Quantum Engi Q and A_Part_48

Quantum Engi Q and A_Part_48 - 142 APPENDIX A ADDENDA...

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142 APPENDIX A. ADDENDA A.21.3 Corrected multipole matrix elements A.21.4 Matrix element ballparks A.21.5 Selection rules A.21.6 Ballpark decay rates A.21.7 Wave functions of definite angular momentum A.21.8 Weisskopf and Moszkowski estimates A.21.9 Errors in other sources

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A.22. DETAILS OF THE ANIMATIONS 143 A.22 Details of the animations A.23 WKB Theory of Nearly Classical Motion A.23.1 Solution wkb-a Question: Use the equation 1 ¯ h integraldisplay x 2 x = x 1 p c ( x ) d x = to find the WKB approximation for the energy levels of a particle stuck in a pipe of chapter 3.5.5. The potential V is zero inside the pipe, given by 0 x x In this case, the WKB approximation produces the exact result, since the classical momentum really is constant. If there was a force field in the pipe, the solution would only be approximate. Answer: Substituting in p c = 2 mE , x 1 = 0 and x 2 = x , 1 ¯ h 2 mEℓ x = and squaring both sides, the energy is found as E = n 2 ¯ h 2 π 2 / 2 mℓ 2 x . That is the same result as in chapter 3.5.5, but obtained in a much easier way.
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