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# pchemII.lecture25.Addition_of_Angular_Momentum - Page 1 of...

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Page 1 of 16 Scott Kirkby Last revised: 17 March 2008 25. Addition of Angular Momentum Suggested Reading: Chapter 13.3 of the text. Introduction: We have seen that the hydrogen-atom electron has angular momentum due to relativistic effects. The amount of angular momentum present from orbital motion depends upon l and m, as described previously. The relativistic spin angular momentum is dependent upon ms. The magnitudes of S 2 and S z for s = 1/2 are given by (25-1) and (25-2) Although we assign different symbols and names to the angular momentum produced by orbital motion and that resulting from relativistic effects, the angular momentum from one source is identical to that from the other. We do not have “green” angular momentum and “purple” angular momentum. Angular momentum is angular momentum. In view of this consideration, we expect the two components of angular momentum to add vectorially, subject to the quantum mechanical quantization S 2 ss 1 + () h 2 = S z m s h =

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Physcal Chemistry I Lecture # 25 Page 2 of 16 Scott Kirkby Last revised: 17 March 2008 constraints suggested by our examination of the rigid rotor. That is, we expect the total angular momentum to be given by the vector sum (25-3) with the magnitude of J quantized in the same manner as and : (25-4) and (25-5) with (25-6) Let us take the hydrogen atom as a simple example. In the ground state, l = m = 0, so there is no orbital angular momentum. As a result, and . Therefore, j = s = 1/2, and both m j and m s can take values of +1/2 or -1/ 2. Now let us examine a hydrogen atom in the 2p excited state. In this eigenstate, we have l = 1, so that m can assume values of -1, 0, and 1. The spin contribution is still described by s = 1/2 and . Since the z components of and are collinear, they add as scalar quantities to produce the z component of the total angular momentum: (25-7) JL S + = L S J 2 jj 1 + () h 2 = J z m j h = jm j j ≤≤ L 0 = JS = m s 12 ± = L S J z L z S z + =
Physcal Chemistry I Lecture # 25 Page 3 of 16 Scott Kirkby Last revised: 17 March 2008 Thus, (25-8) and therefore, (25-9) The possible combinations are summarized in the table below. This table shows that when the z components of and point in the same direction, we have . We know that m j is bounded by j in that it must lie in the range . Therefore, if , we must have a total angular momentum state with j = 3/2. With j = 3/2, m j can have the values -3/2, - 1/2, 1/2 and 3/2. This accounts for four of the six possible angular momentum states listed in the table. We still have two remaining m j states with values 1/2 Table 25-1: Addition of angular momentum for a hydrogen atom in the 2p eigenstate. mm s m j = m + m s 1 1/2 3/2 -1/2 1/2 0 1/2 1/2 -1/2 -1/2 -1 1/2 -1/2 -1/2 3/2 m j hm h m s h + = m j s + = L S m j 32 ± = jm j j ≤≤ m j ± =

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Physcal Chemistry I Lecture # 25 Page 4 of 16 Scott Kirkby Last revised: 17 March 2008 and -1/2. Consequently, there must be a second total angular momentum state with j = 1/2.
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