3502_12_octo07_LG.2

3502_12_octo07_LG.2 - 12-6 The Initial Spherical Harmonics...

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12-6 The Initial Spherical Harmonics Complex form Real (spherical) form Real (cartesian) form Nomenclature < L 2 > < L z > 1 4 ! " # $ % & 1/2 1 4 ! " # $ % & 1/2 1 4 ! " # $ % & 1/2 Y 0,0 ; s 0 0 0 3 8 ! " # $ % & 1/2 sin ( e ) i * Y 1,–1 ; p –1 2 h 2 ! h 3 4 ! " # $ % & 1/2 sin ( sin ) 3 4 ! " # $ % & 1/2 y r Y 1,sin φ ; p y 2 h 2 a 3 4 ! " # $ % & 1/2 cos ( 3 4 ! " # $ % & 1/2 cos ( 3 4 ! " # $ % & 1/2 z r Y 1,0 ; p 0 ; p z 2 h 2 0 3 4 ! " # $ % & 1/2 sin ( cos ) 3 4 ! " # $ % & 1/2 x r Y 1,cos φ ; p x 2 h 2 a 3 8 ! " # $ % & 1/2 sin ( e i ) Y 1,1 ; p 1 2 h 2 h 15 32 ! " # $ % & 1/ 2 sin 2 ( e ) 2 i * Y 2,–2 ; d –2 6 h 2 ! 2 h 15 4 ! " # $ % & 1/2 sin 2 ( sin 2 ) 15 4 ! " # $ % & 1/2 xy r 2 Y 2,sin2 φ ; d xy 6 h 2 a 15 8 ! " # $ % & 1/2 sin ( cos ( e ) i * Y 2,–1 ; d –1 6 h 2 ! h
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12-7 15 4 ! " # $ % & 1/2 sin ( cos ( sin ) 15 4 ! " # $ % & 1/2 yz r 2 Y 2,sin φ ; d yz 6 h 2 a 5 16 ! " # $ % & 1/ 2 3cos 2 ( ) 1 ( ) 5 16 ! " # $ % & 1/ 2 3cos 2 ( ) 1 ( ) 5 16 ! " # $ % & 1/ 2 3 z 2 r 2 ( 1 " # $ $ % & Y 2,0 ; d 0 ; d z 2 6 h 2 0 15 4 ! " # $ % & 1/2 sin ( cos ( cos ) 15 4 ! " # $ % & 1/2 xz r 2 Y 2,cos φ ; d xz 6 h 2 a 15 8 ! " # $ % & 1/2 sin ( cos ( e i ) Y 2,1 ; d 1 6 h 2 h 15 4 ! " # $ % & 1/2 sin 2 ( cos2 ) 15 16 ! " # $ % & 1/ 2 x 2 ( y 2 r 2 Y 2,cos2 φ ; d x 2 y 2 6 h 2 a 15 32 ! " # $ % & 1/ 2 sin 2 ( e 2 i ) Y 2,2 ; d 2 6 h 2 2 h a This real spherical harmonic is not an eigenfunction of L z and thus its expectation value is not tabulated. The spherical harmonics continue through f , g , h , i , and higher functions (can you guess why e is skipped?), but those are not tabulated above. Very nice pictures of the square moduli of the complex spherical harmonics through
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