ch39-p054 - 54. (a) The plot shown below for |200(r)|2 is...

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(b) The extrema of ψ 2 ( r ) for 0 < r < may be found by squaring the given function, differentiating with respect to r , and setting the result equal to zero: = 1 32 24 0 6 () / ra a e π which has roots at r = 2 a and r = 4 a . We can verify directly from the plot above that r = 4 a is indeed a local maximum of 200 2 . r As discussed in part (a), the other root ( r = 2 a ) is a local minimum. (c) Using Eq. 39-43 and Eq. 39-41, the radial probability is Pr r r r a r a e 200 2 200 2 2 3 2 4 8 2 . / == F H G I K J π (d) Let x = r / a . Then 2 2 /2 2 4 3 2 200 3 00 0 0 1 2 (2 ) ( 4 4 ) 88 1 [4! 4(3!) 4(2!)] 1 8 x x rr P r dr e dr x x e dx x x x e dx aa ∞∞ −− ⎛⎞ =− = + ⎜⎟ ⎝⎠ =−+ = ∫∫ where we have used the integral formula 0 z = xe dx n nx ! . 54. (a) The plot shown below for | 200 ( r )| 2 is to be compared with the dot plot of Fig. 39-22. We note that the horizontal axis of our graph is labeled “
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