Lecture_16 - PHYS342 Fall2011 Lecture 16 The Wavefunctions...

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PHYS 342 Fall 2011 Lecture 16: The Wavefunctions for Atomic Hydrogen e - , m U(r ) Lecture 16
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Understanding the allowed energies 2 2 2 2 2 2 2 1 1 1 1 13.6 e e e eV E m mc   2 2 4 2 ' 1 1 2 integer 0; 0 1 o n n n e n n      1,2,... 4 ( , , ) ( ) ( , ) o n l m n l l m c r R r Y      , , , , n m 1 0 0 2 0,1 [0], [-1, 0, +1] 3 0,1,2 [0], [-1, 0, +1], [-2, -1, 0, +1, +2] etc.
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n= The energy spectrum allowed n, ,m (3,0,0); (3,1,-1); (3,1,0); (3,1,-1); 3 E 1 5 V (3,2,-2); (3,2,-1); (3,2,0); (3,2,1); (3,2,2) n=3 E=-1.5 eV (2 0 0); (2 1 1); (2 1 0); n=2 E= 3 4 eV (2,0,0); (2,1,1); (2,1,0); (2,1,-1) E=-3.4 eV (1,0,0) - ground state E=-13.6 eV n=1
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Degeneracy is a Simple Way to Learn Something About a Complicated System Large collection of objects; What are energies? E 1 E 2 E 3 E 4 E 5 E 6 E 7 E 8 E 9 ……… 3 clicks
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What’s it mean? - for example, consider the n=2 state (2 0 0); (2 1 1); (2 1 0); (2 1 -1) Quantum numbers: (2,0,0); (2,1,1); (2,1,0); (2,1, 1) Quantum numbers Energy : 2 2 13.6 3.4 n eV E eV n     2,0,0 2,0 0,0 ( , , ) ( ) ( , ) r R r Y     Wavefunctions: 2,1,1 2,1 1,1 2,1,0 2,1 1,0 ( , , ) ( ) ( , ) ( , , ) ( ) ( , ) r R r Y r R r Y         2,1, 1 2,1 1, 1 ( , , ) ( ) ( , ) r R r Y     Degeneracy: 2 4 n
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Letter codes to specify the angular wavefunctions = 0 1 2 3 code s p d f 2,0,0 2,0 0,0 ( , , ) ( ) ( , ) r R r Y     2s e 2,1,1 2,1 2 1 0 2 1 1 1 0 ,1 ( , , ) ( ) ( , , ) ( ) ( , ) ( , ) Y r R r r R r Y         2p x (linear combination) 2p z generate 2,1,0 2,1 1,0 2,1, 1 2,1 1, 1 ( , , ) ( ( , ) ) Y r R r     2p y (linear combination) Complex wavefunctions; deg ( ) ( ) ( ) r R r Y     1s (ground state) how to handle?
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