In a one dimensional ``box'' of length L,
the ``allowed'' wavelengths are
corresponding to wavenumbers
.
Thus the smallest posssible wavenumber,
and the ``distance'' (in k-space) between successive
allowed wavenumbers, is 
.
There is 
allowed ``state'' per 
.
Put another way, the ``density'' of allowed states
per unit wavenumber is
or, for this one-dimensional (1D) case,
.
Note that n > 0
k > 0.
We are drawing standing waves,
(we choose x=0 at the centre of the box, for symmetry),
for which ``negative'' k values have no independent meaning.