3. This problem to be handed in for grading by 11:59 pm, Fri. April 10th The radial probability density, P(r), is defined such that the probability to find the electron between spheres of radius r and r+ dr is given by P(r)dr. In spherical coordinates with a radially symmetric potential P(r) = p2 Rn1(r)|* . Here the r² term is present because we are working in spherical coordinates. See sections 7.3 and 7.4 of Krane. (a) For an electron in the 2s state of the hydrogen atom, find the probability to find the electron inside the proton. Assume that the proton is a sphere of radius R = 8.414 x 1016 m. Show your work. Radial wave functions can be found in Table 7.1 of Krane. (b) For an electron in the 2p state of the hydrogen atom, find the probability to find the electron inside the proton. Assume that the proton is a sphere of radius R 8.414 x 10-16 m. Show your work. (c) Evaluate these two probabilities for muonic hydrogen. 12 7.3 The Hydrogen Atom Wave Functions 215 TABLE 7.1 Some Hydrogen Atom Wave Functions m1 R(r) O(0) Ф(Ф) 1 -r/ao 1 1 3/per/0 V2n (?- 0. (2a0)3/2 e-r/2a0 do V2 V2n 1 r-r/20 V3(2a0)3/2 ao 3 cos O 1 2 V3 sin 0 1 +1 1 r-r/2a0 V3(2a0)3/2 ao 1 etip /2Tt (1- 2r2 )e-/30 0. 2r (Зар) /2 (3a6)3/2 1 27a Зао V2 2л 8. 3 cos 0 1 9V2(3a0)3/2 (ao e-r/3a0 баз 2л 8. p2 e-r/3a0 бад 3. sin 0 1 ±1 9/2(3aq)³/2 \ao etip 2n 4 -r/3a0 (3 cos20- 1) 1 27V10(3a)3/2 a 2n 4 15 sin 0 cos 0 4 ±1 27V10(3a,)3/2 až e/340 etip V27 12 V15 4 sin e 4 1 e+2ih V2r +2 27V10(3a)3/2 az e340 of Table 7.1. The second excited state (n = 3) can have l = 0 (m = 0), l = 1 (m, = 0, ±1), or l = 2 (m, = 0, ±1,±2). For the n = 2 level, there are four different possible sets of quantum numbers and correspondingly four different wave functions. All of these wave functions correspond to the same energy, so the n = 2 level is degenerate. (Degeneracy was introduced in Section 5.4.) Then = 3 level is degenerate with nine possible sets of quantum numbers. In general, the level with principal quantum number n has a degeneracy equal to n². Figure 7.6 illustrates the labeling of the first three levels. If different combinations of quantum numbers have exactly the same energy, what is the purpose of listing them separately? First, as we discuss in the last section of this chapter, the levels are not precisely degenerate, but are separated by a very small energy (about 10 eV). Second, in the study of the transitions between the levels, we find that the intensities of the individual transitions depend on the quantum numbers of the particular level from which the transition originates. Third, and perhaps most important, each of %3D %3D %3D %3D %3D %3D 5/- It 2. 2. 2. 2. 2. 2. 3. 3. 3. 3. 3. 3.