306
9 Semiclassical Theory: Path Integrals
¯
J = −
∂ 2 R
∂x 0i ∂x j
−1
.
(9.54)
As time, τ , evolves, there will be discrete times, τ = t 0 , t 1 , . . ., when the matrix,
¯
J −1 , becomes singular. These times are called conjugate to the initial time, t 0 . The
conjugate points occur when neighboring trajectories intersect the main trajectories.
When this happens, one or more eigenvalues of the matrix ¯
J become zero. If, for
example, the rank of ¯
J is reduced by two, then there are two conjugate points at
the same time. There can be no more than d conjugate points at a given time. If all
trajectories intersect at a point, that point is called a focus (Gutzwiller 1990). Below
we illustrate these ideas with an example.
Example 9.2 (Conjugate Points for a Harmonic Oscillator)
Consider a harmonic oscillator with Hamiltonian H =
1
2 (p 2 + q 2 ) = e. The solutions to
Hamilton’s equations at time t are q t = q 0 cos(t) + p 0 sin(t) and p t = p 0 cos(t) − q 0 sin(t),
where q 0 and p 0 are the position and momenta, respectively, at time t = 0. Hamilton’s
principal function is
R(q 0 , 0; q t , t) =
1
2 sin(t)
[(q
2
t + q
2
0 ) cos(t) − 2q 0 q t ].
(9.55)
Let us consider two neighboring trajectories that start at the same point, q 0 , but have
slightly different momenta (and energies). We can write q t = q 0 cos(t) + p 0 sin(t) and
q
t = q 0 cos(t) + (p 0 + ) sin(t), where is small. If we plot q t versus t for these two
trajectories, we find that they cross at times t = nπ , where n is an integer. These crossings
are the conjugate points. In fact, there are an infinite number of different trajectories, each
with different initial momentum, and all of them cross at the same time, t = nπ . Thus each
conjugate point is a focus. Let us now note that
∂ 2 R
∂q 0 ∂q t
=
−1
sin(t)
.
(9.56)
Thus, at conjugate points,
∂ 2 R
∂q 0 ∂qt is singular and its inverse, J =
∂qt
∂p 0
= sin(t), goes to zero.
Each time J goes to zero, it changes sign, and this changes the phase of the Green’s function
by a factor (e iπ ) −1/2 = e −iπ/2 . If we write the Green’s function G(q 0 , 0 + ; q τ , τ + ), then κ
is the number of conjugate points between time t = 0 + and time t = τ + .
It is interesting to note that the determinant
D R = Det
∂ 2 R
∂x 0i ∂x j
(9.57)
that appears in Eq. (9.50) is sometimes called the density of classical paths because,
as we shall now show, it satisfies a continuity equation (Choquard 1955) and is
infinite at a focus. Let
P mn =
∂ 2 R
∂x m ∂x 0n
(9.58)
9 Semiclassical Theory: Path Integrals
¯
J = −
∂ 2 R
∂x 0i ∂x j
−1
.
(9.54)
As time, τ , evolves, there will be discrete times, τ = t 0 , t 1 , . . ., when the matrix,
¯
J −1 , becomes singular. These times are called conjugate to the initial time, t 0 . The
conjugate points occur when neighboring trajectories intersect the main trajectories.
When this happens, one or more eigenvalues of the matrix ¯
J become zero. If, for
example, the rank of ¯
J is reduced by two, then there are two conjugate points at
the same time. There can be no more than d conjugate points at a given time. If all
trajectories intersect at a point, that point is called a focus (Gutzwiller 1990). Below
we illustrate these ideas with an example.
Example 9.2 (Conjugate Points for a Harmonic Oscillator)
Consider a harmonic oscillator with Hamiltonian H =
1
2 (p 2 + q 2 ) = e. The solutions to
Hamilton’s equations at time t are q t = q 0 cos(t) + p 0 sin(t) and p t = p 0 cos(t) − q 0 sin(t),
where q 0 and p 0 are the position and momenta, respectively, at time t = 0. Hamilton’s
principal function is
R(q 0 , 0; q t , t) =
1
2 sin(t)
[(q
2
t + q
2
0 ) cos(t) − 2q 0 q t ].
(9.55)
Let us consider two neighboring trajectories that start at the same point, q 0 , but have
slightly different momenta (and energies). We can write q t = q 0 cos(t) + p 0 sin(t) and
q
t = q 0 cos(t) + (p 0 + ) sin(t), where is small. If we plot q t versus t for these two
trajectories, we find that they cross at times t = nπ , where n is an integer. These crossings
are the conjugate points. In fact, there are an infinite number of different trajectories, each
with different initial momentum, and all of them cross at the same time, t = nπ . Thus each
conjugate point is a focus. Let us now note that
∂ 2 R
∂q 0 ∂q t
=
−1
sin(t)
.
(9.56)
Thus, at conjugate points,
∂ 2 R
∂q 0 ∂qt is singular and its inverse, J =
∂qt
∂p 0
= sin(t), goes to zero.
Each time J goes to zero, it changes sign, and this changes the phase of the Green’s function
by a factor (e iπ ) −1/2 = e −iπ/2 . If we write the Green’s function G(q 0 , 0 + ; q τ , τ + ), then κ
is the number of conjugate points between time t = 0 + and time t = τ + .
It is interesting to note that the determinant
D R = Det
∂ 2 R
∂x 0i ∂x j
(9.57)
that appears in Eq. (9.50) is sometimes called the density of classical paths because,
as we shall now show, it satisfies a continuity equation (Choquard 1955) and is
infinite at a focus. Let
P mn =
∂ 2 R
∂x m ∂x 0n
(9.58)