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6. -D. LEE and D. PINES. 1953. Phys. Rev. 92, 883. 7. F. Low and D. PINES. 1955. Phys. Rev. 98, 414. STRONG-COUPLING THEORY OF THE POLARON G. R. ALLCOCK Department of Theoretical Physics, University of Liverpool 1 INTRODUCTION WE have seen that the polaron problem in its idealized form is character- ized by a single dimensionless coupling constant a, which is inversely proportional to the square root of the frequency a) of longitudinal optical lattice vibrations. l. One can first distinguish a dynamic or high-frequency region, in which a) is large, corresponding to the coupling range 0< 6.
K„> = —=n! akn U 2 I 0>. \ n! From equation (13), ak U, = U2ak + U2f(k); moreover the destruction all give zero for operations on the vacuum state I 0> so operators a that 1 0 = f (k 1) (k n n! To obtain <0 I U2 I 0> we use the fact that if the commutator [A, B] of two operators A and B is a c-number, then (k 1 .. k I 0> = > eA+B ) 1 u2 0. EA,B) = eA e B e 2. Hence we can write U2 = exp f(k)— E a kf+ (10_ E k f(k)1 21 2 and finally <0 1 U2 (0> = exP [ Hence the overlap integral 1 f(k)1 2]<0 1 0>.
V2) } Xo. 18) which gives a non-zero contribution is that from the second term in the first bracket and the first term in the second INTRODUCTION TO POLARON THEORY 27 bracket. Hence we find h: = 4rca ew-x S v(1±v 2). 1 (where the electron was regarded as a fixed point source) by the factor (1 +v2 ) in the denominator. 16), replacing the summation over v by an integration and carrying out the angular integration, we find e 2 sin (vRu) di(r, re). 20) where R = r — r el as before. FIG. 20) may be performed by complex integration methods as follows.