A.6 Action-Angle Coordinates
401
and in terms of the transition between classical and quantum mechanics. The action
variable is an adiabatic invariant, and therefore it is a suitable variable to quantize.
In quantum systems, transitions occur in discrete units of ¯
h. If an external field is
applied that is sufficiently weak and slow, it is possible that no changes will occur
in the quantum system because the field is unable to cause a change in the action
of the system by a discrete amount, ¯
h. In the transition from classical to quantum
mechanics, it is the action variables that are quantized in units of ¯
h because they are
adiabatic invariants and have a similar behavior classically (Born 1960; Landau and
Lifshitz 1976). If a slowly varying weak external field (with period much longer than
and incommensurate with the natural period of the system) is applied to a classical
periodic system, the action remains unchanged, whereas the rate of change of the
energy is proportional to the rate of change of the applied field.
Let us consider a one degree of freedom system described in terms of the
usual momentum and position coordinates, (p, q), with Hamiltonian H (p, q).
We introduce a generating function, S(q, J ), which allows us to transform from
coordinates (p, q) to action-angle coordinates, (J, θ ), via the equations
p =
∂S
∂q
J
(A.15)
and
θ =
∂S
∂J
q
.
(A.16)
The generating function is path-independent, so
∂p
∂J
q
=
∂
∂J
∂S
∂q
J
q
=
∂
∂q
∂S
∂J
q
J
=
∂θ
∂q
J
.
(A.17)
We require that H (p, q) = H(J ) so that J = constant and θ = ω(J )t + θ o ,
where ω =
∂H
∂J
and θ o is a constant. Now consider a differential change in S,
dS =
∂S
∂q
J
dq +
∂S
∂J
q
dJ . Find the change in S along a path of fixed J (and
therefore fixed energy), (dS) J =
∂S
∂q
J
dq. Then
S(q, J ) − S(q
, J ) =
S(q,J )
S(q ,J )
dS =
q
q
∂S
∂q
J
dq =
q
q
pdq.
(A.18)
We now define the action as
J =
1
2π
closedpath
pdq.
(A.19)
401
and in terms of the transition between classical and quantum mechanics. The action
variable is an adiabatic invariant, and therefore it is a suitable variable to quantize.
In quantum systems, transitions occur in discrete units of ¯
h. If an external field is
applied that is sufficiently weak and slow, it is possible that no changes will occur
in the quantum system because the field is unable to cause a change in the action
of the system by a discrete amount, ¯
h. In the transition from classical to quantum
mechanics, it is the action variables that are quantized in units of ¯
h because they are
adiabatic invariants and have a similar behavior classically (Born 1960; Landau and
Lifshitz 1976). If a slowly varying weak external field (with period much longer than
and incommensurate with the natural period of the system) is applied to a classical
periodic system, the action remains unchanged, whereas the rate of change of the
energy is proportional to the rate of change of the applied field.
Let us consider a one degree of freedom system described in terms of the
usual momentum and position coordinates, (p, q), with Hamiltonian H (p, q).
We introduce a generating function, S(q, J ), which allows us to transform from
coordinates (p, q) to action-angle coordinates, (J, θ ), via the equations
p =
∂S
∂q
J
(A.15)
and
θ =
∂S
∂J
q
.
(A.16)
The generating function is path-independent, so
∂p
∂J
q
=
∂
∂J
∂S
∂q
J
q
=
∂
∂q
∂S
∂J
q
J
=
∂θ
∂q
J
.
(A.17)
We require that H (p, q) = H(J ) so that J = constant and θ = ω(J )t + θ o ,
where ω =
∂H
∂J
and θ o is a constant. Now consider a differential change in S,
dS =
∂S
∂q
J
dq +
∂S
∂J
q
dJ . Find the change in S along a path of fixed J (and
therefore fixed energy), (dS) J =
∂S
∂q
J
dq. Then
S(q, J ) − S(q
, J ) =
S(q,J )
S(q ,J )
dS =
q
q
∂S
∂q
J
dq =
q
q
pdq.
(A.18)
We now define the action as
J =
1
2π
closedpath
pdq.
(A.19)