
For the basic solution and the dispersion relation
(energy vs. momentum / wave vector) nothing has changed, and we obtain
(independent of boundary conditions) 


y 
= 
æ
ç
è 
1
L 
ö
÷
ø 
3/2 
· 
exp (i · k · r) 
E = 
^{2}
· k^{2}
2 m 




Only the allowed values of the wave vector
k are determined by the boundary conditions; for the fixed
conditions given here we have 


y (x = 0) 
= 
y (x = L) = 0 




Obviously this condition can be satisfied by



k 
= 
n · p
l 



n 
= 
1, 2, 3, ... 



The number of states
Z(k) up to a wave vector k is
generally given by 


Z(k) 
= 
Volume of sphere with radius E(k)
Volume of state 



For fixed boundary conditions we
have 


Z(k) 
= 
1
8 
· 
4/3 (p · k)^{3}
(p/L)^{3} 
= 
(k · L)^{3}
6p^{2} 



This is exactly what we would get for
the periodic boundary conditions  thanks to the factor 1/8. 


Where does this factor come from? Easy  since
the quantum numbers n are restricted to positive integers in this case,
we can not count states in 7 of the 8 octants of the complete
sphere and must divide the volume of the complete sphere by 8. 


This becomes clear if we look at a drawing of the
possible states in phase space: 


