng t o t h is in t e r p r e t a t i on e ve r y obs e r va b le e xis t s in




5 7 3

a s t a t e o f su p e r p o s it io n o f p o s s ib l e s ta t e s , a n d o b s e r v a t io n o r t h ep o t e n t i a l fo r k n o w le d g e c a u s e s t h e w a v e fu n c t io n c o r r e s p o n d in g t o t h ep oss ib i l it i e s to coll apse in to a d e f in i t e .

Ac cor d in g t o t he qu a n t u m m e c ha n ic a l v ie w, a m ovi ng pa r t i cl e isr e ga r d e d a s a wa ve g r ou p . To r e ga r d a m ovin g p a r t i cle a s a wa ve g r ou p

im p lie s t h a t t h e r e a r e f un d a m e n t a l lim it s t o t he a c cu r a c y wit h wh ic hs u c h " p a r t i cle " p r o p e r t ie s a s p o s it io n a n d m o m e n t u m c a n b e m e a s u r e d .Q u a n t u m p r e d i ct s t h a t t h e p a r t ic le m a y b e l o ca t e d a n y w h e r e w it h i n i t swa ve gr ou p wi t h a p r ob a b i lit y Ψ 2

. An i so l a t e d wa ve g r ou p i s t h e r e s u l to f su pe r p os in g a n i n f in it e nu m be r o f wa ve s wit h d iff er e n t wa ve le n g t hs .T h e n a r r o w e r t h e w a v e g r o u p , t h e g r e a t e r r a n g e o f wa v e le n g t h s i n v o lv e d .A n a r r ow de Br og l ie wa ve g r oup t h us m e a n s a wel l-d e f in e d p os it ion ( ∆ xs m a l le r ) b u t a p o o r ly d e fin e d w a ve le n g t h a n d a l a r ge u n c er t a in t y ∆p int h e m o m e n t u m o f t h e p a r t i cle t h e gr o u p r e p r e s en t s . A w id e wa v e g r o u pm e a n s a m o r e p r e c is e m o m e n t u m b u t a le s s p r e c is e p o s i tio n . T h e

in f a m o u s He is e n b e r g u n c er t a i n t y p r i n c ip l e is a f o r m a l st a t e m e n t o f t h es t a nd a r d d e v ia t i ons o f p r op e r t i e s im p lic it in t h e p r o ba b i lit y m od e l o f fu n d a m e n t a l p a r t ic le s.

∆ x∆p ≥h

2( 3 8 . 6 4 )

Ac co r d in g t o t h e st a n d a r d in t e r p r e t a t io n o f q u a n t u m m e ch a n ic s, t h e a c to f m e a su r i n g t h e p o s it io n o r m o m e n t u m o f a q u a n t u m m e ch a n ic al en t it yc o lla p s e s t h e wa ve -p a r t i c le d u a l it y b e c a us e t h e p r i nc ip le f o r b i ds bo t hq u a n t it ie s t o b e s im u l t a n e o u s ly k n o w n w it h p r e c is io n .

T h e W a v e -P a r t i c le D u a l i t y i s N o t D u e t o t h e U n c e r t a i n t yP r i n c i p l e

Qua n t u m e n t it ie s c a n b e h a ve like pa r t ic le s o r wa ve s , d e p e n d in g onh o w t h e y a r e o b s e r ve d . T h e y ca n b e d iffr a c t e d a n d p r o d u c e in t e r fe r e n c ep a t t e r n s ( w a ve b e h a v io r ) w h e n t h e y a r e a l lo w e d t o t a k e d iffe r e n t p a t h sfr o m s o m e s o u r c e t o a d e t e c t o r --in t h e u s u a l e xa m p l e , e le c t r o n s o rp h o t o n s g o t h r o u g h t w o s lit s a n d f o r m a n in t e r fe r e n c e p a t t e r n o n t h es cr e e n b e h i n d . On t h e o t h e r h a n d , w it h a n a p p r o p r i a te d e t ec to r p u ta lo n g o n e o f t h e p a t h s ( a t a s lit , s a y ), t h e q u a n t u m e n t i t ie s ca n b ed e t e c t e d a t a p a r t i c u l a r p l a c e a nd t i me , a s i f t h e y a r e p o i n t - like p a r t i c le s .

Bu t a n y a t t e m p t t o d e t e r m i n e w h ic h p a t h is t a k e n b y a q u a n t u m o b je ctd e s t r oys t h e in t e r f e r e nc e pa t t e r n . Ric ha r d Fe yn m a n d e s cr i be d t h i s a st h e c e n t r a l m y s te r y o f q u a n t u m p h y s ic s.

Boh r c a lle d t h is va gue p r in c ip le ' c omp le m e n t a r y ' , a nd e xp l a in e d i tin t e r m s o f t h e u n c e r t a in t y p r in c ip l e , p u t f o r wa r d b y W e r n e r H eis e n b e r g ,h is p o s t d o c a t t h e t im e . In a n a t t e m p t t o p e r s u a d e Ein s te in t h a t w a ve -p a r t i c le du a l it y is a n e s s e n t i a l p a r t o f qu a n t u m m e c ha n ic s, Boh rc o n s t r u c t e d m o d e l s o f q u a n t u m m e a su r e m e n t s t h a t sh o w e d t h e fu t i lit y




5 7 4

o f t r y in g t o d e t e r m i n e w h i ch p a t h w a s t a k e n b y a q u a n t u m o b j ec t in a nin t e r fe r e n c e e xp e r im e n t . As so o n a s e n o u g h i n f o r m a t io n is a cq u ir e d f o rt h i s d e t e r m i n a t i o n , t h e q u a n t u m in t e r f e r e n c e s m u s t v a n i sh , s a id Bo h r ,b e c a u s e a n y a c t o f o b s e r v in g w ill im p a r t u n c o n t r o l la b l e m o m e n t u mki cks t o t he qu a n t u m ob je c t . Th i s is qu a n t if ie d by He is e nb e r g ' s

u n c e r t a in t y p r i nc ip le , wh i ch r e l a t e s u n c e r t a in t y in po s it ion a lin fo r m a t io n t o u n c e r t a in t y in m o m e n t u m --w h e n t h e p o s it io n o f a n e n t it yis co n s t r a in e d , t h e m o m e n t u m m u s t b e r a n d o m iz ed t o a c er t a in d e g r e e .

M o r e t h a n 6 0 y e a r s a ft e r t h e f a m o u s d e b a t e b e t w e e n N ie ls Bo h ra n d Alb e r t Ein s t e in o n t h e n a t u r e o f q u a n t u m r e a lit y , a q u e s t io n c en t r a lt o t h e i r d e b a t e --t h e n a t u r e o f q u a n t u m in t e r fe r e n c e --h a s r e su r f a ce d .The u s ua l t e xt bo ok e xp l a na t ion o f wa ve -p a r t i c le du a l it y i n t e r m s o f u n a v o id a b le ' m e a s u r e m e n t d is t u r b a n c es ' is e xp e r i m e n t a lly p r o v e nin c o r r e ct b y a n e x p e r i m e n t r e p o r t e d in t h e Se p t e m b e r 3 , 1 9 9 8 i ss u e o f N a tu r e [2 4 ] b y D u r r , No n n , a n d Re m p e . D u r r , N o n n , a n d Re m p e r e p o r t

o n t h e i n t e r f e r e n c e fr i n g e s p r o d u c ed w h e n a b e a m o f c o ld a t o m s isd if fr a c t e d b y s t a nd in g wa ve s o f ligh t . The i r in t e r f e r om e t e r d i s p l a ye dfr i n g e s o f h ig h c o n t r a s t --b u t w h e n t h e y m a n ip u la t e d t h e e le c tr o n ic s ta t ewit h i n t h e a t om s wit h a m ic r owa ve fi eld a c cor d in g t o whi ch p a t h wa st a k e n , t h e fr i n g e s d is a p p e a r e d e n t ir e ly . T h e in t e r fe r o m e t e r p r o d u c e d as pa t i a l d is t r ibu t ion o f el ec t r on ic pop u la t i ons whi c h we r e ob s e r ve d v i af lu or e s c e nc e . The m ic r owa ve fi el d c a nc e le d t h e sp a t i a l d is t r i bu t i on o f e le c t r on ic p op u la t i ons . The ke y t o t h i s n e w e xpe r i m e n t wa s t h a ta lt h o u g h t h e in t e r fe r e n c e s a r e d e st r o y e d , t h e i n it ia lly i m p o s e d a t o m icm o m e n t u m d i st r ib u t io n le ft a n e n v e lo p e p a t t e r n ( in w h i ch t h e fr in g e s

u s e d t o r e s id e ) a t t h e de t e c t o r . A c a r e f u l a n a l ysis o f t h e pa t t e r nd e m o n s t r a te d t h a t i t h a d n o t b e e n m e a su r a b l y d ist o r t ed b y a m o m e n t u mki ck o f t he t yp e in voke d by Boh r , a nd t h e r e f o r e t h a t a n y l oca l ly r e a l is t icm o m e n t u m k ic ks im p a r t e d b y t h e m a n i p u l a tio n o f t h e i n t e r n a l a t o m i cs t a t e a cc o r d i n g t o t h e p a r t ic u l a r p a t h o f t h e a t o m a r e t o o s m a l l t o b er e s p o n s ib l e f o r d e s t r o y in g i n t e r f e r e n c e .

T h e C o r r e s p o n d e n c e Pr i n c i p l e Do e s N o t H o ld

Re c e n t e xpe r i m e n t a l r e s u l t s a ls o d i sp e l a n o t h e r d oc t r i ne o f q u a n t u m m e ch a n ic s [2 5 , 2 6 ] . Bo h r p r o p o s e d a r u le o f t h u m b c a lle d t h e

c or r e s p on d e n c e p r i nc i p l e [ 27] . A f o r m of t h e p r i nc i p l e wid e ly r e pe a t e din t e xt b o o k s a n d le c tu r e h a lls st a t e s t h a t p r e d i ct io n s o f q u a n t u mm e c h a n ic s a n d c la s sic a l p h y s ic s sh o u l d m a t c h f o r t h e m o s t e n e r g e t icc a s e s .

In th e Nov. 22 Ph ysica l Review Let te rs [2 5] , Bo Gao ca lcu la tesp os s ib l e e ne r gy s t a t e s o f a n y c h ill e d , t wo-a t om m ol e cu l e , s uc h a ss o d i u m , t h a t ' s v ib r a t i n g a n d r o t a t in g a lm o s t t o t h e b r e a k i n g p o i n t . H ep e r f o r m s t h e c a lc u la t io n s v ia q u a n t u m m e ch a n ic a l a n d s o c a lle d




5 7 5

s e m i cla s sic a l m e t h o d s a n d c o m p a r e s t h e r e s u l t s. In s t ea d o f t h e r e s u lt sa g r e e i ng be t t e r fo r in c r e a s in g ly e ne r ge t i c s t a t e s . The opp os it e h a p p e n s .

C la s s i c a l So l u t i o n o f t h e Sc h r o d i n g e r Eq u a t i o n

Mills ha s so l ve d a n d p u b l is he d a s o lu t ion o f a Sc hr od i n g er ty p e

e qu a t i on ba s e d o n f ir s t p r i n c ip le s [ 1 ]. The c e n t r a l fe a t u r e o f t h i s t h e or yi s t h a t a l l p a r t i c le s ( a t om i c-s iz e a n d m a c r os c op i c p a r t i c le s ) obe y t h esam e p h ys ica l l aws. Wher eas Sch r o d i n g er p o s tu l at e d a b o u n d a r yc o n d i t i o n : Ψ → 0 a s r → ∞ , t h e b o u n d a r y c o n d i t io n in M ills ' t h e o r y w a sd e r i ve d f r om Ma xwe ll ' s e qu a t i ons [ 28] :

Fo r n o n -r ad i a tiv e s t at e s , t h e c u r r e n t -d e n s it y f u n c t io n m u s t n o t

p o s se s s sp a ce -t im e Fo u r ie r c o m p o n e n t s t h a t ar e s y n c h r o n o u s

wi t h w aves t rave lin g a t t he sp eed o f ligh t .

Ap p lic a t ion o f t h is bou n d a r y c on d it ion le a d s t o a p h ys ic a l m od e l o f p a r t i c le s , a t om s , m ol e cu le s , a n d , in t h e fi na l a na l ysis , c os mol ogy . Thec lo s e d -fo r m m a t h e m a t ic a l s o lu t i o n s c o n t a i n fu n d a m e n t a l c o n s t a n t s o n ly,a n d t h e c a lc u l a t e d va l ue s fo r p h ys ic a l qu a n t it ie s a g r e e wit h e xp e r i me n t a lob s e r va t ion s . In a d d it ion , t h e t h e or y p r e d ic t s t h a t Eq . ( 38 .3 ) , s hou ld ber e p l a ce d b y Eq . (38 .65 ) .

n = 1,2,3,..., and , n =1

2,1

3,1

4,... ( 3 8 . 6 5 )

So m e r e v is io n s t o s t a n d a r d q u a n t u m t h e o r y a r e im p l ie d . Q u a n t u mm e ch a n ic s b e c o m e s a r e a l p h y s ic a l d e s c r ip t i o n a s o p p o s ed t o a p u r e ly

m a t h e m a t ic a l m o d e l wh e r e t h e o ld a n d t h e r e v is e d v e r sio n s a r ein t e r ch a n g e a b l e b y a Fo u r ie r T r a n s fo r m o p e r a t i o n [1 ] .

The t h e or i e s o f Boh r , Sc hr od in ge r , a nd p r e s en t ly Mills a ll g ive t h ei de n t ic a l e qu a t i on f o r t h e p r in c ip a l e ne r gy l e ve ls o f t h e on e e le c t r ona t o m .

E ele = −Z 2e2

8 on2a H

= −Z 2

n2 X 2.1786 X 10−18 J = − Z 2 X

13.598

n2 eV ( 3 8 . 6 6 )

The Mills t he or y s o lve s t he t wo d im e n s ion a l wa ve e qu a t i on f o r t h ec h a r g e d e n s it y fu n c t io n o f t h e e l ec t r o n . An d , t h e Fo u r ie r t r a n s fo r m o f t h e c ha r ge d e n s it y fu n c t ion is a so l u t i on o f t he t h r e e d im e n s ion a l wa ve

e q u a t io n i n fr e q u e n c y k ,( ) s pa c e . Whe r e a s , t h e Sc hr od i n g e r e q u a t io ns o lu t ion s a r e t h r e e d i m e n s ion a l in s pa c e t im e . The e n e r gy is g ive n b y

H −∞

∞

∫ dv = E 2

−∞

∞

∫ dv ; ( 3 8 .6 7 )

2

−∞

∞

∫ dv = 1 ( 3 8 . 6 8 )




5 7 6

T h u s ,

H −∞

∞

∫ dv = E ( 3 8 . 6 9 )

In t h e c a se t h a t t h e p o t e n t i a l e n e r g y o f t h e H a m i lt o n i a n , H , is a c on s t a n tt i me s t h e wa ve nu m be r , t he Sc hr o d in ge r e qu a t i on i s t he we ll kn ownBe s se l e qu a t i on . The n wit h on e o f t h e s o lu t i ons fo r , Eq . ( 38 .69) i se qu iva le n t t o a n i nve r s e Fou r i e r t r a n s fo r m . Ac cor d in g t o t h e du a l it y a nds ca le c h a n g e p r o p e r t i e s o f Fo u r ie r t r a n s fo r m s , t h e e n e r g y e q u a t i o n o f t h e p r e s en t t h e o r y a n d t h a t o f q u a n t u m m e ch a n ic s a r e id e n t ic a l, t h ee n e r gy o f a r a d ia l D ir a c d e lt a fu n c t ion o f r a d iu s e qua l t o a n in t e ge rm u l t ip l e o f t h e r a d iu s o f t h e h y d r o g e n a t o m ( Eq . (3 8 .6 6 ) ) . An d , Bo h ro b t a i n e d t h e s a m e e n e r g y f o r m u la b y p o s t u la t in g n o n r a d i a t iv e st a t e sw it h a n g u la r m o m e n t u m

L z = mh ( 3 8 . 7 0 )a n d s o lv in g t h e e n e r gy e q u a t ion c la s sic a lly .

T h e m a t h e m a t ic s fo r a l l th r e e t h e o r i es c o n v e r g e t o Eq . ( 3 8 . 6 6 ) .However , th e ph ys ics i s qu i t e d i ff e ren t . Only th e Mi ll s th eor y is d e r ivedfr o m fir s t p r in c ip le s a n d h o ld s o v e r a s c a le o f sp a c e tim e o f 4 5 o r d e r s o f m a g n i t u d e : it c o r r e ct ly p r e d i ct s t h e n a t u r e o f th e u n iv e r se fr o m t h es ca le o f t h e q u a r k s t o t h a t o f t h e c o s m o s .

Mill s r e v is ion s t r a n s fo r m Sc hr od i n g e r 's a n d He is en b e r g' s q u a n t u mt h e o r y in t o w h a t m a y b e t e r m e d a c lassi cal quantum t heory . Physicald e s cr i p t i ons flow r e a d il y f r om t h e t he or y . For e xa m p le , in t h e o ldq u a n t u m t h e o r y t h e sp i n a n g u l a r m o m e n t u m o f t h e e le ct r o n i s c a lle d t h e" in t r i ns i c a n gu l a r m om e n t u m " . Th i s t e r m a r i s e s be c a u s e it is d if fi cu l t t o

p r ov i de a ph ys ic a l in t e r p r e t a t i on f o r t he e le c t r on ' s s p i n a n gu l a rm o m e n t u m . Q u a n t u m Ele ct r o d y n a m ic s p r o v id e s s o m e w h a t o f a p h y s ic a lin t e r p r e t a t io n b y p r o p o s in g t h a t t h e " va c u u m " co n t a i n s flu c t u a t i n ge le c t r ic a nd m a gn e t ic fie ld s . In c on t r a s t , in Mills ' t he or y , sp in a n gu l a rm o m e n t u m r e s u lt s fr o m t h e m o t io n o f n e g a t iv e ly ch a r g e d m a s s m o v in gs y st e m a t i ca lly , a n d t h e e q u a t i o n fo r a n g u l a r m o m e n t u m , r × p , c a n b ea p p l ie d d ir e c t ly t o t h e w a v e fu n c t io n ( a c u r r e n t d e n s it y fu n c t io n ) t h a td e sc r ib e s t h e e le ct r o n . An d , q u a n t i za t io n is ca r r i e d b y t h e p h o t o n ,r a t h e r t h a n p r o b a b ilit y w a v e s o f t h e e le ct r o n .

Fr a c t i o n a l Q u a n t u m En e r g y Le v e l s o f H y d r o g e nThe n on r a d ia t i ve s t a t e o f a t om ic hyd r oge n whi c h i s h is t o r ic a lly

c a lle d t h e " gr o u n d s t a te " fo r m s t h e b a s is o f t h e b o u n d a r y c o n d it io n o f Mill s t h e or y [ 1 ] t o s o lve t h e wa ve e qu a t i on . Mills fu r t h e r p r e d ic t s [ 1 ]t h a t c e r t a i n a t om s o r ion s se r ve a s c a t a lys t s t o r e le a s e e ne r gy fr omh y d r o g e n t o p r o d u c e a n i n c r e a se d b in d in g e n e r g y h y d r o g e n a t o m c a lle da h y d r in o a to m h a v in g a b i nd in g e ne r gy o f




5 7 7

Binding Energy =13.6 eV

n2 ( 3 8 . 7 1 )

w h e r e

n =1

2,1

3,1

4,...,

1

p( 3 8 . 7 2 )

a n d p is a n in t e g e r g r e a t e r t h a n 1 , d e s ig n a t e d a s H a H

p

w h e r e a H is t h e

r a d iu s o f th e h y d r o g e n a t o m . ( Alt h o u g h i t is p u r e ly m a t h e m a t ic a l, t h e s es t a b l e e ne r gy l e ve ls a r e a l so g ive n by bo t h Boh r ' s a n d Sc hr od i n g e r ' st h e or i e s by p os t u l a t in g in t e ge r va lu e s o f t h e c en t r a l ch a r ge . J us t if ic a t ionm a y b e b a s e d o n n o t io n s su c h v ir t u a l p a r t ic le s wh ic h a r e a c ce p t a b le i no t h e r a p p li ca t i ons o f Sc hr od i n g e r ' s e q u a t io n .) Hy d r in o s a r e p r e d i ct e dt o fo r m b y r e a c t in g a n o r d in a r y h y d r o g e n a t o m w it h a c a ta ly s t h a v in g an e t e n t h a lp y o f r e a c t io n o f a b o u t

m ⋅ 27.2 eV ( 3 8 . 7 3 )

w h e r e m is a n in t e ge r . Th i s c a t a lys is r e le a s e s e n e r gy fr om t h e hyd r oge na t o m w it h a c o m m e n s u r a t e d e c r e a s e in s iz e o f t h e h y d r o g e n a t o m ,r n = na H . For exam p le , th e ca ta lys is of H (n = 1) t o H (n = 1/2) r e leases

40.8 eV , a n d t h e h y d r o g e n r a d iu s d e cr e a s e s fr o m a H t o1

2a H .

It is t a u g h t i n t e x t b o o k s t h a t a t o m ic h y d r o g e n c a n n o t g o b e l o w t h eg r o u n d s t a t e. At o m ic h y d r o g e n h a v in g a n e x p e r im e n t a l g r o u n d s t a t e o f 13 .6 e V c a n on ly e xis t in a va c u u m or i n i so l a t ion , a n d a t o m ic hyd r oge nc a n n o t go be l ow t h is g r ou n d s t a t e in i so l a t ion . Howe ve r , t h e r e is n ok n o w n c o m p o s it io n o f m a t t e r co n t a in in g h y d r o g e n in t h e gr o u n d s t a t e

o f 13 .6 e V. At om ic h yd r oge n i s r a d ic a l a n d i s ve r y r e a c t ive . It ma y r e a c tt o fo r m a h y d r i d e io n o r c o m p o s it io n s o f m a t t e r . It is a ch e m ic a lin t e r m e d ia t e wh i ch m a y b e t r a p p e d a s m a n y c h e m ic a l in t e r m e d i a t es m a yb e b y m e t h o d s su c h a s is o la t io n o r c r y o g en i ca lly . A h y d r in o a t o m m a yb e c o n s id e r e d a c h e m ic a l in t e r m e d ia t e t h a t m a y b e t r a p p e d i n v a cu u mo r i so la t io n . A h y d r in o a t o m m a y b e v e r y r e a c t iv e t o fo r m a h y d r id e i o no r a n o v e l c o m p o s it io n o f m a t t e r . H y d r o g e n a t p r e d i c te d l o we r e n e r g yle ve ls , h yd r i no a t o m s , h a s be e n id e n t ifie d in t h e e xt r e m e u lt r a v i o le te m is sio n s p e c t r u m fr o m in t e r s te lla r m e d i u m . In a d d i t io n , n e wc o m p o s it io n s o f m a t t e r c o n t a i n i n g h y d r o g e n a t p r e d i ct e d l o we r e n e r g y

le ve ls ha ve r e c e n t ly be e n obs e r ve d in t h e la bo r a t o r y , wh ic h e n e r gy l eve lsa r e a c h ie ve d u s in g t he n ove l c a t a lys t s .

The e xcit e d e n e r gy s t a t e s o f a t om ic hyd r oge n a r e a l so g ive n b y Eq .( 3 8 . 7 1 ) e x ce p t t h a t

n = 1,2,3,... ( 3 8 . 7 4 )Th e n = 1 s t a t e is t h e " g r o u n d " s t a t e fo r " p u r e " p h o t o n t r a n s it io n s ( th en = 1 s t a t e c a n a b s o r b a p h o t o n a n d g o t o a n e xc it e d e l e ct r o n ic st a t e , b u t




5 7 8

it c a n n o t r e l e a se a p h o t o n a n d g o t o a l o w er -e n e r g y e le c tr o n ic st a t e ) .H o we v e r , a n e le c tr o n t r a n s it io n fr o m t h e g r o u n d s t a t e t o a lo w e r -e n e r g ys t a t e is pos s ib l e by a n on r a d ia t i ve e ne r gy t r a n s fe r s uc h a s mu lt ip o l ec oup li ng o r a r e son a n t co l lis ion m e c ha n is m . The s e lowe r - e ne r gy s t a t e s

h a v e fr a c tio n a l q u a n t u m n u m b e r s , n=

1

integer. Pr oc e s s e s t h a t oc c ur

w it h o u t p h o t o n s a n d t h a t r e q u ir e co llis io n s a r e co m m o n . Fo r e xa m p l e ,t h e e x o t h e r m ic ch e m ic a l r e a c t io n o f H + H t o f o r m H 2 d o e s n o t o c c u rwit h t h e e mi s si on o f a ph o t on . Ra t h e r , t h e r e a c t ion r e qu ir e s a co l lis ionw it h a t h i r d b o d y , M , t o r e m o v e t h e b o n d e n e r gy - H + H + M → H 2 + M *[ 2 9 ] . T h e t h i r d b o d y d i st r ib u t e s t h e e n e r g y fr o m t h e e xo t h e r m icr e a c t io n , a n d t h e e n d r e s u lt is t h e H 2 m o le c u le a n d a n in c r e a se in t h et e m p e r a t u r e o f t h e s y st e m . So m e co m m e r cia l p h o sp h o r s a r e b a se d o nn on r a d ia t i ve e ne r gy t r a n s fe r in vo lv in g mu lt ip o le cou p lin g [30 ] . Fore xa m p l e , t h e s t r o n g a b s o r p t i o n s t r e n g t h o f Sb3+ ion s a lon g wit h t h e

e f fic ie n t n on r a d ia t i ve t r a n s fe r o f e xc it a t i on f r om Sb3+ t o Mn2+ , a r er e s p o n s ib l e fo r t h e s t r o n g m a n g a n e s e lu m in e s c e n c e fr o m p h o s p h o r scon ta in ing th ese ions . Simi la r ly , th e n = 1 st a t e of h y d r o g e n a n d t h e

n =1

integers t a te s o f h y d r o g e n a r e n o n r a d i a t iv e , b u t a t r a n s it io n b e t w e e n

t wo n on r a d ia t i ve s t a t e s is pos s ib l e v ia a n on r a d ia t i ve e n e r gy t r a n s fe r , sa yn = 1 t o n = 1/ 2 . In t h e s e c a s e s, d u r in g t h e t r a n s it ion t h e e le c t r on c ou p le st o a n o t h e r e l e ct r o n t r a n s it io n , e le c t r o n t r a n s fe r r e a c t io n , o r i n e l a st ics ca t t e r in g r e a ct io n w h i ch c a n a b s o r b t h e e xa c t a m o u n t o f e n e r g y t h a tm u s t b e r e m o v e d fr o m t h e h y d r o g e n a t o m . T h u s , a ca t a ly s t p r o v id e s a

n e t p o s it iv e e n t h a lp y o f r e a ct io n o f m ⋅ 27.2 eV ( i .e . it a b s o r b s m ⋅ 27.2 eV w h e r e m is a n i n t e ge r ) . Ce r t a i n a t om s o r ion s se r ve a s c a t a lys t s wh i c hr e s o n a n t ly a c ce p t e n e r g y f r o m h y d r o g e n a t o m s a n d r e le a s e t h e e n e r g y t ot h e s u r r o u n d in g s t o e ffe c t e le ct r o n ic t r a n s it io n s t o f r a c tio n a l q u a n t u menergy l eve l s .

Onc e fo r m e d h yd r i nos ha ve a b i nd in g en e r gy g ive n b y Eqs . ( 38 .71-3 8 .7 2 ) ; th u s , t h e y m a y s e r v e a s ca t a ly s ts w h ic h p r o v id e a n e t e n t h a l p y o f r e a c t i on g i ve n by Eq . ( 38 .73 ) . Als o , t h e s im u l t a ne ou s ion iz a t ion o f t woh yd r oge n a t om s m a y p r ov i de a n e t e n t h a lp y give n by Eq . ( 38 .73) . Sin c et h e s u r f a c es o f st a r s c o m p r i se s ig n i fic a n t a m o u n t s o f a t o m i c h y d r o g e n ,

h y d r in o s m a y b e fo r m e d a s a s o u r c e t o in t e r s te lla r s p a c e w h e r e fu r t h e rt r a n s it io n s m a y o c cu r .

A n u m b e r o f e xp e r i m e n t a l o b s e r v a t io n s l ea d t o t h e c o n c lu s io n t h a ta t o m ic h y d r o g e n c a n e x is t in fr a c t io n a l q u a n t u m s t a t es t h a t a r e a t l o we re n e r g ie s t h a n t h e t r a d it io n a l " gr o u n d " ( n = 1) s t a t e . For e xa m p l e, t h ee xis t en c e o f fr a c t io n a l q u a n t u m s t a t es o f h y d r o g e n a t o m s e xp l a in s t h es p e c t r a l o b s e r v a t io n s o f t h e e xt r e m e u lt r a v io le t b a c k gr o u n d e m is sio n




5 7 9

fr o m in t e r s t ella r s p a c e [2 ] , w h i ch m a y c h a r a c t e r iz e d a r k m a t t e r a sd e m o n s t r a t ed i n T a b le 2 . ( In t h e se ca s e s, a h y d r o g e n a t o m in a

fr a c t io n a l q u a n t u m s t a t e, H ni( ) , co l lides , for exam p le , wi th a n =1

2

h y d r o ge n a t o m , H 1

2

, a n d t h e r e s u l t is a n e ve n l ower -e n e r gy hyd r oge n

a t o m , H n f ( ), a n d H 1

2

i s ionized .

H ni( ) + H 1

2

→ H n f ( ) + H + + e− + photon ( 3 8 . 7 5 )

The e n e r gy r e l e a se d , a s a ph o t on , is t he d iff er e n c e be t we e n t h e e ne r g ie so f t he i n i t ia l a nd f in a l st a t e s g ive n by Eqs . (38 .71 -38 .72 ) m in us t h e

i on i za t i on e n e r gy o f H 1

2

, 54.4 eV .) The c a t a l ysis o f a n e n e r gy s t a t e o f

h y d r o g e n t o a l o we r e n e r g y s t a te w h e r e i n a d iffe r e n t lo w e r e n e r g y s t a t ea t o m o f h y d r o g e n s er v e s a s t h e c a t a ly s t is c a lle d d is p r o p o r t io n a t io n b yMills [1].

Id e n t i f i ca t i o n o f Lo w e r - En e r g y H y d r o g e n b y So f t X- r a y s f r o m

D a r k In t e r s t e l l a r M e d i u m

T h e f ir s t s o ft X-r a y b a c k gr o u n d w a s d e t e c te d a n d r e p o r t e d [ 3 1 ]a b ou t 25 ye a r s a go . Qu i t e n a t u r a l ly , i t wa s a s s um e d t h a t t h e s e s o ft X-r a ye m is sio n s we r e fr o m io n iz e d a t o m s wit h in h o t g a se s . In a m o r e r e c e n tp a p e r , a g r a z in g in c id e n c e sp e c t r o m e t e r w a s d e s ig n e d t o m e a su r e a n d

r e c o r d t h e d i ffu s e e xt r e m e u l t r a v io le t b a c k gr o u n d [2 ] . T h e in s t r u m e n tw a s c a r r ie d a b o a r d a s o u n d in g r o c k e t a n d d a t a w er e o b t a i n e d b e t w ee n80 Å a n d 650 Å ( d a t a p o in t s a p p r o xim a t e ly e v e r y 1.5 Å ) . He r e a ga i n , t h ed a t a w e r e in t e r p r e t e d a s e m i ss io n s fr o m h o t g a se s . H o we v e r , t h ea u t h o r s le ft t h e d o o r o p e n f o r so m e o t h e r in t e r p r e t a t io n w i th t h efo llo w in g s t a t e m e n t f r o m t h e i r i n t r o d u c t io n :

" It i s n ow gen era l ly be l ieved th a t th i s d i ffuse soft X-ra y ba ckgrou n dis p r o d u c e d b y a h i g h -t e m p e r a t u r e co m p o n e n t o f t h e in t e r s te lla rm e d i u m . H o we v e r , e v id e n c e o f t h e t h e r m a l n a t u r e o f t h i s e m is sio n isin d ir e c t in t h a t i t is ba s e d n o t on obs e r va t i on s o f l in e e m i ss ion , bu t

o n in d ir e c t ev id e n c e t h a t n o p l a u s ib l e n o n -t h e r m a l m e ch a n i sm h a sb e e n s u g g e st e d w h ic h d o e s n o t c o n f lic t wit h s o m e co m p o n e n t o f t h eob s e r va t ion a l e v id e n c e ."

The a u t h or s a l so s t a t e t h a t " if t h is in t e r p r e t a t i on is co r r e c t , ga s a ts e ve r a l t e m p e r a t u r e s is p r e s e n t . " Sp e c if ic a lly , e m i ss ion s we r ea t t r i bu t e d t o ga s e s in t h r e e r a n ge s : 5 .5 < log T < 5 .7 ; log T = 6 ; 6 .6 < logT < 6 .8 .




5 8 0

T h e e x p la n a t io n p r o p o s ed h e r e in o f t h e o b s e r v e d d a r k i n t e r s t ella rm e d i u m s p e c tr u m h in g e s o n t h e p o s s ib i lit y o f e n e r g y s t a t e s b e l o w t h en = 1 s t a t e , a s g ive n b y Eqs . ( 38 .71 -38 .72 ) . Thu s , lowe r - e ne r gyt r a n s it ion s o f t h e t ype ,

∆E =1

n f 2 −

1

ni2

X 13.6 eV − 54.4 eV n = 1,

1

2 ,

1

3,

1

4 ,..., a n d ni > n f ( 3 8 . 7 6 )

in d u c ed b y a d i sp r o p o r t io n a t io n r e a c tio n w it h H a H

2

o u g h t t o o cc u r .

The wa ve l en g t h i s r e la t e d t o ∆E b y

(in Å) =1.240 X 104

∆E (in eV )( 3 8 . 7 7 )

The e n e r g ie s a n d wa ve l e ng t h s o f s e ve r a l o f t h e s e p r op os e d t r a n s it ion sa r e s hown i n Ta b l e 1 . No t e t h a t t he lowe r e ne r gy t r a n s it ion s a r e in t h esof t X-ra y r egion .

Tab le 1 . En ergies (Eq. (38 .76 ) ) of seve ra l f r ac t ion a l -s ta te t r a n s it ion s

c a t a lyz e d b y H a H

2

.

n i n f ∆E (eV) λ ( Å)

1

2

1

3 1 3 .6 9 1 2

13

14 4 0 .8 0 3 0 3 .9

14

15 6 8 .0 0 1 8 2 .4

15

16 9 5 .2 0 1 3 0 .2

16

17 1 2 2 .4 1 0 1 .3

17

18 1 4 9 .6 8 2 .9

T h e D a t a A n d I t s In t e r p r e t a t i o n




5 8 1

In th e i r an a lys is of th e d a ta , Lab ov an d Bowyer [2 ] e stab l isheds e ve r a l t e s ts t o s e p a r a t e e m is sio n f e a t u r e s fr o m t h e b a c k gr o u n d . T h e r ewe r e se ve n f e a t u r e s ( pe a ks ) t h a t pa s s e d t h e ir cr i t e r ia . The wa ve le n g t h sa n d o t h e r a s p e c ts o f t h e s e p e a k s a r e sh o w n i n T a b le 2 . Pe a k s 2 a n d 5w e r e in t e r p r e t e d b y La b o v a n d Bo w y e r a s in s tr u m e n t a l s ec o n d -o r d e r

i ma ge s o f p e a ks 4 a n d 7 , r e s pe c t i ve ly . Pe a k 3 , t he s t r on ge s t f ea t u r e , isc le a r l y a he l iu m r e s ona n c e l in e : He (1s 12 p 1 → 1 s2) . At is sue h e re , i s th ein t e r p r e t a t io n o f p e a k s 1 , 4 , 6 , a n d 7 . It is p r o p o s e d t h a t p e a k s 4 , 6 , a n d

7 a r is e fr o m t h e13 →

14 ,

14 →

15 , a n d

16 →

17 h y d r o g e n a t o m s

t r a n s it ion s give n b y Eq . ( 38 .76) . It is a ls o p r o p os e d t h a t pe a k 1 a r i s e s

f r om i ne l a s t ic h e liu m s c a t t e r i ng o f p e a k 4 . Tha t is , t h e13 →

14

t r a n s it ion y ie ld s a 40.8 eV p h o t o n ( 303.9 Å) . W h e n t h i s p h o t o n s t r ik e s He (1s2 ) , 21.2 eV is a bs or b e d i n t h e e xcit a t i on t o He (1s12 p1 ) . Th is leaves a

19.6 eV p h o t o n (632.6 Å) , p e a k 1 . Fo r t h e s e fo u r p e a k s , t h e a g r e e m e n tb e t w e e n t h e p r e d i ct e d v a l u e s ( Ta b l e 1 ) a n d t h e e x p e r i m e n t a l v a lu e s( T a b le 2 ) i s r e m a r k a b le .




5 8 2




5 8 3

O n e a r g u m e n t a g a i n s t t h i s n e w i n t e r p r e t a t io n o f th e d a t a is t h a t

t h e t r a n s it ion15 →

16 i s mi s sin g—p r e d ic t e d a t 130.2 Å by Eqs . (38 .76 -

38 .77 ) . Th i s m is sin g pe a k c a nn o t be e xp la i ne d i n t o e x is t e nc e , bu t ar e a s o n a b le r a t i o n a l e ca n b e p r o v id e d fo r w h y i t m i g h t b e m is sin g f r o m

t h e s e d a t a . T h e d a t a o b t a i n e d b y La b o v a n d Bo w y er a r e o u t s t a n d in gw h e n t h e r e g io n o f t h e s p e c t r u m , th e t im e a l lo t t e d fo r d a t a c o lle c tio n ,a n d t h e log is t ic s a r e c on s id e r e d . None t h e le s s, i t is cle a r t h a t t h e s igna l-t o -n o is e r a t io i s lo w a n d t h a t c o n s id e r a b l e e ffo r t h a d t o b e e x p e n d e d t od if fe r e n t i a t e e m is sion f e a t u r e s fr om t h e ba c kgr ou n d . Th i s p a r t i c u l a r

p e a k ,15 →

16 , is like ly t o be on ly s ligh t l y s t r onge r t h a n t h e

16 →

17

p e a k ( t h e in t e n s it i es , Ta b l e 2 , a p p e a r t o d e c r e a s e a s n d e c r e a s e s ), wh i c hh a s low in t e ns i t y . La bov a n d Bowye r p r ov id e d t h e ir d a t a ( wa ve le ng t h ,c o u n t , c o u n t e r r o r , b a c k gr o u n d , a n d b a c k gr o u n d e r r o r ) . T h e co u n t sm in u s b a c k g r o u n d v a lu e s fo r t h e r e g io n o f in t e r e st , 130.2 ± 5 Å , a r e s h owni n Ta b l e 3 ( t he c on f id e n c e lim it s fo r t h e wa ve le n g t h o f a b ou t ±5 Å a r et h e s in g le -s id e 1 c on fid e n c e le ve ls a nd in c lu d e bo t h t h e u n c e r t a in t ie s int h e fit t in g p r oc e d u r e a nd u n c e r t a in t ie s in t h e wa ve le n g t h c a lib r a t i on ) .N o t e t h a t t h e la r g e st p e a k ( c o u n t – b a c k g r o u n d ) i s a t 129.64 Å a n d h a s acounts − background = 8.72 . The counts − background f o r t h e s t r on ge s t signa l o f t h e o t h e r h y d r i n o t r a n s it io n s a r e : n = 1/3 t o n = 1/4 , 20.05 ; n = 1/ 4 t on = 1/5 , 11.36; n = 1/ 6 t o n = 1/7 , 10.40 . Thu s , t h e r e is fa i r a gr e e m e n t w it ht h e wa ve l en g t h a n d t h e st r e n g t h o f t h e signa l. Th i s , o f c our s e , d oe s n o tm e a n t h a t t h e r e is a p e a k a t 130.2 Å. Howe ve r , i t i s n o t un r e a s on a b le t o

c o n c lu d e t h a t a s p e ct r u m w it h a b e t t e r s ig n a l -t o -n o is e r a t io m ig h tu n c o ve r t h e m is sin g p e a k .




5 8 4

Ta b l e 3 . Da t a ( La b ov & Bowye r ) n e a r t h e p r e d ic t e d1

5→

1

6t r a ns i t i on

(130.2 Å) .

l counts background counts – background(Å)

1 2 5 .8 2 2 6 2 1 .5 8 4 .4 2

1 2 7 .1 0 2 2 2 1 .3 2 0 .6 8

1 2 8 .3 7 1 8 1 9 .5 0 –1 .5 0

1 2 9 .6 4 2 9 2 0 .2 8 8 .7 2

1 3 0 .9 0 1 8 1 9 .7 6 –1 .7 6

1 3 2 .1 5 2 0 1 9 .5 0 0 .5 0

1 3 3 .4 1 1 9 1 9 .5 0 –0 .5 0

1 3 4 .6 5 1 9 2 0 .8 0 –1 .8 0

An o t h e r , a n d m o r e im p o r t a n t , a r g u m e n t a g a in s t t h i s n e win t e r p r e t a t io n is t h e f a ct t h a t t h e p r o p o s e d f r a ct io n a l-q u a n t u m -s t a t eh y d r o g e n a t o m s h a v e n o t b e e n d e t e ct e d b e fo r e . T h e r e a r e s e ve r a le xp l a na t i on s . Fir s t ly , t h e t r a n s it i ons t o t he s e fr a c t i ona l st a t e s mu s t bef o r b i dd e n o r m u s t ha ve ve r y h igh a c t i va t i on e n e r g ie s —ot h e r wis e a llh y d r o g e n a t o m s w o u ld q u ic k ly g o t o t h e s e lo w e r e n e r g y s t a t e s (a ne s t im a t e d t r a n s it ion p r ob a b i lit y , ba s e d on t h e La bo v a nd Bowye r d a t a , isb e b e t w ee n 10

−15 a n d 10−17s−1 ) . In a c t ua l it y , a c a t a lys t is r e qu ir e d in o r d e r

t o o b t a in e m is sio n . Se co n d l y , t h e n u m b e r o f h y d r o g e n a to m s ( n = 1) , t h eh y d r o g e n -a t om d e n s it y , a n d t h e p r e s en c e o f a n a c t iv e ca t a ly s t u n d e r a n y

c ond it ion s on Ea r t h is exce e d i n g low. The c om bi na t ion o f e x t r e me l y lowp o p u la t io n a n d e xt r e m e ly l o w tr a n s it io n p r o b a b ilit y m a k e s t h e d e t e ct io nof these t r an s it ions e sp ec ia l ly d i ff icul t . Th i rd ly , th i s is a ve ryt r o u b l e so m e r e g io n o f t h e e le c t r o m a g n e t i c s p e c t r u m fo r d e t e ct io nb e c a u s e t h e s e w a v e le n g t h s d o n o t p e n e t r a t e e v e n m illim e t e r s o f t h ea t m os ph e r e ( i.e . t h is r e g ion is t he va c uu m u lt r a v i o le t wh i ch r e qu ir e swin d owl es s s pe c t r os c op y a t va c uu m f o r de t e c t i on ) . La s t ly , n o one




5 8 5

p r e v i ous ly ha s be e n a c t ive ly s e a r c h i ng fo r t h e s e t r a n s it ion s . TheCh a n d r a X-r a y Obs e r va t o r y is sc he d u le d t o p e r f o r m s im ila r e xpe r i m e n t sw it h d e t e c t io n a t m u c h b e t t e r s ig n a l t o n o i se t h a n o b t a i n e d b y La b o v a n dBowyer .

H y d r o g e n C a t a l y s t sThe c a t a l ysis o f h yd r oge n in vo lve s t he n on r a d ia t i ve t r a n s fe r o f

e n e r g y f r o m a t o m ic h y d r o g e n t o a c a t a ly s t wh ic h m a y t h e n r e le a se t h et r a n s fe r r e d e n e r g y b y r a d i a t iv e a n d n o n r a d i a t iv e m e c h a n is m s . As ac o n s e q u e n c e o f t h e n o n r a d ia t iv e e n e r g y t r a n s fe r , t h e h y d r o g e n a t o mb e c o m e s u n s t a b le a n d e m it s fu r t h e r e n e r g y u n t i l it a c h i e ve s a lo w e r -e ne r gy n on r a d ia t ive s t a t e ha v in g a p r i nc ipa l e n e r gy l e ve l give n by Eqs .( 3 8 . 7 1 - 3 8 . 7 2 ) .

Po t a s s iu m ion s ca n p r ov i de a ne t e n t h a l py o f a m u lt ip l e o f t h a t o f t h e p o t e n t i a l e n e r g y o f t h e h y d r o g e n a t o m . T h e se co n d i o n i za t io n

e n e r gy o f p o t a s s iu m is 31.63 eV ; a n d K +

r e leases 4.34 eV whe n it i sr e d u c ed t o K . T h e co m b i n a t i o n o f r e a c t io n s K + t o K 2+ a n d K + t o K ,t h e n , h a s a n e t e n t h a lp y o f r e a ct io n o f 27.28 eV , wh ic h is e qu iva l en t t om = 1 in Eq . ( 38 .73 ) .

27.28 eV + K + + K + + H a H

p

→ K + K 2 + + H

a H

( p +1)

+ [( p +1)2 − p2

] X 13.6 eV ( 3 8 . 7 8 )

K + K 2+ → K + + K + + 27.28 eV ( 3 8 . 7 9 )

The ove r a l l r e a c t ion is

H a H

p

→ H

a H

( p +1)

+ [( p +1)2 − p2

] X 13.6 eV ( 3 8 . 8 0 )

Ar gon ion is a ca t a l ys t . The s e c ond ion iz a t ion e n e r gy is 27.63 eV .

27.63 eV + Ar + + H a H

p

→ Ar 2+ + e− + H

a H

( p +1)

+ [( p +1)2 − p2

] X 13.6 eV ( 3 8 . 8 1 )

Ar 2+ + e− → Ar + + 27.63 eV ( 3 8 . 8 2 )

An d , t he ove r a l l r e a c t i on i s H

a H

p

→ H

a H

( p +1)

+ [( p +1)2 − p2

] X 13.6 eV ( 3 8 . 8 3 )

An a r g o n i o n a n d a p r o t o n c a n a ls o p r o v id e a n e t e n t h a l p y o f am u l t ip l e o f t h a t o f t h e p o t e n t i a l e n e r g y o f t h e h y d r o g e n a t o m . T h e t h i r di on i za t i on e n e r gy o f a r gon i s 40.74 eV , a n d H + r e leases 13.6 eV whe n it i s




5 8 6

r e d u c ed t o H . T h e co m b i n a t i o n o f r e a c t io n s o f Ar 2+ t o Ar 3+ a n d H + t o H ,t h e n , h a s a n e t e n t h a lp y o f r e a ct io n o f 27.14 eV , wh i ch is e qu iva le n t t om = 1 in Eq . ( 38 .73 ) .

27.14 eV + Ar 2+ + H + + H a H

p

→ H + Ar 3+ + H

a H

( p +1)

+ [( p +1)2 − p2

] X 13.6 eV ( 3 8 . 8 4 )

H + Ar 3+ → H + + Ar 2 + + 27.14 eV ( 3 8 . 8 5 )An d , t he ove r a l l r e a c t i on i s

H a H

p

→ H

a H

( p +1)

+ [( p +1)2 − p2

] X 13.6 eV ( 3 8 . 8 6 )

An o m a l o u s T h e r m a l Br o a d e n i n g o f t h e A t o m i c Hy d r o g e n

Em i s s io n Sp e c t r u m i n a G a s D i s c h a r g e Ce l l b y t h e P r e s e n c e o f

A r g o n

The ob s e r va t ion by Kur a ic a a n d Kon je v ic [3 ] o f hyd r oge n Ba l me r

lin e s in a r g o n -h y d r o g e n , n e o n -h y d r o g e n , a n d p u r e h y d r o g e n m ixt u r e sr e v e a le d in t e n s e win g d e v e lo p m e n t s w it h a r g o n p r e se n t i n t h e p la s m a o f n e ga t ive g low of a g low d i sc ha r ge ir r e s p e c t ive o f ca t h od e m a t e r i a l( c a r bo n , c opp e r , a n d s ilve r ) . The a u t h or s o ff e r a t e n t a t i ve e xp l a na t ionfo r h y d r o g e n l in e s h a p e s in t h e p r e s en c e o f a r go n w h i ch is b a s ed o n aq u a s ir e s o n a c e ch a r g e t r a n s fe r b e t w e e n m e t a st a b l e a r g o n i o n s a n dh y d r o g e n m o le c u le s a n d t h e f o r m a t i o n o f a h y d r o g e n m o le c u la r io n .Ac co r d in g t o th e a u t h o r s ,

" .. . i t is e s sent i a l tha t th e H 2+ o r H 3

+ io n m u s t g a in e n e r g y in t h e

e le c t r i c f ie ld be f o r e d i ss oc ia t i on . O t he r wis e , t h e la r ge e n e r gy o f e xc it e d h y d r o g e n a t o m s (o n t h e a v e r a ge 5 0 e V p e r a t o m ) ca n n o tbe e xp l a in e d " .

T h e s o u r c e o f 50 eV a n o m a lo u s t h e r m a l b r o a d e n i n g o f t h e Ba lm e rlin e s obs e r ve d by Ku r a i ca a n d Kon je v ic [3 ] d u r i ng a g l ow d i sc ha r ge o f h y d r o g e n -a r g o n m ixt u r e s w h i ch w a s n o t o b s er v e d w it h n e o n -h y d r o g e nm ix tu r e s o r p u r e h y d r o g e n ir r e s p e c t iv e o f ca t h o d e m a t e r ia l is a s sig n e dt o lowe r -e n e r gy hyd r oge n t r a n s it ion s . T r a n s it ion s o f h yd r oge n t o lowe r -e n e r gy l e ve ls oc cu r s in a h yd r oge n p la s m a d is ch a r ge c e ll via a c a t a l ys tw it h a n e t e n t h a lp y o f a b o u t 2 7 . 2 e le c tr o n v o lt s in c lu d in g t h e r m a lb r oa d e n in g . Th e c a t a lys t , Ar + i ons ( Eqs . ( 38 .81 -38 .83) ) , o r Ar 2+ io n s a n dp r o t o n s ( Eq s . ( 3 8 . 8 4 - 3 8 . 8 6 ) ) a r e f o r m e d b y t h e d is ch a r g e w h i ch a ls op r o d u c e s r e a c t a n t h y d r o g e n a t o m s . T h e t r a n s it io n s t o lo w e r -e n e r g yle ve ls c a t a lyz e d by a r gon ion s r e s u l t in t h e r e l e a s e o f en e r gy ( Eqs .( 3 8 . 8 1 - 3 8 . 8 6 ) ) . H y d r o g e n a t o m s in fr a c t io n a l q u a n t u m s t a t es c a n u n d e rgo a n a u t oc a t a l y t ic r e a c t i on t o fu r t h e r lowe r e ne r gy st a t e s . Lin e sp e c t r ai n t he e x t r e m e u l t r a v i o le t a nd s o ft x-r a y r e g i ons c o r r e s po n d in g t o t he s e




5 8 7

t r a n s it ion s ha ve be e n ob s e r ve d b y a 4 ° g r a z in g in c id e n c e e x t r e m eu l t r avio le t sp ec t rom ete r a t INP, Gre i fswa ld , Germ an y [7] . The en e rgyh e a t s t h e p la s m a a n d c a u s e s t h e r m a l b r o a d e n i n g o f t h e h y d r o g e ns p e c t r u m .

Ba c k w a r d P e a k i n t h e El e c t r o n Sp e c t r u m f r o m Co l l i s i o n s o f 7 0k e V P r o t o n s w i t h H y d r o g e n At o m s

T h e e n e r g y r e m o v e d b y a s c a t te r in g r e a ct io n is r e so n a n t w it h t h eh yd r oge n e n e r gy r e le a s e d t o s t im ul a t e t h i s t r a n s it ion . Con s id e r t he c a sew h e r e i n a p r o t o n u n d e r g o e s a c o llis io n w it h a n e le c tr o n o f a h y d r o g e na t o m t o e f fe c t a n e le c t r o n i c t r a n s it io n fr o m n = 1 t o n = 1/ 2 . Th e ini t ia la n d f in a l e le c t r on ic st a t e s a r e non r a d ia t i ve ; t h u s , t h e t r a n s it i on m us toc c ur wit h ou t r a d ia t i on . The e n d r e s u l t is a lowe r -e n e r gy s t a t e f o r t heh y d r o g e n a n d b a c k wa r d s ca t t e r e d e le c tr o n w h i ch i s r e q u ir e d t o c o n s e r v ee n e r g y , m o m e n t u m , a n d s a t is fy t h e b o u n d a r y co n d it io n f o r

n o n r a d i a t io n . T h e en e r g y o f t h e h y d r o g e n t r a n s it io n m u s t b et r a n s fe r r e d t o a b a c k w a r d f r e e e le c t r o n . T h e r e a c t io n f o r t h e n = 1 t on = 1/ 2 h yd r oge n t r a n s it i on e f fe c t e d b y i ne l a st i c s ca t t e r i ng o f a 70 ke Vp r o t o n is

H + 70 keV ( ) + H n =1( ) + e−0 eV ( ) → H + 70 keV ( ) + H n =1/ 2( ) + e−

40.8 eV ( ) ( 3 8 . 8 7 )

The b a c kwa r d e le c t r on is Fr a n c k-He r t z [6 ] s ca t t e r e d by f r e e h yd r oge na t om s . For hyd r oge n , t h e Fr a n c k-He r t z s ca t t e r e d p e a ks o f h yd r oge nt r a ns i t i ons , ni → n f is

E =1

n f 2 −

1

ni2

X 13.6 eV −10.2 eV ( 3 8 . 8 8 )

( w h e n t h is e le c t r o n s t r ik e s H (1s1 ) , 10.2 eV is a bs or b e d i n t h e e xcit a t i on t o H (2 p1

)) . For th e n = 1 t o n = 1/ 2 h y d r o g e n t r a n s it io n , a b a c k w a r d 4 0 .8 e Ve l ec t r on i s p r od u c e d wh ic h u n d e r goe s Fr a n c k-He r t z s ca t t e r i n g t o giver i s e t o a 30 .6 e V ba c kwa r d p e a k . Fr a n c k-He r t z s c a t t e r i ng o f t h e 30 .6 e Vp e a k g iv e s r i se t o a 2 0 . 4 e V b a c k w a r d p e a k . O r , t h e b a c k wa r d 4 0 .8 e Ve le c tr o n io n iz e s a h y d r o g e n a t o m t o g iv e r is e t o a 2 7 .2 e V p e a k .

E =

1

n f 2 −

1

ni2

X 13.6 eV −13.6 eV ( 3 8 . 8 9 )( w h e n t h is e le c t r o n s t r ik e s H (1s1 ) , 13.6 eV is a b s o r b e d in t h e io n iz a t io n t o H + + e− ) . O t he r t r a n s it i ons wil l oc c ur t o a le s s e r e x t e n t . D is c on t i n u it i e sa t t h e po s it ion o f t he m a j o r t r a n s it i ons o f 40 .8 e V, 30 .6 e V, 27 .2 e V, a n d2 0 .4 e V w er e o b s er v e d b y Ru d d , e t a l . [4 ] in t h e e le c tr o n s p e c t r u m fr o mc ollis ion s o f 70 ke V p r o t on s wit h h yd r oge n a t om s sh own i n Figur e 1 .




5 8 8

Figur e 1 . Sp e c t r u m of Ru d d e t a l . [ 4 ] o f e le c t r on s e je c t e d a t 1 60 ° fr om7 0 k e V p r o t o n i m p a c t o n t h e t a r g e t g a s fr o m t h e Sle v in h y d r o g e n a t o ms our c e . Ope n c ir c le s , t h e RF e xcit a t i on wa s on a n d t h e me a s ur e dh yd ro gen d i ssoc ia t ion f rac t ion was 93 %; sol id c ir c le s , th e RF exc it a t ion

wa s o ff, m a k i ng t h e t a r ge t pu r e H 2 .

T h e a n g le o f t h e m a x im u m in t e n s it y o f b a c k s ca t t e r in g c a n b ec a lc u la t e d v ia t h e b o u n d a r y c o n d it io n o f n o n r a d ia t io n . T h e c o n d i t io nf o r r a d ia t i on b y a m ovi ng c ha r ge is de r i ve d f r om Ma xwe ll 's e qua t i ons[ 2 8 ] . T h e c o n d it io n f o r n o n r a d ia t io n o f a m o v in g c h a r g e -d e n s it yfu n c t io n is t h a t t h e s p a c e t im e Fo u r ie r t r a n s fo r m o f t h e c u r r e n t - d e n s it y

fu n c t io n m u s t n o t p o s se s s w a v es s y n c h r o n o u s w it h w a v e s t r a v e lin g a tt h e s pe e d o f ligh t , t ha t is syn c hr on ou s wit h n

co r s y n c h r o n o u s w it h

n

c o

whe r e is t h e d ie le c t r ic c ons t a n t o f t h e m e d iu m . The c ond it ion

f o r non r a d ia t i on by a m ovin g po i n t c ha r ge is t ha t i t s s pa c e t im e Fou r i e rt r a n s fo r m d o e s n o t p o s s es s c om p o n e n t s t h a t a r e s yn c h r o n o u s w it hwa ve s t r a ve l in g a t t he s pe e d o f l igh t , a s s hown by Ha u s [28 ] . The Ha usd e r i va t ion a p p lie s t o a m ovi ng c ha r ge -d e n s it y f un c t ion a s we ll be c a u s ec h a r g e o b e y s s u p e r p o s it io n . T h e H a u s d e r i va t io n is su m m a r i ze d b e lo w .

T h e Fo u r i er c o m p o n e n t s o f t h e c u r r e n t p r o d u c ed b y t h e m o v in gc ha r ge a r e de r ive d . The e l ec t r i c f ie ld i s fou n d f r om t h e ve ct o r e qu a t i oni n Fou r i e r s pa c e (k , ω-s pa c e ) . The i nve r s e Fou r i e r t r a n s fo r m is c a r r i e do v e r t h e m a g n i t u d e o f k . T h e r e su l t in g e xp r e ss io n d e m o n s t r a t e s t h a t

t h e r a d ia t i on f ie ld i s p r op or t i on a l t o J⊥ ( cn, ) , whe r e J⊥ (k, ) i s t h e

s p a c e t im e Fo u r ie r t r a n s fo r m o f t h e c u r r e n t p e r p e n d ic u la r t o k a n d




5 8 9

n ≡k

|k |. Specif ically,

E⊥ r,( )d

2=

c

2,Ω( )∫ d d Ω 0

0

n X n X J⊥ cn,

e

ic

n•r

( 3 8 . 9 0 )

Th e f ie ld E⊥ r,( ) d 2 is p r o p o r t io n a l t o J⊥ cn, , na m e ly , t he Fou r i e r

c o m p o n e n t fo r w h i ch k =c

. Fac tor s of th a t m u lt ip ly th e Fou r ie r

c o m p o n e n t o f th e c u r r e n t a r e d u e t o t h e d e n sit y o f m o d e s p e r u n itv o lu m e a n d u n it so lid a n g le . An u n a c ce le r a t e d c h a r g e d o e s n o t r a d ia t ei n f r e e sp a c e , n o t be c a u s e it e xpe r i e nc e s no a c c e le r a t ion , bu t be c a u s e it

h a s n o Fo u r i er c o m p o n e n t J⊥ cn,

.

Con s id e r a 7 0 ke V pr o t o n o f pos it ion r0 t ( ). T h e ch a r g e d e n s it y o f

t h e p r o t o n is d e s cr ib e d b yr, t ( ) = e (r − r0 t ( )) ( 3 8 . 9 1 )w h e r e (r − r0 t ( )) is t h e sp a t ia l u n it im p u l s e fu n c t ion . The cu r r e n t de n s it yis

J r, t ( ) = e•

r0 t ( ) (r − r0 t ( )) ( 3 8 . 9 2 )T h e s p a t i a l Fo u r ie r t r a n s fo r m r e p r e se n t s t h e c u r r e n t d e n s it y a s as up e r p os it i on o f sp a t i a l e xpon e n t ia l s, exp− ik ⋅r .

J k,t ( ) = d 3re•

r0 t ( ) (r − r0 t ( ))exp− ik ⋅r∫ ∫ ∫

= e

•

r0 t ( )e

−ik⋅r 0

( 3 8 . 9 3 )

The f u l l s pa c e t im e Fou r i e r t r a n s fo r m isJ k,( ) = dteve− ik⋅vt + i t ∫ ∫ ∫ = 2 ev − k ⋅ v( )

( 3 8 . 9 4 )

T h e o n ly n o n z er o Fo u r ie r c o m p o n e n t s a r e fo r

k z =v z cos

>c

( 3 8 . 9 5 )

w h e r e is t h e a n g le b e t w e e n t h e w a v en u m b e r v e ct o r , k z , a n d t h eve l oci t y ve c t o r , v z . T h u s , n o Fo u r ie r c o m p o n e n t s t h a t a r e s y n c h r o n o u s

wit h ligh t ve l ocit y wit h t h e p r op a ga t ion c ons t a n t k z = c exis t . Radia t iond u e t o ch a r g e m o t io n d o e s n o t o c cu r w h e n t h i s b o u n d a r y c on d it io n ism e t .

Con s id e r t h e c a se whe r e i n t h e p r o t on u n d e r goe s a co l lis ion wit ht h e e le c t r o n o f a h y d r o g e n a t o m t o e ffe c t a n e l e ct r o n ic t r a n s it io n fr o mn = 1 t o n = 1/ 2 . The i n i t ia l a n d f in a l e le c t r on ic st a t e s a r e no n r a d ia t i ve;t h u s , t h e t r a n s it io n m u s t o cc u r w it h o u t r a d i a t io n . If t h e p r o t o n r e m o v e s




5 9 0

t h e e n e r g y , t h e n r a d ia t io n w ill o cc u r d u r in g t h e c o n c o m i t a n ta c c e le r a t i on . Wit h a c c e le r a t i on , t h e cu r r e n t de n s it y is

J r, t ( ) = e•

r0 t ( ) 1− e− t ( ) (r − r0 t ( )) ( 3 8 . 9 6 )

whe r e is t h e t r a n s it ion d e c a y c ons t a n t . The s pa t i a l Fou r i e r t r a n s fo r m

r e p r e se n t s t h e c u r r e n t d e n s it y a s a s u p e r p o s it io n o f sp a t i a l ex p o n e n t i a ls ,exp− ik ⋅r .

J k,t ( ) = d 3re•

r0 t ( ) 1− e− t ( ) (r − r0 t ( ))exp− ik ⋅r∫ ∫ ∫ = e

•r0 t ( )e−ik⋅r 0 1− e− t ( )

( 3 8 . 9 7 )

The f u l l s pa c e t im e Fou r i e r t r a n s fo r m isJ k,( ) = dteve− ik⋅vt + i t − t ∫ ∫ ∫ = 2 ev

+ i − k ⋅ v( )

( 3 8 . 9 8 )

Fo r Eq . ( 3 8 . 9 8 ) , Fo u r ie r c o m p o n e n t s t h a t a r e s y n c h r o n o u s w i th lig h tv e lo c it y w it h t h e p r o p a g a t io n c o n s t a n t k z =

cd o exis t .

Ho w e ve r , n o n r a d i a t io n c a n b e a c h i ev e d d u r in g t h e t r a n s it io n b yc o u p lin g b e t w e e n t h e p r o t o n a n d a fr e e e le c tr o n . D u r i n g t h e t r a n s it io nf r om n = 1 t o n = 1/ 2 , a t r a n s fe r o f e n e r g y f r o m a h y d r o g e n a t o m c a u s in ga b a c k wa r d e le c tr o n is r e q u i r e d t o s a t is fy t h e n o n r a d ia t iv e b o u n d a r yc o n d it io n . In t h e lim it , t h e c u r r e n t d e n s it y fu n c t io n a p p r o a c h e s t h a t o f ac ha r ge m ovi ng a t c ons t a n t ve loc it y . Th i s c ond it ion a ls o a pp lie s t o t heba c kwa r d e l e ct r on . Thu s , t h e con d it i on fo r t h e fin a l ve loc it y o f t h e

b a c k w a r d e le c tr o n is g iv e n b y Eq s . ( 3 8 . 9 4 ) a n d ( 3 8 . 9 5 ) . T h e b o u n d a r yc o n d it io n fo r t h e v e lo c it y o f t h e p r o t o n , v p , a n d t h e ve loc it y o f t he

b a c k w a r d e le ct r o n , ve , is−ve cos e = v p cos p ( 3 8 . 9 9 )

w h e r e e is t he la bo r a t o r y f r a m e e le c t r on s c a t t e r i ng a n g le a nd p is t h e

l a bor a t o r y fr a m e p r o t on s ca t t e r i ng a ng l e . The ve loc it y , ve , of ane le c tr o n a s a fu n c t io n o f t h e e n e r g y , E e , is

ve =2E eme

( 3 8 . 1 0 0 )

w h e r e me i s t h e ma s s o f t h e e le c t r on . The ve l ocit y , v p , o f a n e le c t r on a s afu n c t io n o f th e e n e r g y , E p , is

v p =2E pm p

( 3 8 . 1 0 1 )

w h e r e m p is t h e m a s s o f t h e p r o t o n . Su b s t it u t i o n o f Eq s . ( 3 8 . 1 0 0 ) a n d

( 38 .101 ) in t o Eq . ( 38 .99 ) g ive s




5 9 1

−2E eme

cos e =2E pm p

cos p ( 3 8 . 1 0 2 )

Fr o m t h e n o n r a d i a t iv e b o u n d a r y c o n d it io n , t h e a n g le o f t h e b a c k wa r de lec t ron i s

= cos−1 − E pme

m p E ecos p

( 3 8 . 1 0 3 )

In t h e c a s e t ha t t h e i n i t ia l a nd fin a l ve loc it y ve c t o r s o f t h e i nc id e n tp r o t o n a r e e q u a l, E p initial

= E p final= 70 keV , a nd t h e i n i t ia l a nd fin a l e ne r g i e s o f

t h e e l e ct r o n a r e E einitial= 0 eV a n d E e final

= 40.8 eV , r e s pe c t ive ly , t he m a x im um

i n t e n s it y o f t he p e a ks is p r e d ic t e d ( Eq . ( 38 .10 3) ) t o b e 16 5°. Th i s is ina g r e e m e n t w it h t h e o b s e r v ed m a x im u m a t a b o u t 1 6 0 ° [4 ] . Fo r s m a l lp r o t o n s ca t t e r i ng a n g le s , Eq . ( 38 .10 3) i s a l so i n a g r e e m e n t w it h t h ea b s e nc e o f t h e ba c kwa r d p e a ks a t 90° [ 4 ]. The m a x im u m i n t e n s it y o f t h e

b a c k w a r d p e a k s fr o m c o llis io n s o f h y d r o g e n a t o m s wit h 3 0 k e V p r o t o n sp r e d i ct e d f r om Eq . (38 .10 3) is 13 0°. Thu s , t h e in t e n s it y o f t h e ba c kwa r dp e a k s a t 1 6 0 ° w it h 3 0 k e V p r o t o n s is p r e d ic t ed t o b e l e ss t h a n t h a t o f 7 0k e V in c id e n t p r o t o n s in a g r e e m e n t w it h Ru d d [4 ] .

N o v e l En e r g y S t a t e s o f H y d r o g e n Fo r m e d b y a C a t a l y t i c

R e a c t i o n

Typi ca l ly t h e e m is sion o f e xt r e m e u lt r a v i o le t l igh t fr om h yd r oge nga s is a c h i e ved v ia a d is ch a r ge a t h igh vo lt a ge , a h igh po we r i nd uc t i ve lyc o u p le d p la s m a , o r a p l a sm a cr e a t e d a n d h e a t e d t o e xt r e m e

t e m p e r a t u r e s by RF c oup lin g ( e .g . > 10

6

K ) wit h c o n f in e m e n t p r o v id e d b ya tor oida l m agn e t ic f ie ld . In ten se EUV em iss ion was obse rve d b y Mil ls e ta l . [ 7 -12 ] a t l ow t e m p e r a t u r e s ( e .g . < 10

3 K ) fr o m a t o m i c h y d r o g en a n d

c e r t a in a t o m iz e d p u r e e le m e n t s o r c e r t a i n g a s e o u s io n s w h i ch io n iz e a tin t e ge r m ul t ip le s o f t h e p o t e n t ia l e n e r gy o f a t om ic h yd r oge n . Ther e l e a se o f e n e r gy fr om h yd r oge n a s e v id e nc e d by t h e EUV e m is sion m us tr e s u l t in a lowe r -e n e r gy st a t e o f h yd r oge n . The lowe r -e n e r gy hyd r oge na t o m c a lle d a h y d r in o a t o m b y M ills [1 ] w o u l d b e e xp e c t e d t od e m o n s t r a t e n o v e l c h e m is t r y . T h e fo r m a t io n o f n o v e l c o m p o u n d s b a s edo n h y d r in o a t o m s w o u ld b e s u b s t a n t ia l e vid e n c e su p p o r t in g c a t a ly s is o f h yd r oge n a s t he m e c ha n is m of t h e obs e r ve d EUV e m is sion . A n ove lh y d r id e i o n c a lle d a h y d r i n o h y d r i d e i o n h a v in g e x t r a o r d in a r y c h e m i ca lp r op e r t i e s g ive n by Mills [1 ] is p r e d ic t e d t o fo r m by t h e r e a c t i on o f a ne le ct r o n w it h a h y d r in o a t o m . Co m p o u n d s co n t a i n in g h y d r in o h y d r id eio n s h a v e b e e n is o la t e d a s p r o d u c t s o f t h e r e a c tio n o f a to m ic h y d r o g e nwith a tom s and ions iden t i fi ed as ca ta lys t s in th e Mill s e t a l . EUV s tu d y[ 1 , 7 -1 2 , 3 2 - 3 7 ] . T h e n o v e l h y d r id e c o m p o u n d s w e r e id e n t ifie da n a l y t ic a lly b y t e c hn iqu e s su c h a s t i me o f fligh t s e c ond a r y i on m a s s




5 9 2

s p e c t r o s co p y , X-r a y p h o t o e le c t r o n s p e c t r o sc o p y , a n d p r o t o n n u c l ea rm a gn e t ic r e s on a n c e s pe c t r os c op y . For e xa m p le , t h e t i me o f f ligh ts e co n d a r y io n m a s s s p e c tr o s co p y s h o w e d a la r g e h y d r i d e p e a k in t h en e g a t iv e sp e c t r u m . T h e X-r a y p h o t o e le c tr o n s p e c tr u m s h o w e d l a r gem e t a l c o r e le ve l s h i ft s du e t o b i nd in g wit h t h e h yd r i de a s we ll a s no ve l

h y d r id e p e a k s . T h e p r o t o n n u c le a r m a g n e t i c r e s o n a n c e s p e c t r u ms howe d s ign i fic a n t l y up fie ld s h ift e d p e a ks wh ic h c o r r e s p on d e d t o a n did e n t i fie d n ove l h yd r i de i ons .

D i s c u s s i o n

The Sc hr od i n g e r e q u a t io n g i ve s t h e o b s er v e d s p o n t a n e o u s lyr a d ia t ive en e r gy le ve ls a n d t h e non r a d ia t ive s t a t e o f h yd r oge n . On t h i sb as i s a lon e , it i s ju s t i fi ed d espi t e i t s in con s is ten cy wi th p h ys ica l s lawsa n d n u m e r o u s e xp e r im e n t a l o b se r v a tio n s s u c h a s

• T h e a p p r o p r ia t e e ig e n v a lu e m u s t b e p o s t u l a te d a n d t h e v a r ia b l eso f t he La gue r r e d if fe r e n t ia l e qu a t i on m u s t be de f in e d a s in t e ge r s in o r d e rt o o b t a i n t h e Ry d b e r g fo r m u l a .

• T h e Sc h r od i n g e r e q u a t io n is n o t Lo r e n t z ia n in v a r i a n t .

• T h e Sc h r od in ge r e qu a t ion v io l a t e s fir s t p r in c ip le s in c lu d in gspe c ia l r e la t iv ity an d Maxwel l' s equ a t ion s [1 , 38 ] .

• T h e Sc h r od in ge r e qu a t i on g ive s n o ba s i s why e xc it e d s t a t e s a r e

r a d ia t i ve a n d t h e 13 .6 e V s t a t e is s t a b l e [ 1 ] . Ma t he m a t i cs d oe s no td e t e r m in e p h ys ic s. It on ly mod e ls ph ys ic s.

• T h e Sc h r od i n g e r e q u a t i o n s o lu t i o n s , Eq . ( 3 8 . 3 6 ) a n d Eq . ( 3 8 . 3 7 ) ,p r e d i ct t h a t t h e g r o u n d s t a t e e le c tr o n h a s z e r o a n g u l a r e n e r g y a n d z er oa n g u l a r m o m e n t u m , r e s p e c t iv e ly .

• T h e Sc h r od i n g e r e q u a t io n s o lu t i o n , Eq . ( 3 8 . 3 6 ) , p r e d i ct s t h a t t h eio n iz ed e l e ct r o n m a y h a v e in f in i t e a n g u l a r m o m e n t u m .

• T h e Sc h r od i n g e r e q u a t i o n s o lu t i o n s , Eq . ( 3 6 ) a n d Eq . ( 3 7 ) ,p r e d i ct t h a t t h e e x cit e d s t a te r o t a t i o n a l e n e r g y l e ve ls a r e n o n d e g e n e r a t ea s a f u n c t io n o f t h e l q u a n t u m n u m b e r e ve n in t h e a b se n c e o f a na p p lie d m a gne t i c f ie ld , a n d t h e p r e d ic t e d e n e r gy is ove r six o r d e r s o f m a g n i tu d e o f t h e o b s er v e d n o n d e g e n e r a t e e n e r g y in t h e p r e s e n c e o f am a gn e t ic f ie ld . In t h e a bs e n c e o f a m a gn e t ic f ie ld , n o p r e fe r r e dd i rec t ion exis t s . In th i s case , th e l q u a n t u m n u m b e r is a fu n c tio n o f th eo r i e n t a t io n o f t h e a t o m w it h r e s p e c t t o a n a r b i t r a r y c o o r d in a t e sy s t e m .




5 9 3

The r e f o r e , t h e non d e ge ne r a c y is n on s e ns ic a l a n d v i o la t e s con s e r va t iono f a n g u la r m o m e n t u m o f t h e p h o t on .

• T h e Sc h r od i n g e r e q u a t io n p r e d i ct s t h a t e a c h o f t h e f u n c t io n st h a t c o r r e s po n d s t o a h i gh l y e xc it e d s t a t e e le c t r on i s n o t in t e g r a b le a nd

c a n n o t be n or m a l iz e d ; t h u s , e a c h i s i n f in i t e.

• T h e Sc h r od i n g e r e q u a t io n p r e d i ct s t h a t t h e i o n i ze d e l ec t r o n i ss in u s o id a l ove r a ll sp a c e a n d c a n n o t b e n or m a l iz e d ; t h u s , it i s in fi n i t e .

• T h e H e is en b e r g u n c e r t a in t y p r in c ip l e a r is es a s t h e s t a n d a r dd e v ia t i on i n t h e e le c t r on p r ob a b ilit y wa ve , bu t e xp e r i me n t a l ly i t is no tt h e b a s is o f wa ve pa r t ic le d u a l it y .

• T h e c o r r e sp o n d e n c e p r in c ip l e d o e s n o t h o l d e xp e r i m e n t a l ly .

• T h e Sc h r od i n g e r e q u a t i o n d o e s n o t p r e d ic t t h e e le c tr o n m a g n e t icm o m e n t a n d m iss es th e s p in q u a n t u m n u m b e r a ll t o ge t h e r .

• T h e Sc h r od i n g e r e q u a t io n is n o t a w a v e e q u a t io n s in c e i t giv e st h e v e lo c it y s q u a r e d p r o p o r t io n a l t o t h e f r e q u e n c y .

• T h e Sc h r od i n g e r e q u a t io n is n o t c o n s is t en t w it h c o n s e r v a t io n o f e n e r gy in a n in ve r s e po t e n t i a l fie ld wh e r e in t he b in d i ng e n e r gy is e qua lt o t h e k in e t i c e n e r g y a n d t h e s u m o f t h e b i n d in g e n e r g y a n d t h e k in e t ic

e n e r gy is equ a l t o t h e po t e n t ia l e n e r gy [14 ] .

• T h e Sc h r od i n g e r e q u a t io n p e r m it s th e e le ct r o n t o e xis t in t h en u c leu s wh ich i s a s t a te th a t i s p h ys ica l ly n on sens ica l with in f in i t ep o t e n t ia l e ne r gy a n d in fin it e n e ga t ive k in e t ic e ne r gy .

• T h e Sc h r od in g e r e q u a t io n in t e r p r e t e d a s a p r o b a b ilit y w a v e o f ap o i n t p a r t i c le c a n n o t e xp la i n n e u t r a l sc a t t e r i ng o f e le c t r on s fr omh y d r o g en [1 ] .

• T h e Sc h r od in g e r e q u a t io n in t e r p r e t e d a s a p r o b a b ilit y w a v e o f ap o i n t p a r t i c le g ive s r is e t o in fin i t e ma gn e t ic a n d e le c t r ic e ne r gy in t h ec o r r e s p o n d in g f ie ld s o f t h e e le c t r o n .

• A m o d ific a t io n o f t h e Sc h r o d i n g e r e q u a t io n w a s d e v e lo p e d b yDir a c t o e xp la i n s p i n wh ic h r e l ie s on t h e un f oun d e d n o t i ons o f n e ga t ivee n e r gy s t a t e s o f t h e va c uu m , v ir t u a l p a r t i c le s , a n d ga m m a fa c t o r s .






5 9 5

3 . Kuraica, M., Konjevic, N., Physical Review A, Volume 4 6, No. 7, Oct ober

(1992), pp. 4429-4432.

4. Rudd, M. E., Gealy, M. W., Kerby, G. W., Hsu, Ying-Yuan, Physical Review

Let t ers, Vol. 68, No. 10, (1 992 ), pp. 15 04 -1506 .

5 . R. Mills, W. Good, J. Phillips, A. Popov, Lower-Energy Hydrogen Met hods

and St ruct ures, US Patent 6,0 24 ,93 5, issued February 1 5, 2 00 0.6 . Beiser, A., Concept s of Modern Physics, Fourt h Edit ion, McGraw-Hill Book

Company, New York, (19 78 ), pp. 31 2-3 54 .

7 . J. P. F. Conrads, R. Mills, "Observat ion of Ext reme Ult raviolet Hydrogen

and Hydrino Emission from Hydrogen-KI Plasmas Produced by a Hollow

Cat hode Discharge" , in progress.

8 . R. Mills, J. Dong, Y. Lu, " Observat ion of Ext reme Ult raviolet Hydrogen

Emission from Incandescent ly Heat ed Hydrogen Gas wit h Cert ain

Catalyst s" , 19 99 Pacif ic Conference on Chemist ry and Spect roscopy

and t he 35t h ACS West ern Regional Meet ing, Ont ario Convent ion Center,

California, ( Oct ober 6-8 , 199 9) .9 . R. Mills, J. Dong, Y. Lu, " Observat ion of Ext reme Ult raviolet Hydrogen

Emission from Incandescent ly Heat ed Hydrogen Gas wit h Cert ain

Catalysts" , Int . J. Hydrogen Energy, accept ed.

10 . J. P. F. Conrads, R. Mills, "Temporal Behavior of Light -Emission in t he

Visible Spect ral Range f rom a Ti-K2CO3-H-Cell', in progress.

11. R. Mills, Y. Lu, and T. Onuma, " Format ion of a Hydrogen Plasma from an

Incandescent ly Heated Hydrogen-Pot assium Gas Mixt ure and Plasma

Decay Upon Removal of Heater Power" , Int . J. Hydrogen Energy,

submitted.

12. R. Mills, M. Nanst eel, and Y. Lu, " Observat ion of Ext reme Ult ravioletHydrogen Emission f rom Incandescent ly Heat ed Hydrogen Gas wit h

St ront ium t hat Produced an Opt ically Measured Power Balance t hat was

40 00 Times t he Cont rol", Int . J. Hydrogen Energy, submit t ed.

13 . Beiser, A ., Concept s of Modern Physics, Fourt h Edit ion, McGraw-Hill,

New York, (1987),. pp. 87-117.

14. Fowles, G. R., Analyt ical Mechanics, Third Edit ion, Holt , Rinehart , and

Winston, New York, (1977), pp. 154-156.

15. McQuarrie, D. A., Quant um Chemist ry , Universit y Science Books, Mill

Valley, CA, (1983), pp. 78-79.

16. Jackson, J. D., Classical Elect rodynamics, Second Edit ion, John Wiley &Sons, New York, (19 62 ), pp. 84 -10 8.

17. McQuarrie, D. A., Quant um Chemist ry , Universit y Science Books, Mill

Valley, CA, (198 3) , pp. 221 -224 .

18. W. Moore, Schrodinger life and t hought , Cambridge Universit y Press,

(1989), p.198.

19. Fowles, G. R., Analyt ical Mechanics, Third Edit ion, Holt , Rinehart , and

Winst on, New York, (19 77 ), pp. 57 -60 .




5 9 6

20. H. Margenau, G. M. Murphy, The Mat hemat ics of Chemist ry and Physics,

D. Van Nostrand Company, Inc., New York, (1943), pp. 77-78.

21. Weisskopf, V. F., Reviews of Modern Physics, Vol. 21, No. 2, (1949), pp.

305-315.

22. Horgan, J., " Quant um Philosophy" , Scient ific American, July, (1992) , pp.

94-104.23. Platt, D. E., Am. J. Phys., 60 (4), April, 1992, pp. 306-308.

24. S. Durr, T. Nonn, G. Rempe, Nature, Sept ember 3 , (1998) , Vol. 39 5, pp.

33-37.

25 . B. Gao, Phys. Rev. Let t ., Vol. 83 , No. 21 , Nov. 22 , (1 99 9) , pp. 4225 -

4228.

26 . Science News, Vol. 156 , November 27 t h 19 99 , p. 34 2, under "Physics

rule of t humb get s thumbs down" 2 nd & 3rd paragraphs.

27 . Science News, 1/ 11 / 86 , p. 26.

28. Haus, H. A., "On the radiat ion from point charges", American Journal of

Physics, 54, (1986), pp. 1126-1129.29. N. V. Sidgwick, The Chemical Element s and Their Compounds, Vo lume I,

Oxford, Clarendon Press, (195 0) , p.17 .

30 . M. D. Lamb, Luminescence Spect roscopy , Academic Press, London,

(19 78) , p. 68.

31. S. Bower, G. Field, and J. Mack, "Det ect ion of an Anisot rophic Sof t X-ray

Background Flux," Nat ure, Vol. 21 7, ( 19 68 ), p. 32 .

32 . R. Mills, B. Dhandapani, N. Greenig, J. He, "Synt hesis and

Charact erizat ion of Pot assium Iodo Hydride" , Int . J. of Hydrogen Energy,

accept ed.

33. R. Mills, "Novel Inorganic Hydr ide" , Int . J. of Hydrogen Energy, Vol. 25 ,(20 00) , pp. 669 -683.

34. R. Mills, " Novel Hydrogen Compounds f rom a Pot assium Carbonat e

Elect rolyt ic Cell" , Fusion Technology, Vol. 37 , No. 2, March, (2 000) , pp.

157-182.

35. R. Mills, J. He, and B. Dhandapani, "Novel Hydrogen Compounds" , 1999

Pacif ic Conference on Chemist ry and Spect roscopy and t he 35 t h ACS

West ern Regional Meet ing, Ontario Convent ion Cent er, California,

(October 6-8, 19 99 ).

36. R. Mills, B. Dhandapani, M. Nanst eel, J. He, " Synt hesis and

Charact erizat ion of Novel Hydride Compounds" , Int . J. of HydrogenEnergy, submit t ed.

37. R. Mills, " Highly St able Novel Inorganic Hydr ides" , Int ernat ional Journal of

Inorganic Mat erials, submit t ed.

38. C. A. Fuchs, A. Peres, " Quant um t heory needs no ' interpretation' " ,

Phys. Today, Vol. 53, March, ( 200 0) , No. 3, pp. 70 -71 .

39 . N. A. Bahcall, J. P. Ost riker, S. Perlmut t er, P. J. St einhardt , Science,

May 28, 19 99 , Vol. 284, pp. 14 81 -148 8.



Randell L. Mills- The Hydrogen Atom Revisited

Documents