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18 119 the key now is to substitute in the

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Unformatted text preview: ing potential V adiabatically, that is so slowly that the system tracks the ground state for all finite times. If we were to turn on the probe sharply, the system would exhibit complicated ringing behavior that we are not interested in. We now return to our model experiment for studying the properties of our system. After ˆ applying some probe via the external potential V , we want to measure the response of some ˆ. We characterize this response through the expectation value of A, ˆ observable of the system A ˆ�: �A ˆ ˆ �A� = �φ (t) |A |φ (t)� ˜ ˆ˜ = �φ (t) |AI |φ (t)� (1.18) (1.19) ˜ The key now is to substitute in the approximation for |φ (t) given by equation (1.17) into ˆ equation (1.19). Since we have only kept terms up to linear order in VI , we must be careful only to keep terms to this order. After performing this substitution, we arrive at �t � ˆ� ≈ �φ0 |A|φ0 � − i ˆ ˆr ˆ �A dt� eηt �φ0 |[AI (�, t) , VI (t� )]|φ0 � (1.20) −∞ � The mysterious factor eηt comes from our “adiabatic switching­on” of the potential. This ensures that the system evolves smoothly from t = −∞ to t. Eventually, we will send η → 0. Since we are interested in positive times t close to 0 when compared with −∞, we don’t need to � worry about the eηt messing anything up. Response Functions 4 ˆr ˆ The other mysterious piece of equation (1.20) is the appearance of the commutator [AI (�, t) , VI (t� )]. ˆI arising from the substitu­ These two terms simply come from the two possible terms linear in V tions �t ˜ ˆ |φ (t)� ≈ |φ0 � − i dt� VI (t� ) |φ0 � −∞ and ˜ �φ (t) | ≈ �φ0 | + i � t ˆ dt� VI (t� ) �φ0 | −∞ ˜ Note that the integration is with respect to t� , since it comes from the expression for |φ(t)� ˆI with respect to t� . The observable A is also a function of space ˆ which involves an integration of V and time, but there is no reason to integrate over it at this point. This is one way to remember what to integrate over if you forget some day. 1.2 Response Functions ˆ ˆ What we’re really interested in, however, is not �A� itself, but the change in �A� relative to the unperturbed state: ˆ ˆr �δ A� = �δ A(�, t)� − �φ0 |δ A|φ0 � ˆ �t � ˆr ˆ = lim −ieηt dt� eη(t −t) �φ0 |[AI (�, t) , VI (t� )]|φ0 � η →0 (1.21) (1.22) −∞ Now is when we will specialize to the specific type of probe potential describe in the previous example. For concreteness, we consider the potential of equation (1.2): � ˆ V= d�ρ (�) U (�, t) rˆ r r V ˆ U (�, t) commutes with the Hamiltonian, so the interaction picture representation of V is r given by � �� � � ˆ ˆ ˆ �ˆ� � VI = eiH t dr� ρ r� U r� , t e−iH t (1.23) V � � � ˆ � ˆ ˆ� � = dr� eiH t ρ(r� )e−iHt U r� , t (1.24) �V � � �ˆ � � = dr� ρI (r�...
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This note was uploaded on 02/07/2014 for the course PHYS 8.512 taught by Professor Patricklee during the Fall '09 term at MIT.

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