8.69) and Table 8.1. 8.34 If you haven't already done so, do parts (a) and (b) of Problem 8.33, and then do part (c), but for the five spherical harmonics with/ 2. r momentum l expansion to . (This shows SECTIONS 8.7 and 8.8 (The Energy Levels of the Hydrogen Atom and Hydrogenic Wave Functions) m mechanics el.) r the special 8.35 Prove that the degeneracy of the nth level in the hy- drogen atom is n'; that is, verify the result (8.77). (But be aware that this number gets doubled because of the electron's spin, as we describe in Chapter 9.) constant d solution is ow that this acceptable). r differential mbination of n, and prove 8.36 .It is known that a certain hydrogen atom has a defi- nite value of l. (a) What does this statement tell you about the angular momentum? (b) What are the allowed energies consistent with this information? nstant. 8.37 The mean value (or expectation value) of 1/r for any state is (1/r) the 1s state of hydrogen. Comment. [Hint: See the integrals in Appendix B.] = J (1/r) P (r) dr. Find (1/r) for r the case ution. (Any sin 0 is the e complete ependence m = -1. (a) It is known that a certain hydrogen atom has 8.38 2. How many different states are n = 5 and m consistent with this information? (b) Answer the same question (in terms of n and m) for arbitrary values ofn and m leave it as 12 a maxi- approximation) k V+y2 = absolute value (or modulus) of z OME INTEGRALS r-iy=complex conjugate of z gerals of the form x"e Av? dx P- A is a positive number, occur positive integer, their value can be found from the following: frequently in several branches of physics. When n ere IGONOMETRIC RELATIONS T lo 4A TT 12= 21 1 I3 14 2A2 1613 (Euler's relation) cos + i sin 0 e TT 8 A +e cos 8 = 2 dIn-2 sin 6 = ice that the integral e Ax dx equals 21, when n is even, but is zero if n is odd. 2i Another common integral is the indefinite integral fre. Jn "e"x/b dx hat is analytic near the point z = a can be expanded in a Taylor is a small integer, this is easily evaluated by parts, for example, J4-(b +bx)exb J2=(2b +2bx + bx)eb 4-6e1 (z- a) (a)( - a) + a) (Taylor's series) 2! general 6 t special cases of this expansion are 22 ite in particular, that 2! 3! reb dx 2b xelb dx = b2 In(1 + z) z 3 aso that cos z 1 2! 4! νTο Vxelb dx 2 sin z z 31 5! n 1) n(n - 1(n- 2) + + + 1 1 1.

8.69) and Table 8.1. 8.34 If you haven't already done so, do parts (a) and (b) of Problem 8.33, and then do part (c), but for the five spherical harmonics with/ 2. r momentum l expansion to . (This shows SECTIONS 8.7 and 8.8 (The Energy Levels of the Hydrogen Atom and Hydrogenic Wave Functions) m mechanics el.) r the special 8.35 Prove that the degeneracy of the nth level in the hy- drogen atom is n'; that is, verify the result (8.77). (But be aware that this number gets doubled because of the electron's spin, as we describe in Chapter 9.) constant d solution is ow that this acceptable). r differential mbination of n, and prove 8.36 .It is known that a certain hydrogen atom has a defi- nite value of l. (a) What does this statement tell you about the angular momentum? (b) What are the allowed energies consistent with this information? nstant. 8.37 The mean value (or expectation value) of 1/r for any state is (1/r) the 1s state of hydrogen. Comment. [Hint: See the integrals in Appendix B.] = J (1/r) P (r) dr. Find (1/r) for r the case ution. (Any sin 0 is the e complete ependence m = -1. (a) It is known that a certain hydrogen atom has 8.38 2. How many different states are n = 5 and m consistent with this information? (b) Answer the same question (in terms of n and m) for arbitrary values ofn and m leave it as 12 a maxi- approximation) k V+y2 = absolute value (or modulus) of z OME INTEGRALS r-iy=complex conjugate of z gerals of the form x"e Av? dx P- A is a positive number, occur positive integer, their value can be found from the following: frequently in several branches of physics. When n ere IGONOMETRIC RELATIONS T lo 4A TT 12= 21 1 I3 14 2A2 1613 (Euler's relation) cos + i sin 0 e TT 8 A +e cos 8 = 2 dIn-2 sin 6 = ice that the integral e Ax dx equals 21, when n is even, but is zero if n is odd. 2i Another common integral is the indefinite integral fre. Jn "e"x/b dx hat is analytic near the point z = a can be expanded in a Taylor is a small integer, this is easily evaluated by parts, for example, J4-(b +bx)exb J2=(2b +2bx + bx)eb 4-6e1 (z- a) (a)( - a) + a) (Taylor's series) 2! general 6 t special cases of this expansion are 22 ite in particular, that 2! 3! reb dx 2b xelb dx = b2 In(1 + z) z 3 aso that cos z 1 2! 4! νTο Vxelb dx 2 sin z z 31 5! n 1) n(n - 1(n- 2) + + + 1 1 1.