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| classes:2009:fall:phys4101.001:lec_notes_1207 [2009/12/08 09:49] – myers | classes:2009:fall:phys4101.001:lec_notes_1207 [2009/12/15 23:58] (current) – ely |
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| ===== Dec 07 (Mon) ===== | ===== Dec 07 (Mon) ===== |
| ** Responsible party: John Galt, Dark Helmet ** | ** Responsible party: John Galt, Dark Helmet, Esquire ** |
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| **To go back to the lecture note list, click [[lec_notes]]**\\ | **To go back to the lecture note list, click [[lec_notes]]**\\ |
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| ====Chapter 6: Time Indepent Perturbation Theory==== | ====Chapter 6: Time Indepent Perturbation Theory==== |
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| | We do it becuase it is a useful tool |
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| | Shroedinger's equation is the must fundamental tool for QM. |
| | We take solutions and eigenstates/eigenvectors to get energy levels |
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| ===Non-Degenerate Case=== | ===Non-Degenerate Case=== |
| | This is the simplest case |
| | single energy->single equation |
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| | <math> H_0|\psi_n^{(0)}>=E_n^{(0)}|\psi_n^{(0)}> </math> |
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| | <math> <\psi_m^{(0)}|\psi_n^{(0)}>=\delta_{mn} </math> |
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| | <math> H=H_0 +\lambda H' </math> ⇒ <math> H|\psi_n>=E_n|\psi_n> </math> |
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| | The goal is to seek an approximation of this new Hamiltonian expression. Specifically we want... |
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| | <math> E_n=E_n^{(0)}+\lambda E_n^{(1)}+\lambda^{2} E_n^{(2)} </math> |
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| | We define <math> <\psi_n^{(0)}|\psi_n^{(1)}>=<\psi_n^{(0)}|\psi_n^{(2)}>=0 </math> |
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| | A Fourier expansion can be used to express <math> |\psi_n^{(1)}>=\Sigma C_{mn}|\psi_n^{(0)}> </math> where m≠n |
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| | Plugging this into the new Hamiltion yields |
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| | <math> (H_0+\lambda H')(|\psi_n^{(0)}>+\lambda|\psi_n^{(1)}>)=(E_n^{(0)}+\lambda E_n^{(1)})(|\psi_n^{(0)}>+\lambda|\psi_n^(1)>) </math> |
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| | <math> H_0|\psi_n^{(0)}>+\lambda(H'|\psi_n^{(0)}>+H_0|\psi_n^{(1)}>)=E_n^{(0)}|\psi_n^{(0)}>+\lambda(E_n^{1}|\psi_n^{(0)}>+E_n^{(0)}|\psi_n^{(1)}>) </math> |
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| | <math> H'|\psi_n^{(0)}>+H_0|\psi_n^{(1)}>=E_n^{(1)}|\psi_n^{(0)}>+E_n^{(0)}|\psi_n^{(1)}> </math> |
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| | Now using the Fourier expansion expression |
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| | <math> H'|\psi_n^{(0)}>+\Sigma C_{nm}H_0|\psi_m^{(0)}>=E_n^{(1)}|\psi_n^{(0)}>+E_n^{(0)}\Sigma C_{nm}|\psi_m^{(0)}> </math> |
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| | <math> H'|\psi_n^{(0)}+\Sigma C_{nm}E_m^{(0)}|\psi_m^{(0)}>=E_n^{(1)}|\psi_n^{(0)}>+E_n^{(0)}\Sigma C_{nm}|\psi_m^{(0)}> </math> |
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| <math> H_0|\psi_0^{(0)}>=E_n^{(0)}|\psi_n^{(0)}> </math> | Using this, one can find an expression for the expectation of the new Hamiltonian as follows |
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| | <math> <\psi_n^{(0)|H'|\psi_n^{(0)}>+\Sigma C_{nm}E_m^{(0)}<\psi_n^{(0)}|\psi_m^{(0)}>=E_n^{(1)}<\psi_n^{(0)}|\psi_n^{(0)}>+E_n^{(0)}\Sigma C_{nm}<\psi_n^{(0)}|\psi_m^{(0)}> </math> |
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| | <math><\psi_n^{(0)|H'|\psi_n^{(0)}>=E_n^{(1)} </math> |
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| | Now one can introduce a new parameter l≠n but l can equal m and show |
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| | <math><\psi_l^{(0)|H'|\psi_n^{(0)}>+\Sigma C_{nm}E_m^{(0)}<\psi_l^{(0)}|\psi_m^{(0)}>=E_n^{(1)}<\psi_l^{(0)}|\psi_n^{(0)}>+E_n^{(0)}\Sigma C_{nm}<\psi_l^{(0)}|\psi_m^{(0)}> </math> |
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| | <math><\psi_l^{(0)|H'|\psi_n^{(0)}>+C_{nl}E_l^{(0)}=C_{nl}E_n^{(0)} </math> |
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| | ⇒<math>C_{nl}=<\psi_l^{(0)|H'|\psi_n^{(0)}>/(E_n^{(0)}-E_l^{(0)}) </math> |
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| | ⇒<math>E_n^{(2)}=\Sigma|<\psi_l^{(0)|H'|\psi_n^{(0)}>|^2/(E_n^{(0)}-E_l^{(0)}) </math> |
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| | This was all i had for notes as well-Dark Helmet |
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