Question

2. [Total: 18 pts] In this problem you will analyze the probability that an electron in the ground state of hydrogen will be found inside the nucleus. a) [6 pts) Let b be the radius of nucleus, first calculate the exact answer using the ground state wave function. b) [6 pts Expand your result from part a) as a power series in the small number e = 2b/a, and show that the lowest-order term is P = (4/3)(b/a)'. This should be a suitable approximation, provided that b « a (which it is). c) [3 pts) Alternatively, assume that V(r) is essentially constant over the tiny volume of the nucleus. Thus, approximate the ground state probability density by its value at the origin and check that you get the same answer as in part b). d) [3 pts] Use b = 10-15 m and a = 0.5 x 10-10 m to get a numerical estimate for P. Roughly speaking, this represents the 'fraction of its time that the electron spends inside the nucleus'.

Answer 1

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