Consider a one-dimension (1D) metal of length "
*
a
*
"
containing N electrons. Treat the metal as having a uniform potential within
the box. You are to compare and contrast two cases:

(
*
i
*
) The allowed values of the electron wave
vector,
*
k
_{
x
}
*
, are determined by requiring "periodic
boundary conditions."

(
*
ii
*
) The allowed values of
*
k
_{
x
}
*
are
determined by having the potential outside the box infinitely high (and
repulsive) so that the electron wave function must vanish at the boundaries (

(a) Find the allowed values of
*
k
_{
x
}
*
for
the two cases.

(b) If one plots the allowed values of
*
k
_{
x
}
*
as points on a line in

(c) If all the states are filled by the
*
N
*
electrons (i.e. at absolute zero,
*
T
*
= 0), what is the energy of the
highest energy electron in the two cases? Give your answer in terms of N,
*
Ä§
*
,
*
m
*
, and
*
a
*
. Please note that the answers should be identical!

(d) Derive the density of states in energy
*
g
*
(
*
E
*
)
for the two cases, and show that they are identical.