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Physics For Scientists & Engineers
Found in: Page 1296

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Short Answer

Prove that the radial probability density peaks at r=aB for the 1s state of hydrogen.

pr(r) has its maximum peak at r=aB

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Step by Step Solution

Given information:

Radial probability density pr(r)=4πr2Rnl(r)2

For the 1s state where role="math" localid="1648542807295" n=1,l is s



When pr(r) is maximum, then ddrpr(r)=0



At r=aB,pr(r) has either a maximum or a minimum. To determine whether it is a maximum or minimum. we find d2dr2pr(r)


localid="1648543956396" =ddr2re-2raB-ddr2r2aBe-2raB=2e-2raB-4raBe-2raB-4raBe-2raB+4r2aB2e-2raB

Now,at r=aB d2dr2pr(r)=2e-2aBaB-4aBaBe-2aBaB-4aBaBe-2aBaB+4(aB)2aB2e-2aBaB=2e-2-4e-2-4e-2+4e-2=-2e-2<0

pr(r) has its maximum peak at r=aB.

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