2 if { ϕ$_{k}$ } is. The inverse relation is Similarly we define the electron creation operators in the new states ˜ c † k σ in terms of the Bloch creation operators c † k σ by It is clear that our new basis functions do not satisfy Bloch’s theorem, since Nevertheless, I will argue that the new basis has very interesting properties and is relevant to the description of the superconducting state. Consider the case where the Bloch functions are well approximated by plane wave states Then describing standing waves. The values of r for which | ˜ ϕ$_{k}$ ( r ) | 2 is maximum satisfy with n integer, and for those r values It is clear that for small k the situation is the same as for the plane-wave case: the values of R$_{i}$ for which the amplitude is maximum in Eq. (12a) coincide (or nearly coincide) with those where the amplitude in Eq. (12b) is zero. However, this changes when the wavevector k becomes an appreciable fraction of a reciprocal lattice vector G . We have and the new orbitals are given by Similarly, consider a band described by the tight bind- ing approximation with a single orbital φ$_{i}$ ( r ) per site. The Bloch functions are modulation of the wavefunction amplitude for two elec- trons in the same state ˜ ϕ$_{k}$ will give a strong enhancement of the direct Coulomb repulsion in that case compared to the plane wave case. FIG. 1: Wave function amplitudes and charge densities for the standing waves associated with the operators ˜ $_{c}$† k σ (full lines) and ˜ $_{c}$† − k , − σ (dashed lines) , assuming the Bloch states are plane waves. Note that the charge densities avoid each other reducing the Coulomb repulsion. ϕ$_{k}$ ( r ) = ˜ ϕ$_{k}$ ( r ) − i ˜ ϕ$_{−}$$_{k}$ ( r ) √$_{2}$ (5b) ˜ c † k σ = c † k σ + ic † − k σ √$_{2}$ (6a) c † k σ = ˜ c † k σ − i ˜ c † − k σ √$_{2}$ (6b) ˜ ϕ$_{k}$ ( r + R ) = e i k · $^{R}$ϕ$_{k}$ ( r ) + ie − i k · $^{R}$ϕ$_{−}$$_{k}$ ( r ) √$_{2}$ = e i k · $^{R}$˜ ϕ$_{k}$ ( r ) (7) ϕ$_{k}$ ( r ) = 1 √$_{Ω}$ e i k · r (8) ˜ ϕ$_{k}$ ( r ) = 1 + i √$_{Ω}$ cos ( k · r − π 4 ) (9a) ˜ ϕ$_{−}$$_{k}$ ( r ) = 1 + i √$_{Ω}$ cos ( k · r + π 4 ) . (9b) ϕ$_{k}$ ( r ) = 1 √$_{N}$ ∑ i e i k · R i φ$_{i}$ ( r ) (11) ˜ ϕ$_{k}$ ( r ) = 1 + i √$_{N}$ ∑ i cos ( k · R$_{i}$ − π 4 ) φ$_{i}$ ( r ) (12a) ˜ ϕ$_{−}$$_{k}$ ( r ) = 1 + i √$_{N}$ ∑ i cos ( k · R$_{i}$ + π 4 ) φ$_{i}$ ( r ) (12b) k · r − π 4 = nπ (10a) k · r + π 4 = ( n + 1 2 ) π (10b) cos ( k · R$_{i}$ ∓ π 4 ) = 1 √$_{2}$ ( cos ( k · R$_{i}$ ) ± sin ( k · R$_{i}$ )) (13) so that | ˜ ϕ$_{−}$$_{k}$ ( r ) | 2 = 0. Consequently this will cause a strong reduction of the direct Coulomb repulsion between an electron in state ˜ ϕ$_{k}$ and another electron in state ˜ ϕ$_{−}$$_{k}$ compared to the plane wave case where the density is uni- form, as illustrated in Fig. 1. Conversely, the in-phase II. NATURE OF THE WAVE FUNCTIONS