B.1.2. Particle in a Box

Consider a free particle confined in a cubic box of volume V= L 3 .

Its state is the momentum eigenstate which in the r-representation is

        ψ k ( r )=r|k=Cexp( ikr )                   (B.14)

where C is determined by the normalization

        V d 3 r    ψ k ( r ) ψ k ( r )=1

which gives V | C | 2 =1  or C= V 1/2 = L 3/2 .  

If we impose the periodic boundary conditions

        ψ( x,y,z )=ψ( x+L,y,z )=ψ( x,y+L,z )=ψ( x,y,z+L )

we must have

        k i L=2π n i         where      i=x,y,z  and   n i =0,±1,

or

        k= 2π L ( n x , n y , n z )            n i =0,±1,                      (B.12)

Since the allowable values of k are discrete, the orthonormality and completeness conditions become

        k| k' = δ kk'                      k |kk| = 1 ˆ

while those for the position eigenstates become

        r| r' =δ( rr' )              V d 3 r |rr|= 1 ˆ                 (B.13)