Suppose you have a machine that gives you pieces of candy when you push a button. Eighty percent of the time, pushing the button gets you two pieces of candy. Twenty percent of the time, pushing the button yields 10 pieces. The average number of pieces per push is N_avg = 2 × 0.80 + 10 × 0.20 = 3.6. That is, 10 pushes should get you, on average, 36 pieces. Mathematically, the average value when the probabilities differ is N_avg = Σ(Ni × Probability of i). We can do the same thing in quantum mechanics, with the difference that the sum becomes an integral. If you measured the distance of the electron from the proton in many hydrogen atoms, you would get many values, as indicated by the radial probability density. But the average value of r would be

Calculate the average value of r in terms of a_B for the electron in the 1s and the 2p states of hydrogen.

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avg rP,(r) dr 8