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equations.828

equations.828 - j kr = sin kr kr n kr =-cos kr kr l = 1 R I...

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SOME BASIC EQUATIONS Some Spherical Harmonics: H = T + V = P 2 / 2 M + V ( R ) = | < | H | = i h | t H | = E | | u i i < u i | = 1 d | >< | = 1 [ J x , J y ] = i h J z J 2 | jm = j ( j + 1) h 2 | jm J z | jm = m h | jm m = j , j - 1, j - 2, .... , - j J ± = J x ± iJ y 2 = 1 r 2 r 2 r + 1 r 2 sin (sin ) + 1 r 2 sin 2 2 2 L z = h i L 2 = - h 2 { 2 2 + 1 tan + 1 sin 2 2 2 } Y l m ( , ) = N l m e im P l m ( ); P l m - associated Legendre functions cos Y l m = ( l + m + 1)( l - m + 1) (2 l + 1)(2 l + 3) Y l + 1 m + ( l + m )( l - m ) (2 l + 1)(2 l - 1) Y l - 1 m Y 0 0 = 1 4 Y 1 ± 1 = m 3 8 sin e ± i Y 1 0 = 3 4 cos Y 2 ± 1 = m 15 8 sin cos e ± i Y 2 0 = 5 16 (3cos 2 - 1) Y 2 ± 2 = 15 32 sin 2 e ± 2 i P x = h i x X = i h p x [ - h 2 2 m d 2 dr 2 + l ( l + 1) h 2 2 mr 2 + V ( r )] u k , l ( r ) = k 2 h 2 2 m u k , l ( r )

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SCATTERING RELATIONS j 1 ( kr ) = sin( kr ) ( kr ) 2 - cos( kr ) kr n 1 ( kr ) = - cos( kr ) ( kr ) 2 - sin( kr ) kr Asymptotic Relations: Born Approximation: ( ) = b sin db d = ( )2 sin d = - 2 b dr r 2 1 - V ( r )/ E - b 2 / r 2 r min b 2 = r min 2 (1 - V ( r min ) E ) v ( r ) = e ikz + f k ( ) e ikr / r R l ( kr → ∞ ) = sin( kr - l / 2 + l ) kr f ( ) = 1 k 4 (2 l + 1) l = 0 e i l sin l Y l 0 ( ) ( ) = | f ( ) | 2 = 4 k 2 (2 l + 1)sin 2 l = 0 l Y l 0 ( = 0) = 2 l + 1 4 V = 0 R l ( kr ) = cos l j l ( kr ) - sin l n l ( kr ) if constraint at origin is not needed V = constant R l ( kr ) = j l ( k r ) if constraint at origin is needed
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Unformatted text preview: j ( kr ) = sin kr kr n ( kr ) = -cos kr kr l = 1 R I ( ka ) ( dR I dr ) r = a tan l = ( ka ) 2 l + 1 1 (2 l + 1)!!(2 l-1)!! l-l a l + 1 + l a (low energy, finite range) kr → 0 j l ( kr ) → ( kr ) l (2 l + 1)!! n l ( kr ) → -(2 l-1)!! ( kr ) l + 1 kr → ∞ j l ( kr ) → sin( kr-l / 2 ) kr n l ( kr ) → -cos( kr-l / 2 ) kr f ( ) = -2 h 2 sin Kr Kr ∞ ∫ V ( r ) r 2 dr K = 2 k sin 2...
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