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Extra info for Green’s Functions in Quantum Physics
0+ S .... 36) where E(X) - 8(X) - e(-x) 1 ; -1 , and w is real. 37) 30 Since there are three independent 9'S of interest we can define three g's, each as a difference of a pair of 9'S. 2. 28). 40-44) knowledge of g>, g< allows us to determine all g' sand g. 2 we can see that 9->( r,r , ,T ) = co! 45) where ~ ~ O. 30). 50) where the dot denotes differentiation with respect to t and ~(t') is d~/dt for t = t'. 50). Furthermore, ~(t) ~ ~(t') and ~(t) ~ ~(t') as t ~ t'. 26) for an arbitrary time t in terms of ~(t') and ~(t') at a particular time t'.
29, 30). 27,28) and take into account thatK (x) ~ - ~n(x) and (1) + 0 Ho (±x) ~ (2i/TI)~n(x) as x ~ 0 ]. _ Finally, in the 1-0 case the DOS and G(E) approach infinity as 1/1[ in the limit E ~ 0+. 31,32). The above statements connecting the dimensionality with the behavior of G(z), G(E) and the DOS near a band edge were based upon the results for the free particle case. 5 that the same connection exists for almost all cases where the hamiltonian H is periodic. 35) with ~ = cm/~. 27,28,40,49,56).
Depend on the DOS of final and initial states. 1 we plot the DOS versus E for a free particle whose motion in 3-D, Z-D and 1-0 space is governed by the SChrodinger equation. Tn all cases the spectrum has a lower bound (at E = 0) below which the DOS is zero. Energies at which the continuous spectrum terminates are called band edges in solid-states physics. We will adopt this name here. 47 As we shall see later, the behavior of ptE), G(E) and G(z) for E or z near a band edge EB is of great physi~al interest.