HW12_Solutions - PHYS851 Quantum Mechanics I, Fall 2008...

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Unformatted text preview: PHYS851 Quantum Mechanics I, Fall 2008 HOMEWORK ASSIGNMENT 12 1. [20 pts] Using the recursion formula c j +1 = 2( + 1 + j n ) ( j + 1)( j + 2 + 2) c j and the other details from the series solution to the Hydrogen Radial Equation, compute the Radial wave functions: R 1 , , R 2 , , R 2 , 1 , R 3 , , R 3 , 1 , and R 3 , 2 . Finding the normalization constant is not required. Answer: For the hydrogen atom we have ( r,, | n,,m ) = r 1 R n ( r ) Y m ( , ). We made a series of transfor- mations given by: R n ( r ) = u ( r/ n ) = e r/ n parenleftbigg r n parenrightbigg +1 v ( r/ n ) The function v ( x ) is given by v ( x ) = j n summationdisplay j =0 c j x j where j n = n 1, and the coefficients satisfy the above recursion relation. We also found that n = a n . With j 10 = 0 we find: R 10 ( r ) = c e r/a parenleftbigg r a parenrightbigg With j 20 = 1,we need c 1 = 2(0+1+0 2) (0+1)(0+0+2) c = c , so that R 20 ( r ) = c e r/ (2 a ) parenleftbigg r 2 a parenrightbiggbracketleftbigg 1 parenleftbigg r 2 a parenrightbiggbracketrightbigg With j 21 = 0, we find R 21 ( r ) = c e r/ (2 a ) parenleftbigg r 2 a parenrightbigg 2 With j 3 , = 2, we need c 1 = 2(0+1+0 3) (0+1)(0+0+2) c = 2 c and c 2 = 2(0+1+1 3) (1+1)(1+0+2) c 1 = 2 3 c , giving R 30 ( r ) = c e r/ (3 a ) parenleftbigg r 3 a parenrightbigg bracketleftBigg 1 2 parenleftbigg r 3 a parenrightbigg + 2 3 parenleftbigg r 3 a parenrightbigg 2 bracketrightBigg 1 With j 3 , 1 = 1 we need c 1 = 2(1+1+0 3) (0+1)(0+2+2) c = c , so that R 3 , 1 ( r ) = c e r/ ( a 3 ) parenleftbigg r 3 a parenrightbigg 2 bracketleftbigg 1 parenleftbigg r 3 a parenrightbiggbracketrightbigg lastly, with j 3 , 2 = 0, we find R 3 , 2 ( r ) = c e r/ (3 a ) parenleftbigg r 3 a parenrightbigg 3 2 2. [15 pts] Using the previous results, construct the fully-normalized wavefunctions: 2 , , ( r,, ), 2 , 1 , 1 ( r,, ), 2 , 1 , ( r,, ) and 2 , 1 , 1 ( r,, ). Answer: 2 , , ( r,, ) = r 1 R 2 , ( r ) Y ( , ) = r 1 c e r/ (2 a ) parenleftbigg r 2 a parenrightbiggbracketleftbigg 1 parenleftbigg r 2 a parenrightbiggbracketrightbigg 1 4 = c 1 4 a e r/ (2 a ) bracketleftbigg 1 parenleftbigg r 2 a parenrightbiggbracketrightbigg To find the normalization we set c 2 integraldisplay 2 d integraldisplay sin d integraldisplay r 2 dr 1 16 a 2 e r/a bracketleftbigg 1 parenleftbigg...
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HW12_Solutions - PHYS851 Quantum Mechanics I, Fall 2008...

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