6-HW Solutions

6-HW Solutions - HW #6: Atomic Physics Phys320 Spring 2007...

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Unformatted text preview: HW #6: Atomic Physics Phys320 Spring 2007 (Reshchikov) p. 1 Homework #6 Due on Monday, Apr 2 Problem 1 For the hydrogen atom in the 2p state explicitly calculate the probability P(r)r of finding the electron between r = 3.8ao and 4.2ao. Compare your answer with numerical solution by using http://webphysics.davidson.edu/faculty/dmb/hydrogen/default.html. (Section 2). P210 (r )r = r 2 [ R210 (r )] r 2 where r = 4ao and r = 0.4ao from lecture notes with Z =1 R210 (r ) = r a0 e- r 2 a0 3(2 a0 )1.5 R210 (r = 4ao ) = 2 4a0 e-4 a0 2 a0 0.1105 = 1.5 a0 3(2 a0 ) a1.5 0 2 0.1105 0.4a0 ) = 0.078 or 7.8% P210 (r )r = ( 4a0 ) a1.5 ( 0 Numerical integration gives 8%. Problem 2 If a hydrogen atom is in the 5f state, find the possible j, J, and Jz values accounting for the spin-orbit interaction with electron spin s = . How many distinguishable energy levels exist with and without an applied external magnetic field? Note that the 5f state indicates n = 5 and l =3. j = l - s to ( l + s ) = 3 = 7 / 2 or 5 / 2 For j = For j = 5 : 2 7 : 2 J = J = 55 + 1 h = 2.96 h 2 2 77 + 1 h = 3.97 h 2 2 and J= j ( j + 1) h 5 3 1 J z = mh = h, h, h 2 2 2 7 5 3 1 J z = mh = h, h, h, h 2 2 2 2 There are two energy levels without a B field (j = 5/2 and 7/2) and fourteen possible levels with a B field (j = 5/2 split into 6 mj values and j = 7/2 split into 8 mj values). HW #6: Atomic Physics Phys320 Spring 2007 (Reshchikov) p. 2 Problem 3 For l = 4, find the magnitude of the angular momentum L and the possible LZ values in terms of . Draw a vector diagram showing the orientations of L relative to the z axis. How many distinguishable energy levels exist with and without an applied external magnetic field? L = l (l +1) h = 3(3+1) h = 20 h or 4.47 h LZ L = 20h 4.47h Lz = mh = 0, h, 2h, 3h, 4h There is one energy level without a B field (l = 4) and nine levels with a B field (# of ml values). 4 3 2 0 - -2 -3 -4 m=4 m=3 m=2 m=1 m=0 m = -1 m = -2 m = -3 m = -4 Problem 4 Given electron #1 in a 6f state (l1 = 3, s1 = ) and electron #2 in a 4p state (l2= 1,s2 = ), find the jtot values of their coupled angular momenta by using the JJ coupling method: First, find the individual j1 and j2 values and then combine them to find the jtot values. j 1 = l1 - s1 to ( l1 + s1 ) = 3 - 1 to 3 + 1 = 5 ; 7 2 2 2 2 jtot = j 1 - j 2 to j 1 + j 2 by increments of "one" ( ) ( ) j 2 = 1 - 1 to 1 - 1 = 1 , 3 2 2 2 2 ( ) j1 value 5/2 5/2 7/2 7/2 j2 value 1/2 3/2 1/2 3/2 j1 j2 2 1 3 2 j 1 + j2 3 4 4 5 jtot values 2, 3 1, 2, 3, 4 3, 4 2, 3, 4, 5 Problem 5 Solve Problem 4 by using the LS coupling method: Find first total angular and spin momentum values ltot and stot and then combine them to find jtot. ltot = l1 - l2 to ( l1 + l2 ) = 3 - 1 to ( 3 + 1) = 2,3, 4 stot = s1 - s2 to ( s1 + s2 ) = 1 - 1 to 1 + 1 = 0,1 2 2 2 2 jtot = ltot - stot to ( ltot + stot ) by increments of "one" ( ) ltot value 2 2 3 3 4 4 stot value 0 1 0 1 0 1 ltot stot 2 1 3 2 4 3 ltot + stot 2 3 3 4 4 5 jtot values 2 1, 2, 3 3 2, 3, 4 4 3, 4, 5 HW #6: Atomic Physics Phys320 Spring 2007 (Reshchikov) p. 3 Problem 6 Sketch the energy diagram for a system with l = 2 and s = 1/2: before spin-orbit splitting, after spinorbit splitting, after additional application of external magnetic B (label all m values). mJ L=2 S=1/2 J = 3/2 J = 5/2 5/2 3/2 1/2 -1/2 -3/2 -5/2 3/2 1/2 -1/2 -3/2 Spin-orbit B0 Problem 7 Calculate the energy separations between the adjacent levels E in Problem 6 if the magnetic field is 10T. Note that E = g B B , where B is the Bohr magneton (see Lecture notes) and g is the Lande factor: g =1+ j ( j + 1) + s ( s + 1) - l (l + 1) . Note that the splitting for different quantum numbers j is 2 j ( j + 1) different. g = 1+ 2.5(2.5 + 1) + 0.5(0.5 + 1) - 2(2 + 1) = 1.2 1 5(2.5 + 1) and E = 1.21 (5.79 10 -5 ) 10 = 7 10 -4 eV for the splitting of the state with j=5/2 and 1.5(1.5 + 1) + 0.5(0.5 + 1) - 2(2 + 1) = 0.8 3(1.5 + 1) E = 0.8 (5.79 10 -5 ) 10 = 4.63 10 -4 eV for the splitting of the level with j=3/2. g = 1+ and ...
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