2024-08-02 learning physics - liouvilles theorem, incompressibility, and the schrodinger vs heisenberg picture

#update
parents:: learning physics
daily note: 2024-08-02

liouvilles theorem, incompressibility, and the schrodinger vs heisenberg pictures

• classically, hamiltonian mechanics lets us derive the laws of motion from the super compact form of a single, "total energy" like function.

• liouvilles theorem tells us how a distribution of particles in phase space (i.e. represented as points with position and momenta as its degrees of freedom) changes with time

  • $\rho(X,t)$: distribution depending on time, where $X$ is a coordinate
    $$\frac{\del\rho}{\del t}={H,\rho} = \sum_j \frac{\del H}{\del p_j}\frac{\del\rho}{\del q_j} - \frac{\del H}{\del q_j}\frac{\del\rho}{\del p_j}$$
  • The derivation has a very nice physical intuition behind it: just consider a tiny, axis parallel box around a point, and ask how much mass enters in each axis direction
  • This is a consequence of the observation the Hamilton's relations imply that the phase space acts like an incompressible fluid. That is, the divergence of the flow is 0.
  • What's nice is that it implies that the equilibria here are distributions over equi-energy curves
    • Analogous (I think) to how the equilibria of von Neumann equation are the density matrices which are diagonalizable in the

• There's what I think is a sort of dual view to this same picture. Instead of looking at how states change w.r.t. time, one can ask how measurements change with time.

  • $A({p_j,q_j})$: measurement function
    $$\frac{\del A}{\del t}={A,H}$$
  • Derivation for this follows from just chain rule.
  • But what's kind of nice is that this is just the negative of what happens in the distribution setting. That is, the change in a measurement with time can be thought of as how much "measurement value" is leaving this point (as ${A,H}=-{H,A}$ where ${H,A}$ would be how much is entering).

• Everything here is ultimately due to the incompressibility of the phase space "fluid dynamics"

  • [?] In what sense is incompressible flow like a generalization of a unitary transform?
    • Unitaries definitely preserve volumes
    • But incompressible flows don't necessarily have to preserve anything like angles or distances
    • Hm... interesting. So quantum systems must evolve unitarily, yet classical systems just have to evolve incompressibly?
    • Or maybe there's some difference here arising from us having lifted the problem to $2n$ variables?
  • incompressibility enables dual views of measurement evolution vs state evolution

• [?] There's some aesthetic similarity here to the cauchy riemann relations. Is there a formal relation?

• [?] I want to figure out more formally how this relates to the schrodinger vs heisenberg pictures in the quantum setting.