ing peri Optical excitations valence electron X-таy excitations inner electrons PROBLEMS FOR CHAPTER 10 g. 10.17. v line of SECTION 10.2 (The Independent-Particle the 3d wave function is not very treat the outer electron as if it side all the other electrons. (a) what is the potential-energy fur outer electron? (b) In the same should be the energy of an el Compare your answer with t -1.52 eV. Why is the observed estimate? Approximation) ectrum 10.1 Find the electric field & at r = ag in the 1s state of a hydrogen atom. Compare with the breakdown field of dry air, about 3 x 10° V/m. [Hint: Use Gauss's law; treat the atomic electron as a static charge distri- bution with charge density p(r) = -e(r)P and use the result of Problem 8.43.] 10.2 The IPA potential-energy function U(r) is the potential energy "felt" by an atomic electron in the average field of the other Z - 1 electrons plus the nucleus. If one knew the average charge distribution P r) of the Z - 1 other electrons, it would be a fairly simple matter to find U(r). The calculation of an ac- curate distribution p(r) is very hard, but it is easy to 10.6 The ground state of lithium trons in the 1s level and one excited state in which the o been raised to the 3p level. Si tions are not very penetrating energy of this electron by outside both the other electro mation what is the potential- aht mol Chapter 8 The Three-Dimensional Schrödinger Equation 286 8.50 Write down the radial densit 2p states of hydrogen. [See (8 the most probable radius for Hint: If P(r) is maximum, so 8.42 ** 7he average (or expectation) value (r) of the radius for any state isrP(r) dr. Find (r) for the 1s state of hydrogen. Referring to Fig. 8.18, explain the difference between the average and most probable radii. 8.43 ** The probability of finding the electron in the region r > a is P(r) dr. What is the probability that a 1s electron in hydrogen would be found out- side the Bohr radius (r > ag)? SECTION 8.10 (Hydrogen-Like What is the most probable 8.51 the hydrogen-like ion Ni2v 8.44 (a) Write down the radial equation (8.107) for the case that n 2 and / 0 and verify that 8.52 An inner electron in a he tively little by the other e wave function very like th orbit around the same what is the most probable lead? What is this elect energy? = A 2- R2s is a solution. (b) Use the normalization condition (8.86) to find the constant A. (See Appendix B.) Write down the radial equation (8.72) for the case 8 45 #* A hydrogen-like ion M 8.53

ing peri Optical excitations valence electron X-таy excitations inner electrons PROBLEMS FOR CHAPTER 10 g. 10.17. v line of SECTION 10.2 (The Independent-Particle the 3d wave function is not very treat the outer electron as if it side all the other electrons. (a) what is the potential-energy fur outer electron? (b) In the same should be the energy of an el Compare your answer with t -1.52 eV. Why is the observed estimate? Approximation) ectrum 10.1 Find the electric field & at r = ag in the 1s state of a hydrogen atom. Compare with the breakdown field of dry air, about 3 x 10° V/m. [Hint: Use Gauss's law; treat the atomic electron as a static charge distri- bution with charge density p(r) = -e(r)P and use the result of Problem 8.43.] 10.2 The IPA potential-energy function U(r) is the potential energy "felt" by an atomic electron in the average field of the other Z - 1 electrons plus the nucleus. If one knew the average charge distribution P r) of the Z - 1 other electrons, it would be a fairly simple matter to find U(r). The calculation of an ac- curate distribution p(r) is very hard, but it is easy to 10.6 The ground state of lithium trons in the 1s level and one excited state in which the o been raised to the 3p level. Si tions are not very penetrating energy of this electron by outside both the other electro mation what is the potential- aht mol Chapter 8 The Three-Dimensional Schrödinger Equation 286 8.50 Write down the radial densit 2p states of hydrogen. [See (8 the most probable radius for Hint: If P(r) is maximum, so 8.42 7he average (or expectation) value (r) of the radius for any state isrP(r) dr. Find (r) for the 1s state of hydrogen. Referring to Fig. 8.18, explain the difference between the average and most probable radii. 8.43 The probability of finding the electron in the region r > a is P(r) dr. What is the probability that a 1s electron in hydrogen would be found out- side the Bohr radius (r > ag)? SECTION 8.10 (Hydrogen-Like What is the most probable 8.51 the hydrogen-like ion Ni2v 8.44 (a) Write down the radial equation (8.107) for the case that n 2 and / 0 and verify that 8.52 An inner electron in a he tively little by the other e wave function very like th orbit around the same what is the most probable lead? What is this elect energy? = A 2- R2s is a solution. (b) Use the normalization condition (8.86) to find the constant A. (See Appendix B.) Write down the radial equation (8.72) for the case 8 45 #* A hydrogen-like ion M 8.53