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HW9 Solutions (HW9 due Tues, Mar 31) 1. T&M 36.P.38 For ground state of the hydrogen atom, calculate the probability of finding the electron in the region between r and !r = 0.04a0 at the positions r = a0 and r = 1.5a0 . Solution: In the ground state, ! (r) = 1 3 0 "a e# r a0 , and hence the radial 4r 2 #2 r a0 probability density is P(r) = 4! r " (r) = 3 e . The probability of a0 2 2 finding the electron in the small interval between r and r + !r = r + 0.04a0 is approximately P(r)!r . For r = a0 , this results in P(r = a0 )!r = 4 "2 e (0.04a0 ) = 0.0217 , while for r = 1.5a0 , the probability is a0 4(1.5)2 "3 P(r = a0 )!r = e (0.04a0 ) = 0.0179 . a0 2. T&M 36.P.47 Compute the probability that the electron in the ground state of a hydrogen atom is in the region 0 < r < 3.50a0 . Solution: From the previous problem, we know that the probability density 4r 2 #2 r a0 is P(r) = 4! r " (r) = 3 e . The probability that the electron is in a0 2 2 the region 0 < r < 3.50a0 is 3.5a0 ! 0 3.5a0 P(r)dr = ! 0 4r 2 "2 r a0 e dr = 0.97 . a03 3. T&M 36.P.48 The potential energy of a magnetic moment in an external r r magnetic field is given by U = ! µ " B . (a) Calculate the difference in energy between the two possible orientations of an electron in the external field r B = 0.75Tk̂ . (b) If these electrons are bombarded with photons of energy equal to that of the energy difference, “spin-flip” transitions can be induced. Find the wavelength of the photons needed for such transitions. Solution: (a) The difference in energy levels is !E = 2 µ B . The magnetic moment due to spin for the electron is µ = 5.79 ! 10 "5 eV T (Bohr magneton). Hence, !E = 2(5.79 " 10 #5 eV T)(0.75T) = 8.68 " 10 #5 eV . (b) hc . Putting in " hc 1240eV #10 $9 m = = 0.0143m . the numbers, one obtains ! = "E 8.68 % 10 $5 eV The photon wavelength is given by !E = 4. T&M 36.P.49 The total angular momentum of a hydrogen atom in a certain excited state has the quantum number j=3/2. What can you say about the possible values of the orbital angular momentum quantum number l? Solution: The total angular momentum quantum number j is given by j = l ± 12 , and hence l = 2 or l = 1 . 5. T&M 36.P.32 Using the fact that in an atom there are two quantum states for each combination of l and m l , because of the electron spin, find the total number of electron states that have the following values of n. ! ! ! (a) n = 4 (b) n = 2 Solution: 6. T&M 36.P.033 Find the minimum value of the angle θ between and the +z direction for an electron in an atom that has the following values of l . (a) l = 1 (b) l = 4 (c) l = 50 Solution: 7. T&M 36.P.054 What is the ground-state electron configuration of an atom of each of the following: (a) carbon (b) oxygen. Solution: 8. T&M 40.P.015 Calculate the binding energy and the binding energy per nucleon from the masses given in the table for the following. (a) 12C (b) 56Fe (c) 238U. Solution: 9. T&M 36.P.025 The counting rate from a radioactive source is 8000 counts/s at time t = 0, and 10 min later the rate is 1000 counts/s. (a) What is the half-life? (b) What is the decay constant? (c) What is the counting rate after 20 min? Solution: 10. T&M 36.P.039 The rubidium isotope 87Rb is a β emitter that has a half-life of 4.9 x 1010 y that decays into 87Sr. It is used to determine the age of rocks and fossils. Rocks containing the fossils of early animal have a ratio of 87Sr to 87Rb of 0.0100. Assuming that there was no 87Sr present when the rocks were formed, calculate the age of the fossils. Solution: 11. T&M 36.P.028 Use the table below to calculate the energy release in MeV for the α decay of the following: (a) 226Ra (b) 242Pu Solution: