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By Seeger R. J.

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Since the number N and — are canonically conjugate [see Eq. 26)], we have with the use of Eq. 54) which implies that N is an operator of the form From Eqs. 70), the Hamiltonian operator becomes The wave function Schrodinger equation where we write condition for of this planar rotator satisfies the time-dependent simplicity. We solve this equation with the initial Taking into account of the fact that 6, we make the ansatz must be a periodic function of Since the physical interval for the phase variable is the points and 2 are indistinguishable.

The lattice sites of this network (grain centers) are labeled by n. The sites are connected by lattice vectors (links) denoted as . We emphasize that the capacitors Co are essential in producing the short range of the charge interaction of the nearest-neighbor charging model. In fact, as we show shortly, the range of the Coulomb matrix C-1 becomes infinite if C0 is set equal to zero. Even if the array of Fig. 3(a) is fabricated in such a way that the grains have a negligible capacitance with respect to ground, there is still an important contribution to the charging energy associated with the capacitance of isolated grains.

We solve this equation with the initial Taking into account of the fact that 6, we make the ansatz must be a periodic function of Since the physical interval for the phase variable is the points and 2 are indistinguishable. 74) can be replaced by a Fourier integral over the continuous momentum variable. 74) into Eq. 5. 73), we have cn(t = 0) = 1/2 , and the solution of Eq. 75) becomes Inserting this result into Eq. 74), the wave function can be written as The expression on the RHS of Eq. 78) defines the Jacobi theta function (Bellman, 1961).

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