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So I began skimming L. Takhatajan's Quantum Mechanics For Mathematicians, and saw the mathematical formulation of QM that he uses (page 51). (The PDF file is available here.)

I've only taken a basic physics-style quantum mechanics class and was wondering if someone could explain how the operator definition of states (Axiom 3) corresponds to what is typically taught in a physics class (e.g. the $L^{2}$ functions).

I recently saw the post about the rigged Hilbert space formulation of QM and am also interested in how these axioms compare to Takhatajan's. Does Takhatajan's definition of states somehow include (or can be identified with) the eigenfunctions of the position and momentum operators?

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A3. Set of states S of a quantum system with a Hilbert space $\mathscr{H}$ consists of all positive (and hence self-adjoint) $M\in\mathscr{S}_1$ such that $\mathrm{Tr}\,M = 1$. Pure states are projection operators onto one-dimensional subspaces of $H$. For $\phi\in\mathscr{H}$, $||\phi|| = 1$, the corresponding projection is denoted by $P_\phi$. All other states are called mixed states.

The only thing that's different from what's taught in an introductory quantum mechanics class is the terminology, and it's actually not far from more advanced treatments.

In an introductory QM class, one would learn that the physical states are vectors in a Hilbert space over the field of complex numbers, being represented with kets $|\phi\rangle$. At the beginning one might get the impression that the only relevant Hilbert spaces are those that correspond to $L^2$-space of square-integrable functions over some domain (in the position basis, the wavefunctions $\phi(x) = \langle x|\phi\rangle$). But that is not really so, e.g., an electron state would have an ordinary square-integrable wavefunction and spinor components, so the Hilbert space is more complicated than that.

(Occasionally, one would see a footnote about "rigged Hilbert spaces", and that indicates that the vector definition is not quite right. The state kets are actually linear functionals, but as far as most physicists are concerned, that's just mathematician-talk for formally including things like the Dirac delta function and other distributions that are technically not functions. There are other issues that rigged Hilbert spaces address, e.g., restricting the space of states so that one can always apply momentum and position operators as much as necessary.)

Later, one learns that a more general system where the state is uncertain but statistically predictable is described by a density matrix: $$ \rho = \sum_k p_k |\phi_k\rangle\langle\phi_k|,$$ where the system has state ket $|\phi_k\rangle$ with probability $p_k$, thus being a mixed state. The condition of unit trace just means that the probabilities add to $1$, i.e., the system is certainly in some state. So one would say: a system is in a pure state whenever its density matrix is $\rho = |\phi\rangle\langle\phi|$ for some state ket $|\phi\rangle$. In other words, it has probability $1$ of being in state $|\phi\rangle$. See, for example, Sakurai's Modern Quantum Mechanics.

What Takhtajan is doing is simply taking "mixed states" as primary, rather than something introduced fairly long into basic QM. For any state vector, there is a one-dimensional subspace consisting of all scalar multiples of that vector. Thus, this part:

Pure states are projection operators onto one-dimensional subspaces of $H$.

just means that the density matrix is in the form $|\phi\rangle\langle\phi|$, which is the projection operator onto the subspace spanned by $|\phi\rangle$, exactly what one learns in physics class.

If this is unclear, recall from Euclidean vector algebra that $\hat{x}\cdot\vec{y}$ is the compomenent of $\vec{y}$ that lies along a unit vector $\hat{x}$. So the projection of $\vec{y}$ onto $\hat{x}$ is the vector $\hat{x}(\hat{x}\cdot\vec{y})$. In a general Hilbert space, the situation is analogous with the inner product replacing the dot product: the projection of $|\psi\rangle$ onto a (normalized) $|\phi\rangle$ is $|\phi\rangle\langle\phi|\psi\rangle$.

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Takhtajan is using terminology that's standard in quantum statistical mechanics, but slightly out of place in the usual physics-of-atoms-and-molecules introduction. What he's calling a state in axiom 3 is what physicists usually call a density matrix.

He's using the term pure state for a state in the more usual sense. If you have a vector $|v\rangle \in \mathcal{H}$, you get a pure state by constructing the projection operator $P = |v\rangle\langle v|$. Conversely, if you have a projection operator $P$ on to a 1-dimensional subspace $L \subset \mathcal{H}$, you can pick a vector $|v\rangle \in L$ in this subpace. These two constructions are inverse to one another, up to rescaling $|v\rangle$ by a non-zero complex number (which doesn't change the physical state anyways).

More general density matrices are probability distributions on the space of pure states. That's why they come up in quantum stat mech.

None of this has anything to do with rigged Hilbert spaces, which are just a language for dealing rigorously with unbounded operators and their eigenstuff. If you don't like rigged Hilbert spaces, you can say everything that needs to be said using resolutions of the identity or bounded approximations to any unbounded operators.

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Density matrices are equaivalence classes of probability distributions on the space of pure states. The same density matrix is representable in a huge variety of ways as an integral over pure states. –  Arnold Neumaier Nov 30 '12 at 16:43
    
Yes, that's true. –  user1504 Nov 30 '12 at 18:01
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No, Takhatajan in Axiom 3 on p. 51 is not concerned with rigged Hilbert spaces and eigenfunctions for the position and momentum operators.

Instead in Axiom 3, the operator $M\in {\cal S}^1$ is a density operator, also known as a mixed state, and usually denoted with a $\rho$.

Here the set ${\cal S}^1$ is the set of trace class operators.

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