What is the phase space volume in terms of angular momentum?
Answers
Answered by
0
Given a rigid rotor Hamiltonian, defined along the principle axes as
H=∑i=13L2i2IiH=∑i=13Li22Ii
say we would like to compute the classical partition function of this system. Is the proper phase space density give as (?)
1h3∭∞−∞dL1dL2dL3∫2π0dϕ∫π0sinθ dθ1h3∭−∞∞dL1dL2dL3∫02πdϕ∫0πsinθ dθ
I see the dimensionality seems fine, and we are integrating over all possible momentums and rotational positions. Any insight on how we may transform from f(p⃗ ,q⃗ )f(p→,q→) to f(L⃗ ,q⃗ ′)f(L→,q→′) may be helpful.
H=∑i=13L2i2IiH=∑i=13Li22Ii
say we would like to compute the classical partition function of this system. Is the proper phase space density give as (?)
1h3∭∞−∞dL1dL2dL3∫2π0dϕ∫π0sinθ dθ1h3∭−∞∞dL1dL2dL3∫02πdϕ∫0πsinθ dθ
I see the dimensionality seems fine, and we are integrating over all possible momentums and rotational positions. Any insight on how we may transform from f(p⃗ ,q⃗ )f(p→,q→) to f(L⃗ ,q⃗ ′)f(L→,q→′) may be helpful.
Answered by
3
Hello mate here is your answer.
Accidental FourierTransform Because it's a rigid rotor, i.e. described by a point in R3R3 restricted to a sphere of fixed radius. Thus r=x2+y2+z2−−−−−−−−−−√r=x2+y2+z2is constrained, and its momentum vanishes. To fix this you need to go back to the lagrangian and implement the constraint there before going to the hamiltonian formulation.
Hope it helps you.
Accidental FourierTransform Because it's a rigid rotor, i.e. described by a point in R3R3 restricted to a sphere of fixed radius. Thus r=x2+y2+z2−−−−−−−−−−√r=x2+y2+z2is constrained, and its momentum vanishes. To fix this you need to go back to the lagrangian and implement the constraint there before going to the hamiltonian formulation.
Hope it helps you.
Similar questions