对称规范下讨论朗道能级, 量子霍尔效应.pdf

CHAPTER 2. íP(5¡:-Ñ–® 12 ~=A ~ · P~ = B (xPˆy ⌘Ï_ÔÂ(˘ƒ⇤î∫⇢1éP~ · A 2 Hxy = = y Pˆx ) = B2 Lˆz , 1 ˆ eB 2 eB 2 [(Px y) + (Pˆy + x) ] 2µ 2c 2c 2 1 ˆ2 e2 B 2 2 eB ˆ (Px + Pˆy ) + (x + y 2 ) + Lz 2 2µ 8µc 2µc ~ $y:Ô⌃„: µ ~ · B, y1BÒ¸ e2 B 2 (x + y 2 ) 4µc @H e ˆ = Lz @B 2µc µz = :Ò¸¡È. (üPÖË 5P˚®⇤Ùx2 + y 2 ✏ Ô ˝e. ⌘ÏeB„˝œ,Åπ↵⇥ ÷ÅP⌥⇢ ~2 r2 = Px2 + Py2 = ~2 ( @2 1 @ + ) 2 @⇢ ⇢ @⇢ 2 @ 1 @ 2 v- @⇢ 2 + ⇢ @⇢ = P⇢ , :Ñ⌘®œÑsπ⇥ Hxy = ~2 @ 2 ⇢2 @'2 L2 y:“⌘®œsπ ⇢2z . é/ P⇢2 L2 1 2 2 + z 2 + µ!L ⇢ + !L Lˆz , 2µ 2µ⇢ 2 (2.52) eB v-!L = 2µc = !2c , :Larmor ëá. dª y Ÿ/ Ë ✏ 0Hxy *åÙÄ⇣/PÑ»∆œ Lz ˘ ◆ ÷(Hxy , Lz ): å h ∆. æ (⇢, ') = R(⇢)eim' , ö 1 I = ⇢ 2 R, Ôó0Ñ⌘,Åπ↵ ~2 2µ v-E 0 = E ~2 1 1 2 2 (m2 ) + µ!L ⇢ = (E m~!L ) 2µ⇢2 4 2 (m + 12 )(m 12 )~2 2µ 1 2 2 00 + 2 (E 0 µ! ⇢ ) = 0 2 ~ 2µ⇢ 2 L 00 + m~!L . ⇧1/åÙ⇣/PÑÑ⌘–®π↵. Ù-√õøÑ⌘π↵ ”+ 2µ (E ~2 l(l + 1)~2 2µr2 1 2 2 µ! r ) = 0 2 (2.53) ‘É, ⌘Ï↵0˘îs˚r $ ⇢, l $ |m| 12 . ›˘