We consider a cartesian momentum space in coordinates along axes
. The
vector position is momentum space is written
![]() | (A.208) |
We consider the two following curvilinear systems:
The coordinates are defined on the space
-∞≤ p∥ | < ∞ | (A.209) |
0 ≤ p⊥ | < ∞ | (A.210) |
0 ≤ φ | < 2π | (A.211) |
and is related to by
p∥ | = pz | (A.212) | |
p⊥ | = ![]() | ||
φ | = arctan![]() ![]() ![]() |
which is inverted to
px | = p⊥cosφ | (A.213) | |
py | = p⊥sinφ | ||
pz | = p∥ |
The position vector in momentum space then becomes
![]() | (A.214) |
where we define a local orthonormal basis as
![]() | = ![]() | (A.215) |
![]() | = cosφ![]() ![]() | (A.216) |
![]() | = ![]() ![]() ![]() ![]() | (A.217) |
The covariant vector basis is defined in (A.1), which becomes here
e∥ | = ![]() ![]() | (A.218) |
e⊥ | = ![]() ![]() | (A.219) |
eφ | = ![]() ![]() ![]() | (A.220) |
so that we have the covariant basis
![]() | (A.221) |
the scaling factors
![]() | (A.222) |
and the normalized tangent basis
![]() | (A.223) |
The Contravariant vector basis is defined in (A.9), which becomes here
e∥ | = ∇p
∥ = ![]() | (A.224) |
e⊥ | = ∇p
⊥ = ![]() | (A.225) |
eφ | = ∇φ = ![]() | (A.226) |
The relations (A.10-A.11) are here readily verified. The normalized reciprocal basis is
![]() | (A.227) |
which here coincides with the normalized tangent basis, since both bases are orthogonal.
They are defined in (A.12) and become here
gij | = ![]() | (A.228) | |
gij | = ![]() |
As a result
![]() | (A.229) |
and the Jacobian is
![]() | (A.230) |
dl![]() | = dp∥ | (A.231) |
dl![]() | = dp⊥ | (A.232) |
dl![]() | = p⊥dφ | (A.233) |
dS![]() | = p⊥dp⊥dφ![]() | (A.234) |
dS![]() | = p⊥dp∥dφ![]() | (A.235) |
dS![]() | = dp∥dp⊥![]() | (A.236) |
![]() | (A.237) |
They are defined in (A.49) and are all zero here except
![]() |
![]() | (A.238) |
![]() | (A.239) |
![]() | (A.240) |
![]() ![]() | = ![]() ![]() ![]() ![]() ![]() ![]() | (A.241) |
![]() ![]() | = ![]() ![]() ![]() ![]() ![]() | (A.242) |
![]() ![]() | = ![]() ![]() ![]() ![]() | (A.243) |
The coordinates are defined on the space
0 ≤ p | < ∞ | (A.244) |
- 1 ≤ ξ | < 1 | (A.245) |
0 ≤ φ | < 2π | (A.246) |
and is related to by
p | = ![]() | (A.247) | |
ξ | = ![]() | (A.248) | |
φ | = arctan![]() ![]() ![]() |
which is inverted to
px | = p![]() | (A.249) | |
py | = p![]() | ||
pz | = pξ |
Note that we have the following transformation from to
p | = ![]() | (A.250) | |
ξ | = ![]() |
which is inverted to
p∥ | = pξ | (A.251) | |
p⊥ | = p![]() |
The position vector in momentum space then becomes
![]() | (A.252) |
where we define a local orthonormal basis as
![]() | = ![]() ![]() ![]() | (A.253) |
![]() | = ![]() ![]() ![]() ![]() ![]() | (A.254) |
![]() | = -sinφ![]() ![]() | (A.255) |
The covariant vector basis is defined in (A.1), which becomes here
ep | = ![]() ![]() | (A.256) |
eξ | = ![]() ![]() ![]() ![]() | (A.257) |
eφ | = ![]() ![]() ![]() ![]() | (A.258) |
so that we have the covariant basis
![]() | (A.259) |
the scaling factors
![]() | (A.260) |
and the normalized tangent basis
![]() | (A.261) |
The Contravariant vector basis is defined in (A.9), which becomes here
ep | = ∇p = ![]() ![]() | (A.262) |
eξ | = ∇ξ = ![]() ![]() ![]() | (A.263) |
eφ | = ∇φ = ![]() ![]() ![]() | (A.264) |
The relations (A.10-A.11) are here readily verified. The reciprocal basis is
![]() | (A.265) |
and the normalized reciprocal basis is
![]() | (A.266) |
which here coincides with the normalized tangent basis, since both bases are orthogonal.
They are defined in (A.12) and become here
gij | = ![]() | (A.267) | |
gij | = ![]() |
As a result
![]() | (A.268) |
and the Jacobian is
![]() | (A.269) |
dl![]() | = dp | (A.270) |
dl![]() | = ![]() | (A.271) |
dl![]() | = p![]() | (A.272) |
dS![]() | = p2dξdφ![]() | (A.273) |
dS![]() | = -p![]() ![]() | (A.274) |
dS![]() | = ![]() ![]() | (A.275) |
![]() | (A.276) |
They are defined in (A.49) and are all zero here except
![]() | (A.277) |
![]() | (A.278) |
![]() ![]() | = ![]() ![]() ![]() ![]() ![]() ![]() | (A.279) | |
![]() ![]() | = ![]() ![]() ![]() ![]() ![]() ![]() | ||
![]() ![]() | = -![]() ![]() ![]() ![]() ![]() |