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Problem 4.45 What is the probability that an electron in the ground state of hydro- gen will be found inside the nucleus? (a) First calculate the exact answer, assuming the wave function (Equation 4.80) is correct all the way down to r = 0. Let b be the radius of the nucleus. (b) Expand your result as a power series in the small number € = 26/a, and show that the lowest-order term is the cubic: P≈ (4/3)(b/a)³. This should be a suitable approximation, provided that b

Problem 4.45 What is the probability that an electron in the ground state of hydro- gen will be found inside the nucleus? (a) First calculate the exact answer, assuming the wave function (Equation 4.80) is correct all the way down to r = 0. Let b be the radius of the nucleus. (b) Expand your result as a power series in the small number € = 26/a, and show that the lowest-order term is the cubic: P≈ (4/3)(b/a)³. This should be a suitable approximation, provided that b

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Problem 4.45 What is the probability that an electron in the ground state of hydro- gen will be found inside the nucleus? (a) First calculate the exact answer, assuming the wave function (Equation 4.80) is correct all the way down to r = 0. Let b be the radius of the nucleus. (b) Expand your result as a power series in the small number € = 2b/a, and show that the lowest-order term is the cubic: P≈ (4/3)(b/a)³. This should be a suitable approximation, provided that b <a (which it is). (c) Alternatively, we might assume that (r) is essentially constant over the (tiny) volume of the nucleus, so that P≈ (4/3)πb³|y(0)³². Check that you get the same answer this way. (d) Use b≈ 10–¹5 m and à ≈ 0.5 × 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."
Transcribed Image Text:Problem 4.45 What is the probability that an electron in the ground state of hydro- gen will be found inside the nucleus? (a) First calculate the exact answer, assuming the wave function (Equation 4.80) is correct all the way down to r = 0. Let b be the radius of the nucleus. (b) Expand your result as a power series in the small number € = 2b/a, and show that the lowest-order term is the cubic: P≈ (4/3)(b/a)³. This should be a suitable approximation, provided that b <a (which it is). (c) Alternatively, we might assume that (r) is essentially constant over the (tiny) volume of the nucleus, so that P≈ (4/3)πb³|y(0)³². Check that you get the same answer this way. (d) Use b≈ 10–¹5 m and à ≈ 0.5 × 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."
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