Consider the conduction electrons of
a one-dimensional metal (length
*
a
*
) and a two-dimensional metal (length
*
a
*
,
width
*
b
*
) at absolute zero. Approximate the electrons as free particles.

(a) Assuming periodic boundary
conditions, find the allowed values of
*
k
*
_{
x
}
for the one-dimensional metal, and of
*
k
_{
x
}
*

(b) Write the expression for the
total energy of a single electron for the two cases in terms of,
*
ħ
*
,
*
m
*
, and the components of the
wavevector
.

(c) Given that there are N
electrons, calculate
*
k
_{
F
}
*
and

(d) Derive expressions for the
density of states in energy,
*
g
*
(
*
E
*
), for
the two cases.

Draw a qualitatively correct picture
of
*
g
*
(
*
E
*
) vs
*
E
*
to illustrate the
contrast between
*
g
*
(
*
E
*
) for one-, two-, and three-dimensional
metals.

(e) Use
*
g
*
(
*
E
*
)
to find the average electron energy at absolute zero (
*
T
*
= 0) in terms of
*
E
_{
F
}
*
for the one- and two-dimensional solids.