Statistical Physics

198 Boundary ConditionsHere n must be positive, because a wave with a negative value of k is thesame wave as one with a wave number |k|. If there is an upper limit on thewave number k so that k ≤ k M , the number of allowed values of k is k M L x /π.This result can be extended to the three-dimensional case. If the wave isconfined in a box of dimensions L x × L y × L z , the allowed wave vectors aregiven by( π πk = n x , n y , π )n z , (F.3)L x L y L zwhere n x , n y ,andn z are natural numbers.F.2 Periodic Boundary ConditionIn the case of the fixed boundary condition, the existence of the walls isconsidered explicitly. In this case the situation near the walls is somewhatdifferent from that in the interior, far from the walls. However, there is anotherboundary condition that can be used, for which we need not consider theexistence of the walls explicitly. This is the periodic boundary condition. Fora one-dimensional system, we impose the condition thatψ(x + L x ,t)=ψ(x, t) .(F.4)A wave satisfying this condition now has the form of a propagating waveinstead of the standing wave described by (F.1), namelyψ(x, t) =A cos(kx − ωt + α) .(F.5)From the boundary condition (F.4), the allowed values of the wave number k nare given byk n = 2π n,(F.6)L xwhere n is an integer. In this case the wave is a propagating wave, and thedirection of propagation depends on the sign of n. The interval between adjacentvalues of k n in the present case is twice that for (F.2). However, sincea negative value of n describes a different wave, the number of waves thatsatisfy |k| ≤k M is k M L x /π, as before.In the three-dimensional case, the allowed wave vectors are given byk =( 2πL xn x , 2πL yn y , 2πL zn z), (F.7)where n x , n y ,andn z are integers.A periodic boundary condition is difficult to realize experimentally. On theother hand, it is often more convenient for theoretical considerations. When