量子化学 2次元の井戸型モデル - nara-wu.ac.jp

1 1 p97 II ( ) ( )222222,,abcxdy f xyEf xyxy + ++ = (1) ,,,abcd 22ax 22by ( ),f xy 2cx 2dy 2cx 2dy ( ),f xy (1) II ( ),f xy E (1) x y (1) ( ) ( )222222,,acxbdyf xyEf xyxy ++ + = (2) x ( ) Px y ( ) Qy x y 2 ( )222 P xacxx =+ ( )222 Q ybdyy =+ (2) ( ) ()( )( ) ,,PxQ yf xyEf xy += (3) ( ),f xy E x y p102 (3)

2 ( ),f xy x ( )Xx y ( )Yy () ( )( ),f xyX xY y= ()Xx ( )Yy ( ) Px ( ) Qy ( )Xx ( )Yy ( ) ( ) ( ) xPxXx EXx= ( ) ( ) ( ) yQyYy EYy= xE yE ( ) Px ( ) Qy (3) E ( ) Px xE ( ) Qy yE xyEE E=+ (3) ( ),f xy ( ) Px ( ) Qy ( )Xx ( )Yy E xE yE 3 V 0,0 ,0,xVx LVxLV<= =>= (4) 0V= 0xL V= 0xL V 0,0,0xyxLyLV = (5)

3 V= 0V= 0,0xyxLyL V= 0xxL 0yyL (a) y (b) z x y Lx Ly 0 x L (a) (b) 4 0V= V= I IV 222222 22 Hmxmy =- - (6) ( ) ( )222222,,22xyE xymxmyyy -- = (7) ddx ( )

4 ,xyy x y Lx x Ly y I II III IV 5 0x= xL= ( )( )00,0 Lyy== II III IV ( ),00xy= I (8) ( )0,0yy= II (9) ( ),0yxLy= III (10) ( ),0xLyy= IV (11) 0 xL yL x y ( )2,dd1xy xyy - - = (12) 6 0,0xyxLyL ( ),0xyy= x 0 xL y 0 yL ( )200,dd1xyLLxyxyy= (13)

5 (1) H ( ),xyy ( ) () ( ),xyX xY yy= ()Xx 2222mx - ()Yy 2222my - (2) E 2222mx - xE 2222my - yE xyEE E=+ (1) (2) X (x) Y (y) X (x) Y (y) 7 (),xyy ( )Xx ( )Yy ( )2sinxxxnXxxLLn= 2 2222xxxnEmLn= ( )2sinyyynYyyLLn= 2 2222yyynEmLn= ( ) ( ) ( )()2 222 2222,22sinsin221, 2 ,,1, 2 ,,yxxxyyxyyxxyxyxyX xY ynnxyLLLLEEEnnmLmLnnynnnn= = =+=+== ( )() ()

6 ,xyX xY yy= 8 ( )Xx ( )Yy ( )Xx ( )Yy ( ) ( ) ( ),xyX xY yy= (8) (11) ( ) ( ) ( ),000xX xYy== (14) ( ) ( ) ( )0,00y X Yyy== (15) ( )( )( ),0yyxLX xY Ly== (16) ( ) ( ) ( ),0xxL y XL Yyy== (17) (14) ( )( )00X xY= ( )Xx ( )0Xx ( )0Xx= ( ) ( ) ( ),xyX xY yy= ( )0Xx= ( )( )00X xY= ( )00Y= (14) ( )00Y= (15) ( ) ( )00X Yy= ()Yy ( )0Yy ( ) ( )00X Yy= ( )00X= (15) ( )00X= (16) (17) ( )0yYL= ( )0xXL= (14) (17) ( )Xx ( )Yy ( )00Y= (18) ( )00X= (19) ( )0yYL= (20) ( )0xXL= (21) 9 ( )Xx ( )Yy ( )Xx ()Yy (13) ( ) ( ) ( ),xyX xY yy= ( )( ) ( )( ) ( ) ( ) ( )() ()( ) ( )()( )( )()

7 220000**00**0022002200,dddddddddddd1xyxy xyxyxyxyLLLLLLLLLLLL xyxyX xY yxyX xY y X xY y x yX x X xY yY y x yX x Y yxyXx xYy yy==== = = ()20dxLXx x ()20dyLYy y ( )Xx ( )Yy ( ),xyy ()20dxLXx x ( )20dyLYy y ( )20dxLXx x = 2 ( )20dyLYy y 12 ( )Xx ( )Yy ( )Xx ( )Yy ( )20d1xLXx x= (22) ( )20d1yLYyy= (23) 10 ( )Xx ( )Yy ( )Xx ( )Yy (7) ( ) ( ) ( ),xyX xY yy= ( ) ( )222222,,22xyE xymxmyyy -- = ( ) ( ) ( ) ( )22222222X xY yEX xY ymxmy --= (24) ( ) ( )( ) ( ) ( ) ( )22222222X xY yX xY yEX xY ymxmy --= (25) ( )( )( )( ) ( ) ( )222 22222 YyXxXxYy EXxYymxm y --= (26) (26) ( ) ( )1X xY y ( )( )()( )( )( )()()222222dd2d2dYyXxXxYy EmX xY y xmX xY y y--= (27) ( )( )( )( )

8 2 222221d1d2 d2dXxYy Em X x xmY y y- -= (28) (28) x ( )( )( )( )2 22222d1d1ddd2d2ddEXxYyxm X x xmY y yx --= (29) E ( )Yy x ( )Yy ( )Xx 11 x ( )( )( )( )2 22222d1dd1ddd2dd 2ddEXxYyxm X x xxmY y yx -+-= ( )( )222d1d0d2dXxxmX x x -= (30) (30) x ()( )222d1dd0dd2dXx xxxmX x x -= ()( )2221d2dxXxEmX x x-= (31) xE (31) ( )Xx ( )Xx ( )( )222d2dxXxEXxmx-= (32) (28) y ( )Xx E y ( )( )2221d2dyYyEmY y y-= (33) yE (33) ( )Yy ( )Yy ( )( )222d2dyYyEYymy-= (34) (31) (33) (28) xyE EE+= (35) xE yE xyE EE+= ( )Xx x 0 12 ( )Xx ()( )222d2dxXxEXxmx-= ( )Xx ( )00X= ( )0xXL= ( )Xx ( )20d1xLXx x= ( )Xx ( )

9 Yy ( )( )222d2dyYyEYymy-= ( )Yy ( )00Y= ( )0yYL= ( )Yy ()20d1yLYyy= ( )Yy ( )