Waves in ID

Let us take a closer look of a Bloch function corresponding to the vector fc  

Electronic Motion in the Mean Field Periodic Systems 

Let me stress that 4 k represents a function of position r in the 3D space and only the periodicity has a ID character. The function is a linear combination of the hydrogen atom D orbitals. The linear combination depends exclusively on the value of k. Eq. (9.28) tells us that the allowed k e (0, ), or alternatively k e (-, ). If we exceed the FBZ length then we would simply repeat the Bloch functions. For = 0 we get 

The function looks like a chain of buoys floating on a perfect water surface. If we ask whether j)Q represents a wave, the answer could be, that if it does then its wave length is oo. What about A = - In such a case  

If we decide to draw the function in space, we would obtain Fig. 9.5.b. When asked this time, we would answer that the wave length is equal to A = 2a, which by the way is equal to There is a problem. Does the wave correspond to A = f or A = - f It corresponds to both of them. Well, does it contradict the theorem that the FBZ contains all different states No, everything is OK. Both functions are from the border of the FBZ, their A values differ by (one of the inverse lattice vectors) and therefore both functions represent the same state. 

---

Fig. 9.5. Waves in ID. Shadowed (white) dicles mean negative (positive) value of the function. Despite the fact that some waves are complex, in each of the cases (a)-(f) we are able to determine their wave length.

---