p40_059 - 59. (a) 210 is real. Squaring it, we obtain the...

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59. (a) ψ 210 is real. Squaring it, we obtain the probability density: | ψ 210 | 2 = r 2 32 πa 5 e r/a cos 2 θ. Each of the other functions is multiplied by its complex conjugate, obtained by replacing i with i in the function. Since e e = e 0 = 1, the result is the square of the function without the exponential factor: | ψ 21+1 | 2 = r 2 64 πa 5 e r/a sin 2 θ and | ψ 21 1 | 2 = r 2 64 πa 5 e r/a sin 2 θ. The last two functions lead to the same probability density. (b) The total probability density for the three states is the sum:
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