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# Recall that t 0 for t 0 and t 1 for t 0 this ensures

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Unformatted text preview: ) U r� , t (1.25) V ˆ Substituting this expression for VI back into equation (1.22), we obtain: �t � � ˆr ˆr �δ A(�, t)� = lim −ieηt dt� d� eη(t −t) �φ0 |[AI (�, t) , ρI (�� , t� )]|φ0 � U (�� , t� ) r ˆr r η →0 −∞ (1.26) V ˆ r, We deﬁne the response function χ as the kernel of this expression for �δ A(� t)�: �∞ ˆr �δ A(�, t)� = dt� d�� χ(�, �� , t − t� ) U (�� , t� ) r rr r −∞ (1.27) Electron Density Response to an Applied Electric Potential 5 ˆ χ is a function of (t − t� ) only, since H is independent of time. The interpretation of equation � (1.27) is that if we “shake” the system with an external potential U (r� , t� ), then the response of ˆ r the system in terms of some observable A at the point � and time t is modulated by the response function χ(�, �� , t − t� ). rr Thus from comparing this deﬁnition with equation (1.26), we see that χ(�, �� , t − t� ) ≡ rr (1.28) � ˆr − i �φ0 |[AI (�, t) , ρI (�� , t� )]|φ0 � eη(t −t) θ(t − t� ) ˆr Note that in equation (1.27) we extended the limits of integration from −∞ to ∞ for conve­ rr nience, and thus have added the Heaviside step function θ(t − t� ) to our deﬁnition of χ(�, �� , t − t� ). Recall that θ(t) = 0 for t &lt; 0 and θ(t) = 1 for t &gt; 0. This ensures causality in our deﬁnition of χ, since the system should not be able to respond to the perturbation before it happens. Notice also that based on this deﬁnition, the response function is purely a function of the ˆ system’s unperturbed Hamiltonian H ; U does not appear anywhere in the expression. Thus investigations of χ can reveal information about the systems Hamiltonian. In this deﬁnition, the electron density ρI (�� , t� ) appears because we specialized to the case ˆr of an applied external electric potential that couples to the system’s charge density. For a probe that couples to some other density, such as magnetization density m(r� , t� ), we can simply replace ˆ� �� � ˆ� ρI (� , t ) by mI (r� , t ) in deﬁni tion (1.28). ˆr 1.3 Electron Density Response to an Applied Electric Potential In this section, we will specialize further to the case where we observe the response of the electron ˆˆ density to an applied potential that couples to the density. Thus we are picking A = ρ. We begin by taking the Fourier transform of equation (1.28) with respect to time: t�� = t� − t � χ(�, �� , ω ) = − i rr 0 �� dt�� e−(iω−η)t �φ0 |[ρI (�, 0), ρI (�� , t�� )]|φ0 � ˆr ˆr (1.29) −∞ ˆ Recall that we have a complete set of eigenstates of H : � ˆ H |n� = En |n� |n��n| = ˆ 1 n Inserting this complete set of states into the commutator � [ρI (�, 0), ρI (�� , t�� )] = [ρI (�, 0), ˆr ˆr ˆr |n��n|ρI (�� , t��...
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