314
9 Semiclassical Theory: Path Integrals
where ¯
ρ(e) is the average density of states and is defined as
¯
ρ(e) = −
1
π
Im
dx 0 ¯
G(x 0 , x 0 ; e)
,
(9.91)
and ρ osc (e) gives oscillations about the average density of states and is defined as
ρ osc (e) = −
1
π
Im
dx 0 G osc (x 0 , x 0 ; e)
.
(9.92)
In Eqs. (9.91), ¯
G(x 0 , x 0 ; e) is the contribution to the Green’s function from very
short orbits and G osc (x 0 , x 0 ; e) contains contributions from orbits of finite length.
The average density of states can be found in two ways. It can be computed
directly from the energy Green’s function, G(x 0 , x 0 ; e), or it can be computed by
counting states in phase space, as we now show. Let us consider a system with d
degrees of freedom. We can estimate the average number of energy levels, ¯
N(e),
with energy less than e. A single quantum state has phase space volume h d (h
is Planck’s constant). The total volume of phase space with energy less than e is
given by
=
d
d pd
d q q(e − H (p, q)).
(9.93)
Thus, the average number of states (and therefore energy levels if we assume no
degeneracy) with energy less than e is
¯
N(e) =
(e)
h d ,
(9.94)
and the average density of states in the neighborhood of energy e is
¯
ρ(e) =
1
h d
dd
de
.
(9.95)
In the examples below, we illustrate these ideas for a free particle and for a particle
in a potential well.
Example 9.4 (Density of States for a Free Particle)
If we combine Eqs. (9.20) and (9.64), the energy Green’s function for a free particle can be
written
G(x 0 , x; e) =
m
2πi ¯
h
∞
0
dt
1
√
t
exp
i
e
¯
h
t +
im(x − x 0 ) 2
2 ¯
h(t − t 0 )
=
1
i ¯
h
m
2e
exp
2i
e
¯
h
|x − x 0 |
.
(9.96)
9 Semiclassical Theory: Path Integrals
where ¯
ρ(e) is the average density of states and is defined as
¯
ρ(e) = −
1
π
Im
dx 0 ¯
G(x 0 , x 0 ; e)
,
(9.91)
and ρ osc (e) gives oscillations about the average density of states and is defined as
ρ osc (e) = −
1
π
Im
dx 0 G osc (x 0 , x 0 ; e)
.
(9.92)
In Eqs. (9.91), ¯
G(x 0 , x 0 ; e) is the contribution to the Green’s function from very
short orbits and G osc (x 0 , x 0 ; e) contains contributions from orbits of finite length.
The average density of states can be found in two ways. It can be computed
directly from the energy Green’s function, G(x 0 , x 0 ; e), or it can be computed by
counting states in phase space, as we now show. Let us consider a system with d
degrees of freedom. We can estimate the average number of energy levels, ¯
N(e),
with energy less than e. A single quantum state has phase space volume h d (h
is Planck’s constant). The total volume of phase space with energy less than e is
given by
=
d
d pd
d q q(e − H (p, q)).
(9.93)
Thus, the average number of states (and therefore energy levels if we assume no
degeneracy) with energy less than e is
¯
N(e) =
(e)
h d ,
(9.94)
and the average density of states in the neighborhood of energy e is
¯
ρ(e) =
1
h d
dd
de
.
(9.95)
In the examples below, we illustrate these ideas for a free particle and for a particle
in a potential well.
Example 9.4 (Density of States for a Free Particle)
If we combine Eqs. (9.20) and (9.64), the energy Green’s function for a free particle can be
written
G(x 0 , x; e) =
m
2πi ¯
h
∞
0
dt
1
√
t
exp
i
e
¯
h
t +
im(x − x 0 ) 2
2 ¯
h(t − t 0 )
=
1
i ¯
h
m
2e
exp
2i
e
¯
h
|x − x 0 |
.
(9.96)