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Operator for an electron with momentum k returning to

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Unformatted text preview: c� (1.8) � r = � c k −q � �k � k Equation (1.7) is the first quantized form of ρ (�), and equation (1.8) is the second quantized ˆq † form with c� the creation operator for an electron with momentum1 � − � and c� the destruction kq k k −q � �. operator for an electron with momentum k ˆ Returning to equation (1.1), we’d like to think about V as a perturbation on the external ˆ . This leads us naturally to consider H as the unperturbed Hamil­ ˆ field­free system Hamiltonian H ˆ tonian within the interaction picture representation. Recall that this H is a very complicated beast with all of the electron­electron repulsions included, but for our purposes we just take as a given that there are a set of eigenstates and energies that diagonalize this Hamiltonian. Recall the formulation of the interaction representation: ih ¯ ∂ ˆ ˆ |φ (t)� = (H + V )|φ (t)� ∂t (1.9) ˆ We can “unwind” the natural time dependence due to H from the state ket |φ (t)� to form an ˜ (t)�I by interaction representation state ket |φ ˆ ˜ |φ (t)�I = eiHt |φ (t)� ˆ˜ |φ (t)�I = e−iHt |φ (t)� (1.10) (1.11) ˆ Note that in the absence of V , these interaction picture state kets are actually the Heisenberg h picture state kets of the system. Also, we have now officially set ¯ = 1. After substituting (1.11) into (1.9), we obtain i¯ h 1 ∂ ˆˆ ˆ˜ = eiHt V e−iHt |φ (t)� ∂t ˆ˜ = VI |φ (t)� (1.12) (1.13) � and q are actually wavevectors, which differ from momenta by a factor of ¯ . When in doubt, assume h = 1. k � h ¯ Response Functions and the Interaction Representation 3 where we have set ˆˆ ˆ ˆ VI = eiH t V e−iHt (1.14) Thus the interaction picture state ket evolves simply according to the dynamics governed solely ˆ by the interaction picture perturbing potential VI . More generally, we can write any observable (operator) in the interaction picture as ˆˆ ˆ ˆ AI = eiHt Ae−iHt We can integrate equation (1.12) with respect to t to get �t ˜ (t)� = |φ0 � − i ˜ ˆ |φ dt� VI (t� ) |φ (t� )� (1.15) (1.16) −∞ At first it seems like we have not done much to benefit ourselves, since all we have done is ˆ to convert the ordinary Schrodinger equation, a PDE, into an integral equation. However, if VI is small, then we can iterate equation (1.16): �t ˜ ˆ |φ (t)� ≈ |φ0 � − i dt� VI (t� ) |φ0 � + · · · (1.17) −∞ ˆ The essence of linear response theory is that we focus ourselves on cases where VI is suffi­ ciently weak that the perturbation series represented by equation (1.17) has essentially converged ˆ after including just the first non­trivial term listed above. This term is linear in VI . Throughout this discussion, we will be working at T = 0, so |φ0 � is simply the ground state ˆ of the non­perturbed total system Hamiltonian H . Note that we have taken our initial time, i.e. the lower limit of integration in equation (1.16), to be −∞. This is because we want to ˆ imagine turning on the prob...
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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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