Koopman–von Neumann classical mechanics is a formulation of classical mechanics in terms of Hilbert space techniques that are normally associated with quantum mechanics. It's a fascinating example of how the mathematical machinery of quantum mechanics can be applied to classical systems.
This approach was introduced in a series of papers by Bernard Koopman and John von Neumann in the early 1930s. Here's a basic outline of the formulation:
1. States as Functions: In classical mechanics, the state of a system is given by a point in phase space, which is spanned by position $q$ and momentum $p$. In the Koopman–von Neumann formulation, states are instead described by complex-valued functions over phase space.
2. Observables: Physical quantities or observables are represented by self-adjoint operators on a Hilbert space of functions.
3. Evolution: The time evolution of the state is given by a unitary operator. This is in contrast to the standard Hamiltonian or Lagrangian formulations of classical mechanics, where the time evolution is described by deterministic trajectories in phase space.
4. Expectation Values: The expectation value of an observable in a given state can be computed using an integral over phase space, similar to how expectation values are computed in quantum mechanics.
5. Measurements: In the Hilbert space and operator formulation of classical mechanics, the Koopman von Neumann–wavefunction takes the form of a superposition of eigenstates, and measurement collapses the KvN wavefunction to the eigenstate which is associated the measurement result, in analogy to the wave function collapse of quantum mechanics. However, it can be shown that for Koopman–von Neumann classical mechanics _non-selective measurements_ leave the KvN wavefunction unchanged
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Author of the notes: Antonio J. Pan-Collantes
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