Question

Why is it not possible to normalize the free-particle wave functions over the whole range of motion of the particle?

Answer 1

Answer:

Free particles still exist in a box of dimensions  Lx×Ly×Lz  with periodic boundaries:  ψ(x+Lx,y,z)=ψ(x,y,z)  ,  ψ(x,y+Ly,z)=ψ(x,y,z)  , and  ψ(x,y,z+Lz)=ψ(x,y,z)  . The eigenstates are planewaves:

ψ(x,y,z)=1V√eikxxeikyyeikzz  ,

where  V=LxLyLz  is the volume. They are thereby normalized. Importantly, the periodic boundary conditions imply that only the following wave numbers are allowed:  kx=2πnx/Lx ,  ky=2πny/Ly ,  kz=2πnz/Lz , where  nx ,  ny , and  nz  are integers. You can get the “whole range of motion” in the thermodynamic limit:  Lx ,  Ly ,  Lz→∞ . Notice that the allowed momenta approach the continuum of phase space in such case