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Unformatted text preview: PHY 251 Fall 2009: homework problem set 8, due in the PHY 251 drop box in room A129 by noon on Friday, Nov. 13. 1. Serway 8.7 Answer: This problem is separable, in that one can solve the Schrodinger equation for each dimension separately. Lets assume ( x, y, z ) = A sin( n 1 x L 1 ) sin( n 2 y L 2 ) sin( n 3 z L 3 ) and integrate to normalize: 1 = A 2 [ integraldisplay L 1 x =0 sin 2 ( n 1 x L 1 ) dx ] [ integraldisplay L 2 y =0 sin 2 ( n 2 y L 2 ) dy ] [ integraldisplay L 3 z =0 sin 2 ( n 3 z L 3 ) dz ] Now all three integrals have identical form. Lets solve for x , using a = n 1 /L 1 : integraldisplay L 1 x =0 sin 2 ( ax ) dx = [ x 2 sin(2 ax ) 4 a ] vextendsingle vextendsingle vextendsingle vextendsingle x = L 1 x =0 = [ L 1 2 sin(2 n 1 L 1 /L 1 ) 4 n 1 /L 1 ] [ 2 sin(2 n 1 /L 1 ) 4 n 1 /L 1 ] = [ L 1 2 0] [0 0] = L 1 2 because sin(2 n 1 ) = 0 and sin(0) = 0 . We thus have 1 = A 2 L 1 2 L 2 2 L 3 2 or A = radicalBig 8 / ( L 1 L 2 L 3 ) . 2. Write out the full wavefunction for the 2 p state of hydrogen with L z = h . Whats the Bohr model energy for this state? Answer: The 2 p state with L z = h has n = 2 , = 1 , and m = 1 , so the wavefunction is = R n =2 , =1 ( r ) Y m = 1 =1 ( , ) = parenleftbigg Z 2 a parenrightbigg 3 / 2 Zr 3 a e Zr/ 2 a 1 2 radicalBigg 3 2 sin( ) e + i where we have used Table 8.4 on p. 280 for R n ( r ) and Table 8.3 on p. 269 for Y m ( , ) . 3. Calculate numerical values for the net angular momentum, and for all possible z axis angular momenta, for the 2 p and 4 d states of hydrogen. Answer: The 2 p state has n = 2 and = 1 , with m = 1 , , +1 . Numerical values are  L  = h radicalBig ( + 1) = (1 . 055 10 34 J s ) radicalBig 1(1 + 1) = 1 . 492 10 34 J s and L z = { 1 , , 1 } 1 . 055 10 34 J s. The 4 d state has n = 4 and = 2 with m = 2 , 1 , , 1 , 2 , so  L  = h radicalBig ( + 1) = (1 . 055 10 34 J s ) radicalBig 2(2 + 1) = 2 . 584 10 34 J s and L z = { 2 , 1 , , 1 , 2 } (1 . 055 10 34 J s ) ....
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 Fall '01
 Rijssenbeek
 Physics, Work

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