一维周期场中电子运动近自由电子近似

1、)(xVV VVxV)( 22 0 2 2 d HV m dx 000 2 022 2 EV dx d m NaL V V m k Ek 2 22 0 0 1 ( ) ikx k xe L Na lk 2 0 00 * kk L kk dx 0 HHH ( )HV xVV 22 0 2 2 d HV m dx 0() 11 ( ) ikxik x Na k xee LL 2lkNa . )2()1(0 kkkk EEEE (1) 0 11 ( ) L ikxikx k EeV xe dxV LL (1) 0 11 ( ) L ikxikx k EeV xVe dx LL 0 )1( k E (

2、1) | k Ek HkkVxVk|)(| 2 (2) 00 | k k kk kHk E EE L xkki dxxVe L kxVk 0 )( )( 1 | )(| |( )|kHkkV xV k 1 0 )1( )( )( 1 | )(| N n an na xkki dxxVe Na kxVk nax kk |( )|kV xk 1 0 0 )()( )( 1 | )(| N n a kkinakki dVee Na kxVk 1 ( )( ) 0 0 11 |( )|( ) N a i kki kk na n k V xkeVde aN 2 kkn a 1 ( ) 0 1 1 N i

3、 kk na n e N )()(naVVnax ( ) 0 1 |( )|( ) a i kk n k V xkeVdV a )2( Na l k )2( Na l k 0 1 11 )( )( akki Nakki e e N |( )|0k V xk a nkk 2 1 ( )( ) 0 0 11 |( )|( ) N a i kki kk na n k V xkeVde aN ( ) 1 ( ) ( ) 0 11 1 1 i kk Na N i kk na i kk a n e e NNe 2 | 2 |0 n kknkHkV a kknkHk a V m k Ek 2 22 0 V

4、m k Ek 2 22 0 2 (2) 00 | k k kk kHk E EE 2 (2) 2 22 (2 ) 2 n k n V E n kk ma 2 |0kknkHk a 2 | n kknkHkV a 2 22 2 22 2 (2 ) 2 n k n Vk EV nm kk ma ( ) 0 1 ( ) a i kk n VeVd a . )2()1(0 kkkk EEEE0 )1( k E n n k a n kk m V E )2( 2 22 2 2 )2( V m k Ek 2 22 0 .)()()( )1(0 xxx kkk 0 1 ( ) ikx k xe L (1)0

5、00 | kk k kk kHk EE V m k Ek 2 22 0 V m k Ek 2 22 0 2 |0kknkHk a 2 | n kknkHkV a (2) (1) 2 22 1 (2 ) 2 n i kx n a k n V e nL kk ma 2 (1) 2 22 1 (2 ) 2 n ix ikx n a k n V ee nL kk ma 2 2 22 11 ( ) (2 ) 2 n ix ikxikx n a k n V xeee nLL kk ma 2 2 22 1 ( )1 (2 ) 2 n ix ikx n a k n V xee nL kk ma x a n i

6、 n n k e a n kk m V xu 2 22 2 )2( 2 1)( ()( ) kk uxmaux 1 ( )( ) ikx kk xe ux L 2 2 22 11 ( ) (2 ) 2 n ix ikxikx n a k n V xeee nLL kk ma 2 a n kk )2( 2 22 2 a n kk m Vn (2) 2 22 1 (2 ) 2 n i kx n a n V e nL kk ma a n k )2( 2 22 2 a n kk m Vn 2 2 22 1 (2 ) 2 n ix ikx n a n V ee nL kk ma 2 2 22 11 (

7、) (2 ) 2 n ix ikxikx n a k n V xeee nLL kk ma 0 1 ( ) ikx k xe L (2 ) 0 1 ( ) n i kx a k xe L a n k a n a n kk 2 2 2 22 11 ( ) (2 ) 2 n ix ikxikx n a k n V xeee nLL kk ma V m k Ek 2 22 0 V m k Ek 2 22 0 00 kk EE n n k a n kk m V E )2( 2 22 2 2 )2( 22 )2( a n kk )2( k E a n k a n a n kk 2 a n k 10 (1

8、) n k a (1) n k a 即即 a n kk 2 00 ( ) kk xab 0 1 ikx k e L 0 1 ik x k e L 0 ( )( )( )HxHxEx 22 0 2 2 d HV m dx ( )HV xVV 000000 00 kkkkkk HEand HE 0)()( 0 0 00 kkkk VEEbVEEa * 0 k * 0 k 0kVkkVk 0)()( 0 0 00 kkkk VEEbVEEa * | n k HkkHkV 0*0 ()0 8 2 1 ; 6 2 1 ; 4 2 1 ; 2 2 1 a a a a k ng VVVVE2,2,2,2 3

9、21 m k Ek 2 22 ), 3 , 3 (), 2 , 2 (),( aaaaaa ,2,2,2 321 VVVE g ),(),(),( 321 kEkEkE )( 1 kE aa a 2 )( 2 kE aa 2 aa 2 a 2 )( 3 kE aa 2 3 aa 3 2 a 2 Na lk 2 Na lk 2 kkk k Na l 2 a k 2 N a Na 2 2 aa k k Na lk 2 km a k 2 2 2 22 11 ( ) (2 ) 2 n ix ikxikx n a k n V xeee nLL kk ma ( )( ) ikx k xeu x 2 2 22 1 ( )(1) (2 ) 2 n ix n a n V u xe nL kk ma km a k 2 2 ()2 2 22 11 ( )() (2 ) 2 n im k xix n aa k n V xee nLL kk ma ( )( ) ikx k xeu x 2 2 2 22 11 ( )() (2 ) 2 n imxix ikx n aa k n V xeee nLL kk ma 2 2 2 22 11 ( )() (2 ) 2 n imxix n aa n V u xee nLL kk ma )()(xuex xk i k 2 2 2 22 11 (