4

I know that states in quantum mechanics are positive trace class operators acting on a separable complex Hilbert space $\mathcal H$ and having trace = 1. Specifically, pure states are one-dimensional orthogonal projections that of course can be identified with the one-dimensional subspace itself. So the set of pure spaces is the projective Hilbert space $\mathcal P(\mathcal H)$. My question is about the term referring to the Hilbert space $\mathcal H$ itself. Is it "state space", "structure space" or something else?

Wikipedia writes:

Specifically, in quantum mechanics a state space is a complex Hilbert space in which the possible instantaneous states of the system may be described by unit vectors.

I also found the term "state space" e.g. in this answer. However I haven't found this term in any quantum mechanics book except in the book of Claude Cohen-Tannoudji referred to in the Wikipedia article, not even the other book referenced there (Griffiths). Is "state space" a generally accepted term? References are welcome.

mma
  • 802

1 Answers1

8

I sometime use “state space”, but literally this name is wrong. The pure states are the rays of the Hilbert space, namely the unit vectors up to phases. And it is true only when all bounded selfadjoint operators represent observables. A better terminology would be “the Hilbert space of the system”.

Valter Moretti
  • 80,330