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