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Pure and Applied Mathematics Journal2013; 2(4): 146-148Published online August 30, 2013 (http://www s!ien!epublishingg"oup !o#/$/p%#$)doi: 10 11648/$ p%#$ 20130204 12

Derivation of Schrdinger equation from a variational

principleSami. H. Altoum

# Al-'u"% &ni e"sit *+ A-&ni e"sit ollege o. Al- un.ud%h, ip !ode 21 12 * bo 110

Email address:s%#i%ltou# hot#%il !o#

To cite this article:%#i Altou# 5e"i %tion o. !h" dinge" 7'u%tion ."o# % %"i%tion%l P"in!iple Pure and Applied Mathematics Journal.ol 2, 9o 4, 2013, pp 146-148 doi: 10 11648/$ p%#$ 20130204 12

Abstract: he %i# o. this "ese%"!h is to de"i e !h" dinge" e'u%tion ."o# !%l!ulus o. %"i%tions ( %"i%tion%l p"in!iple),so we use the #ethodolog o. !%l!ulus o. %"i%tions he %"i%tion%l p"in!iple one o. g"e%t s!ienti.i! signi.i!%n!e %s the

p"o ide % uni.ied %pp"o%!h to %"ious #%the#%ti!%l %nd ph si!%l p"oble#s %nd ield .und%#ent%l e plo"%to" ide%s

Keyword: !h" dinge" 7'u%tion, %"i%tin%l P"in!iple, %#iltoni%n- %!obi 7'u%tion

1. Introduction he !%l!ulus o. %"i%tions is % .ield o. #%the#%ti!%l

%n%l sisth%t de%ls with #% i#iethe .un!tion%l %tt%in % #% i#u# o" #ini#u# %lueo" st%tion%" .un!tions, those whe"e the "%te o. !h%nge o.the .un!tion%l is

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Pu"e %nd Applied D%the#%ti!s ou"n%l 2013; 2(4): 146-148 14E

i

i

y F

dxdp

= ni , ,21= (4)

9ow, i. the %!obi%n

( )0

, ,,, ,,

21

21

n

y y y

y y y D F F F D n ,

hen the s ste# o. e'u%tion (3) !%n be sol ed %s

( )1 1, , , , , ,i i n n y x y y p p =

Chen these %"e substituted into (4) ,we get % s ste# o..i"st-o"de" e'u%tions %s

( )nnii p p y y xdxdy

, ,,, ,, 11 = ,i

i

y F

dxdp

= with ni , ,21= (@)

en!e.o"w%"d the p%"entheses in the se!ond e'u%tion o.(@) signi. th%t i y in F %"e "epl%!ed b i Ce nowint"odu!e the %#iltoni%n .un!tion

( ) =

=n

iiinn F p p p y y x H

111 , ,,, ,, (6)

hen the s ste# (6) !%n be w"itten %s

i

i

p H

dxdy

= ,

i

i

y H

dxdp

= ni , ,21= (E)

his s ste# is "e.e""ed to %s the %#iltoni%n (!%noni!%l)s ste# o. 7ule" s e'u%tions %nd o. 2n-o"din%" e'u%tions in2n un>nown .un!tions ( ) x y i %nd i p

1.2. 2-The Hamiltonian-Jacobi Equation

onside" the .un!tion%l in (1), the 7ule" e'u%tion .o" this.un!tion%l %d#it o. solutions in ol ing 2n %"bit"%"!onst%nts e"e spe!i.i!%tion o. two points A %nd F in thesp%!e o. %"i%bles n y y x , ,, 1 th"ough whi!h %n e t"e#%l#ust p%ss gi es p"e!isel 2n e'u%tions .o" dete"#iningthese !onst%nts en!e in the gene"%l !%se the"e %ppe%"s %dis!"ete set o. e t"e#%ls $oining these points Bet AB I bethe %lue o. the .un!tion%l on e%!h o. these e t"e#%l,A

being "eg%"ded %s the initi%l %nd F %s the te"#in%l point BetA be .i ed while ( )n y y y x B , ,,, 21 is "eg%"ded %s %#o %ble point hen AB I is % .un!tiono. ( )n y y y x , ,,, 21 %nd we w"ite

( )n AB y y y xS I , ,,, 21= (8)

?. F !h%nges its position (6) gi es

=

+=n

iii dy p HdxdS

1

Chi!h in tu"n le%ds to

H xS =

, i

i

p yS =

ni , ,21=

?t then .ollows th%t S s%tis.ies the .ollowing p%"ti%ldi..e"enti%l e'u%tion o. .i"st o"de"

0, ,,, ,,,1

21 =

+

nn y

S yS

y y y x H xS

( )

Chi!h is >nown %s %#iltoni%n * %!obi e'u%tion

1.3. 3-Schrdin er Equation and Variatinal principle

9ow we de"i e the .und%#ent%l e'u%tion o. 'u%ntu##e!h%ni!s ( !h" dinge" e'u%tion) ."o# % %"i%tion%l

p"in!iple=i"st we de.ine %n ope"%to" >nown %s the %#iltoni%n

ope"%to" %s .ollows:

( ) z y xV H ,,2 + (10)

e"e ( )mh 22 8/ = , whe"e h %nd m st%nd .o" thePl%n> s !onst%nt the #%ss o. the p"in!iple whose #otion is!onside"ed in % .ield o. potenti%l ene"g V Ce now see> %w%"e .un!tion

Possibl !o#ple e t"e#i

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148 %#i Altou# 5e"i %tion o. !h" dinge" 7'u%tion ."o# % %"i%tion%l P"in!iple

=+ V 2 (13)

his is w"itten %s = H ?. we #ultipl this b %nd integ"%te o e" the do#%in

o. ,,, the le.t side be!o#es the st%tion%" integ"%l (11)

whi!h depend b " en!e b (12) we h% e " = , so(13) "edu!es to !h" dinge" e'u%tion ?t is wo"th pointingout he"e th%t the"e is %n inte"esting %nd i#po"t%nt!onne!tion between %#iltoni%n-$%!obi e'u%tion .o"!l%ssi!%l s ste# %nd the !h" dinge" e'u%tion .o" %'u%ntu# #e!h%ni!%l s ste# ?n .%!t ,i. we put the w% e

.un!tion ( )S hie /= ,whe"e S is the %!tion .un!tion o. the!l%ssi!%l s ste# (8),then the !h" dinge" e'u%tion "edu!esto the %#iltoni%n * %!obi e'u%tion( ) p"o ided S is #u!hl%"ge" th%n Pl%n> s !onst%nt h hus in the li#it o. l%"ge%lues o. %!tion %nd ene"g ,the su".%!es o. !onst%nt ph%se.o" the w% e .un!tion "edu!e to su".%!es o. !onst%nt%!tion S .o" the !o""esponding !l%ssi!%l s ste# ?n this!%se, w% e #e!h%ni!s "edu!es to !l%ssi!%l #e!h%ni!s $ust%s w% e opti!s "edu!es to geo#et"i!%l opti!s in the li#it o.e" s#%ll w% elength ?t #% be noted th%t the +lein-Go"don e'u%tion

,01

22

22

2 =

hmc

t c

(!H elo!it o. light) "ep"esenting % possible w% ee'u%tion .o" % "el%ti isti! p%"ti!le (though it is not !o""e!t.o" %n ele!t"on o" p"oton)!%n be !onst"u!ted in

12

22

2

2

+

=

hmc

t cmh

#

2. Conclusion he #%in "esult o. this "ese%"!h we dedu!ed !h" dinge"

e'u%tion b using %"i%tion%l p"in!iple %nd %ddition%l "esultwe dedu!ed !h" dinge" e'u%tion is "edu!es to the%#iltoni%n * %!obi e'u%tion

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!io"ra#hy$r. Sami Ha%a&i 'usta a: "e!ei ed his Ph5deg"ee in 5i..e"enti%l Geo#et" ."o#Alneel%in &ni e"sit , ud%n in 200E e w%s% he%d o. 5ep%"t#ent o. D%the#%ti!s inA!%de# o. 7nginee"ing !ien!es in ud%n

9ow he is Assist%nt P"o.esso" o. D%the#%ti!s,&ni e"sit !ollege o. Al'un.udh%, # Al

u"% &ni e"sit , %udi A"%bi%