# Is every state an eigenstate of an observable?

#### atyy

atyy,

Not sure whether I should get involved in this thread, but I get the feeling you're looking at all this in a cart-before-horse manner...

To construct a quantum model of a particular physical scenario, one starts with a physically-meaningful dynamical group (which maps solutions of the equations of motion into other solutions), and attempts to construct a unitary representation thereof (i.e., a representation of the group as operators on a Hilbert space, or generalization thereof). One way to do this is to find the group's Casimirs, and a maximal commuting set of generators within the dynamical algebra of group generators, determine their joint spectrum, and construct a Hilbert state space from that spectrum (verifying of course that the rest of the generators and the whole group are also satisfactorily represented on the state space so-constructed).

[Edit: In more general cases, it may be more physically-sensible to construct a POVM on the spectrum, since a POVM is closer to what real apparatus does. Cf. http://en.wikipedia.org/wiki/POVM]

So what I'm trying to say is: the dynamical group comes first, followed by the states which are (in a sense) derived from the dynamical group and the requirement of unitary representation. However, in this thread, it seems you're trying to do it the other way around, therefore encountering some difficulty with the physical interpretation.

BTW, Ballentine chapters 1-3 give a better introduction to this way of looking at things than the Preskill notes (imho).
If one follows this approach, is every state also the eigenvector of some "mathematical observable" (ie. is the condition for a mathematical observable the same as in micromass's approach, that it just has to be self-adjoint, or are there additional conditions from the dynamical group)?

#### strangerep

If one follows this approach, is every state also the eigenvector of some "mathematical observable"
Given a state, one can always find a rank-1 operator, as others have pointed out above. That wasn't really the reason for my interjection though. I was really just curious why you're quite intensely focused on this question? Is there a specific case you have in mind?

(ie. is the condition for a mathematical observable the same as in micromass's approach, that it just has to be self-adjoint, or are there additional conditions from the dynamical group)?
The generators of the dynamical group are usually represented as self-adjoint operators (or perhaps a suitably extended version of "self-adjoint" in the case of rigged Hilbert space). More generally, the finite group elements should be represented as unitary operators.

(Strictly speaking, one can construct more general cases than Hermitian or self-adjoint, such as in the case of unstable systems -- there one can have non-Hermitian Hamiltonians and the imaginary part of the energy eigenvalues correspond to decay lifetimes.)