Koopman–von Neumann classical mechanics
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The Koopman–von Neumann (KvN) theory is a description of classical mechanics as an operatorial theory similar to quantum mechanics, based on a Hilbert space of complex, square-integrable wavefunctions. As its name suggests, the KvN theory is loosely related to work by Bernard Koopman and John von Neumann in 1931 and 1932, respectively.[1][2][3] As explained in this entry, however, the historical origins of the theory and its name are complicated.
History
Ergodic theory
The origins of the Koopman–von Neumann theory are tightly connected with the rise[when?] of ergodic theory as an independent branch of mathematics, in particular with Boltzmann's ergodic hypothesis.
In 1931 Koopman and André Weil[citation needed] independently observed that the phase space of the classical system can be converted into a Hilbert space. According to this formulation, functions representing physical observables become vectors, with an inner product defined in terms of a natural integration rule over the system's probability density on phase space. This reformulation makes it possible to draw interesting conclusions about the evolution of physical observables from Stone's theorem, which had been proved shortly before. This finding inspired von Neumann to apply the novel formalism to the ergodic problem. Subsequently, he published several seminal results in modern ergodic theory, including the proof of his mean ergodic theorem.
Historical Misattribution
The Koopman-von Neumann theory is often used today to refer to a reformulation of classical mechanics in which a classical system's probability density on phase space is expressed in terms of an underlying wavefunction, meaning that the vectors of the classical Hilbert space are wavefunctions, rather than physical observables.
This approach did not originate with Koopman or von Neumann, for whom the classical Hilbert space consisted of physical observables, rather than wavefunctions. Indeed, as noted in 1961 by Thomas F. Jordan and E. C. George Sudarshan:
It was shown by Koopman how the dynamical transformations of classical mechanics, considered as measure preserving transformations of the phase space, induce unitary transformations on the Hilbert space of functions which are square integrable with respect to a density function over the phase space. This Hilbert space formulation of classical mechanics was further developed by von Neumann. It is to be noted that this Hilbert space corresponds not to the space of state vectors in quantum mechanics but to the Hilbert space of operators on the state vectors (with the trace of the product of two operators being chosen as the scalar product).[4]
The practice of expressing classical probability distributions on phase space in terms of underlying wavefunctions goes back at least to the 1952–1953 work of Mário Schenberg on statistical mechanics.[5][6] This method was independently developed several more times, by Angelo Loinger in 1962,[7] by Giacomo Della Riccia and Norbert Wiener in 1966,[8] and by E. C. George Sudarshan himself in 1976.[9]
The name "Koopman-von Neumann theory" for representing classical systems based on Hilbert spaces made up of classical wavefunctions is therefore an example of Stigler's law of eponymy. This misattribution appears to have first shown up in a paper by Danilo Mauro in 2002.[10]
Definition and dynamics
Derivation starting from the Liouville equation
In the approach of Koopman and von Neumann (KvN), dynamics in
The KvN wavefunction is postulated to evolve according to exactly the same Liouville equation as the classical probability density. From this postulate it can be shown that indeed probability density dynamics is recovered.
In classical statistical mechanics, the probability density (with respect to
- Remark
- The last step of this derivation relies on the classical Liouville operator containing only first-order derivatives in the coordinate and momentum; this is not the case in quantum mechanics where the Schrödinger equation contains second-order derivatives.
Derivation starting from operator axioms
Conversely, it is possible to start from operator postulates, similar to the Hilbert space axioms of quantum mechanics, and derive the equation of motion by specifying how expectation values evolve.[14]
The relevant axioms are that as in quantum mechanics (i) the states of a system are represented by normalized vectors of a complex Hilbert space, and the observables are given by self-adjoint operators acting on that space, (ii) the expectation value of an observable is obtained in the manner as the expectation value in quantum mechanics, (iii) the probabilities of measuring certain values of some observables are calculated by the Born rule, and (iv) the state space of a composite system is the tensor product of the subsystem's spaces.
The above axioms (i) to (iv), with the
- ,
- The expectation value of an observable at time is
- The probability that a measurement of an observable at time yields is , where . (This axiom is an analogue of the Born rule in quantum mechanics.[15])
- (see Tensor product of Hilbert spaces).
These axioms allow us to recover the formalism of both classical and quantum mechanics.[14] Specifically, under the assumption that the classical position and momentum operators commute, the Liouville equation for the KvN wavefunction is recovered from averaged Newton's laws of motion. However, if the coordinate and momentum obey the canonical commutation relation, the Schrödinger equation of quantum mechanics is obtained.
We begin from the following equations for expectation values of the coordinate x and momentum p
aka, Newton's laws of motion averaged over ensemble. With the help of the operator axioms, they can be rewritten as
Notice a close resemblance with Ehrenfest theorems in quantum mechanics. Applications of the product rule leads to
into which we substitute a consequence of Stone's theorem and obtain
Since these identities must be valid for any initial state, the averaging can be dropped and the system of commutator equations for the unknown is derived
-
(commutator eqs for L)
Assume that the coordinate and momentum commute . This assumption physically means that the classical particle's coordinate and momentum can be measured simultaneously, implying absence of the uncertainty principle.
The solution cannot be simply of the form because it would imply the contractions and . Therefore, we must utilize additional operators and obeying
-
(KvN algebra)
The need to employ these auxiliary operators arises because all classical observables commute. Now we seek in the form . Utilizing KvN algebra, the commutator eqs for L can be converted into the following differential equations:[14][16]
Whence, we conclude that the classical KvN wave function evolves according to the Schrödinger-like equation of motion
-
(KvN dynamical eq)
Let us explicitly show that KvN dynamical eq is equivalent to the classical Liouville mechanics.
Since and commute, they share the common
-
(xp eigenvec)
with the resolution of the identity Then, one obtains from equation (KvN algebra)
-
(KvN dynamical eq in xp)
The quantity is the probability amplitude for a classical particle to be at point with momentum at time . According to the axioms above, the probability density is given by . Utilizing the identity
-
(Liouville eq)
Moreover, according to the operator axioms and (xp eigenvec),
Projecting KvN dynamical eq onto the common eigenvector of the operators and (i.e., -representation), one obtains classical mechanics in the doubled configuration space,[20] whose generalization leads [20] [21] [22] [23] [24] to the
As in the derivation of classical mechanics, we begin from the following equations for averages of coordinate x and momentum p
With the help of the operator axioms, they can be rewritten as
These are the Ehrenfest theorems in quantum mechanics. Applications of the product rule leads to
into which we substitute a consequence of Stone's theorem
where was introduced as a normalization constant to balance dimensionality. Since these identities must be valid for any initial state, the averaging can be dropped and the system of commutator equations for the unknown quantum generator of motion are derived
Contrary to the case of classical mechanics, we assume that observables of the coordinate and momentum obey the canonical commutation relation . Setting , the commutator equations can be converted into the differential equations [14][16]
whose solution is the familiar quantum Hamiltonian
Whence, the Schrödinger equation was derived from the Ehrenfest theorems by assuming the canonical commutation relation between the coordinate and momentum. This derivation as well as the derivation of classical KvN mechanics shows that the difference between quantum and classical mechanics essentially boils down to the value of the commutator .
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.[12]
KvN vs Liouville mechanics
The KvN dynamical equation (KvN dynamical eq in xp) and Liouville equation (Liouville eq) are first-order linear partial differential equations. One recovers Newton's laws of motion by applying the method of characteristics to either of these equations. Hence, the key difference between KvN and Liouville mechanics lies in weighting individual trajectories: Arbitrary weights, underlying the classical wave function, can be utilized in the KvN mechanics, while only positive weights, representing the probability density, are permitted in the Liouville mechanics (see this scheme).
Quantum analogy
Being explicitly based on the Hilbert space language, the KvN classical mechanics adopts many techniques from quantum mechanics, for example,
The KvN approach is fruitful in studies on the
-
The time evolution of the classical KvN wave function for the Morse potential: . Black dots are classical particles followingNewton's law of motion. The solid lines represent the level set of the Hamiltonian. This video illustrates the fundamental difference between KvN and Liouville mechanics.
-
Quantum counterpart of the classical KvN propagation on the left: The Wigner function time evolution of the Morse potential in atomic units (a.u.). The solid lines represent the level set of the underlying Hamiltonian. Note that the same initial condition used for this quantum propagation as well as for the KvN propagation on the left.
See also
- Classical mechanics
- Statistical mechanics
- Liouville's theorem
- Quantum mechanics
- Phase space formulation of quantum mechanics
- Wigner quasiprobability distribution
- Dynamical systems
- Ergodic theory
References
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Mauro, D. (2002). "Topics in Koopman–von Neumann Theory". arXiv:quant-ph/0301172. PhD thesis, Università degli Studi di Trieste.
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- ^ Blokhintsev, D.I. (1940). "The Gibbs Quantum Ensemble and its Connection with the Classical Ensemble". J. Phys. U.S.S.R. 2 (1): 71–74.
- ^ Blokhintsev, D.I.; Nemirovsky, P (1940). "Connection of the Quantum Ensemble with the Gibbs Classical Ensemble. II". J. Phys. U.S.S.R. 3 (3): 191–194.
- ^ Blokhintsev, D.I.; Dadyshevsky, Ya. B. (1941). "On Separation of a System into Quantum and Classical Parts". Zh. Eksp. Teor. Fiz. 11 (2–3): 222–225.
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Further reading
- Mauro, D. (2002). "Topics in Koopman–von Neumann Theory". arXiv:quant-ph/0301172. PhD thesis, Università degli Studi di Trieste.
- H.R. Jauslin, D. Sugny, Dynamics of mixed classical-quantum systems, geometric quantization and coherent states[permanent dead link], Lecture Note Series, IMS, NUS, Review Vol., August 13, 2009
- The Legacy of John von Neumann (Proceedings of Symposia in Pure Mathematics, vol 50), edited by James Glimm, John Impagliazzo, Isadore Singer. — Amata Graphics, 2006. — ISBN 0821842196
- U. Klein, From Koopman–von Neumann theory to quantum theory, Quantum Stud.: Math. Found. (2018) 5:219–227.[1]