First make a prediction and then verify the prediction. The radial probability d

Question

First make a prediction and then verify the prediction.

The radial probability density is P)Ra)2, where the radial wave functions Rndr) are given in Table 7.1. Example 7.11 in the text explains that the most probable radius occurs when the probability density Pn(r) has a maximum. The example also calculates this maximum to be at r = a for the 1s state and r = 4 ao for the 2p state. According to the simple predictions from the Bohr planetary model, the radii for the different energy levels was given by r = n2 ao. Look at Figure 7.12, what state appears to fulfill the simple Bohr prediction for n -3? Verify your prediction by calculating this most probable value for r. Does it match r = 9 a0 (Note that the most probably value for r is not the same as the average value of r given by in Example 7.12.)

Explanation / Answer

For n=1, the max value of Pnl is observed at radius (a0) = 1.5

For n=3, for l=0,1,2 the max values and thus the radius can be observed at 14, 12, 9

For n=3, l =0 fulfils Bohr's prediction.

for n = 3 , r = 9a0

14 = 9 *.15

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