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I've been reading about Liouville's equation and am now trying to understand Von Neumann's equation, which seems to be more or less the quantum mechanical version. Both cases involve an ensemble of states of varying statistical probability being described by either a probability distribution function (classical) or density operator (quantum).

The difference I'm wondering about is that the classical probability function is continuous i.e. there are an infinite number of possible states. In quantum however, The density operator I've been reading about is defined as:

$\rho=\sum_{j}p_j|\psi_j\rangle\langle\psi_j|$

where $|\psi_j\rangle$ is a pure state and $p_j$ its associated probability. Here we are dealing with discrete states each with a particular probability.

Is it possible to have a continuous distribution in quantum as in classical? e.g. if a pure state is represented by $|\psi\rangle=\sum c_n|n\rangle$, can we have some function of all the $c_n$ that gives a probability distribution across Hilbert space?

I understand the notion of a statistical ensemble being used to represent uncertainty about a state, e.g. in classical mechanics one might say that a particle is at a particular point in phase space with some uncertainty, given by the probability distribution function. This distribution can then be evolved with Liouville's equation. This idea does not seem to carry over into von Neumann's equation, since the nature of the density operator implies one has to know a particular set of possible states. However, my understanding is that there are infinite possible states, described by different combinations of $c_n$.

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  • $\begingroup$ Maybe you are thinking of Wigner functions en.m.wikipedia.org/wiki/Wigner_quasiprobability_distribution, which are continuous functions (of $x$ and $p$ in the simplest cases) that can be used as (quasi)probability densities for a quantum system. $\endgroup$ – ZeroTheHero May 26 '18 at 12:48
  • $\begingroup$ I had come across that. It certainly makes the most sense to compare quantum to classical with a phase space representation, however I'm still wondering if there could be such probability distributions in Hilbert space. $\endgroup$ – Dogtard May 27 '18 at 5:46

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