1/17 第13章電子とディラック方程式

1 1/17 I. III. Dirac Paul Adrien Maurice Dirac (1902 8 8 - 1984 10 20 1933 Erwin Rudolf Josef Alexander Schr dinger( 1887 8 12 - 1961 1 4 ) x p E (),tyx ()()( ),,, ,Etitttyy = x pxx( ) ())

2 , ,Etx p m ()20,2,mEt=px p ()2 22 40, ,cEm ct=+px p ()( )2 22 40,,ticm cttyy =+ xpx( ) ()()123:, ,, ,zyx=p p p p p p p 2/17 132:, ,,,xyxyzz = = + += xixxjxk( ) ( )( )( )( )( )( )( )( ),,,,,,,,yzxtittttzxiittityyyyyyyyy = == = p xxxx p xxp xxpx( ) x ()xp=p ( )xp (),tyx i = px( ) x p ( )( )[]()()( )( )[ ]( )( )( ) ,,, ,,,,, ,,, ,,,0xijijyijztittittttitztyxiyyyyyyydyy = = = = = p xxp xxxxxpxxx pxp( ) ( ) i = xp ()()

3 , ,,ijijtityyd= xpp p x p ( ), tiy = xxxp( ), tiy = pppx( ) Werner Karl Heisenberg (1901 12 5 - 1976 2 1 3/17 0 10001001 0201020 , 02030203220 030003 0xxiww - == -- - xp ( )2 Ewnp= = 2 x p ( ) ( )( )( )( ) ,, , ,, , , :iiiiijijttttiiyyyyd == = :: x x x x p xxxx px p ( ) ( ) [] ,xxi= x p , iix p ( )( )( )( ) ,, , ,, , iiiiiAAiittttiAA BByyyy == * ** == * ** = *.)

4 Xx xxxxxxxppxx p x p ( ) 2222222222222222222222 pxyzzixyi == -= - + + + +== -= - pxpxppp( ) ( ) ()( )( )( )( )( )( )22 422222 40022 402,,, ,,,,tiEtcm ctcm ctttiEtcm cttyyyyyyy ==+=+ ==+- pxppxxxpp( ) 4/17 ()2222422 4020cm ccm c- +=+ p 2 ( )222 420???cm c+=p ( )22 420???cm c+=p ()?

5 ?? 2 22 40 Ecm c=+p( ) ( ) ( )()( )()()( )2 22 40,,??? , , , , , ,Ettittcm cEtEtyy = = += x p px pxppp( ) 2 22 40cm c+p (),typ 2 22 40cm c+p 0 1 ()??? 1/2 II. 0 ( )22 22 40???cm c+=p p ()f x p ( )( )ipf xf xx= 5/17 p* * = * * ( ) ( )( )( )( )( )( )2222 22 2002 202 22022202?

6 ????,,,?,cm ctm cticm cttctm cm cyyyy ++ =+= ++ = : :p pppppppp( ) 0 200m= ( )( )( )( )( )( )2222222,,,?,??ctticttctyyyy == = : : xppxppppp( ) 2 2 () :1 00 1I I = ( ) 222221 00 100I === pp p pp( ) x ()22: , 0, 0pp =pp ( ) 222200ppp Ip == ( ) 2II=( ) ( )2222 2 IpppIpI=== ( ) pIp= ( )6/17 K K ( ) ( )( )2222pIKIpp Ip= =( ) 2KI=( ) pKp= ( ) 2KI= K ( )2KI= 2 2 K a bKc d = ( ) ( ) a babKpppppppc dcd === ( ) , , ,a b c d ( )2KI= 2a ba bKIc dc d == ( )

7 221001ab bcba ba baca dbcc dc dcdd+ + == + +( ) 4 221, 100ca daab bddcbcbc + == == +++ ( ) ( ) 4 22001122111 00 110011 001b cb caaaaa bc aaab bdAca dcdcbbbcaca= == == ==-= = + =+ = = + =+ = = - - ( )7/17 0220111,0 11 0010100a dbcb cbcbi cdiaa bc abcab bdAca dccaba bcdiia= = == == == =- = = + =+ = = + =+ = =- ( ) ( ) ( ) 3 ()()123.

8 , , , , xyz=s s s s s s s( ) 31210 11010000ii- - = = = sss( ) K 123, ,K=sss( ) ( ) 2 23131I===sssss s( ) ( )333311112222123 1 23 1iii= - == -=* = - = ss ssssss ss sssss( ) 333222111132000+=+=+=sss sssss sss s( ) ( ) 0 3 3 ()()()()123123 :, , , , :, ,, ,xyzxyz==s s s s s s sA A A A A A A( ) 3 () 1, 2,3ii=A 8/17 31iii= = ssA A( ) 123, , s s s ( ) ( ) ( )()()23222221231231223231113iii= ==++++= + + = AA A A A A A A A A A Assssssss( ) ( ) ( )()22222123 = = + +A A A A As( ) ( ) 3121231231122331 00 11 00001iiii- -- =++ =++= +- A A AA A A A AAAA A Assss( ) 312123ii- = +- A A AAA A As( ) ( ) ( )

9 22 =A As ( )22222123 = = + +A A A A As( ) 123, , s s s ( ) ( ) ()()()()22123222111 1111231 21 33223 33333222 32222221123 =++=++++++++= + +A A A AA A AAAAAA AA A Ass sssssss ss ssss sssssss( ) 1 1113 333332 22222110I===+=+=+=s ssss sss ssss sssss s( ) 2 2 ( ) ( ) ( ) 9/17 ()222112233I==++ p p p p psss( ) ( ) ( )( )() ( )122213322,,,cttitctyyy = ++ : ppp ppppppsss( ) ( )() ( )123312,,ticttyy =++ pp p p psss( ) 0 4 4 III.

10 ( ) 3 2 2 4 4 ( )132, ,a babKc dcpppKdpppp ==== sss( ) 2 2 2 2 213, ,s ss 4 4 213, ,a aa ( ) 22333331111111122233321232201 00101001aba bKcdc dabiia bKcdc daba bKcdc d ==== = ==== = ==== = -- 4 4 4 4 4 4 s sss sa ss ssssa ss ss sss sa ss ss s( ) K 3 213, ,s ss 13K = - ss( ) 1 2,3K s 2 22 21 2,3K s10/17 ( )