In quantum mechanics, the state space is a separable complex Hilbert space.
By definition, a Hilbert space is a complete inner product space.
The term complete means that any Cauchy sequence of elements (vectors) belonging to the Hilbert space converges to an element which also belongs to the space. In other words, completeness means that the limits of convergent sequences of elements belonging to the space are also elements of the space. Intuitively, we can say that Hilbert spaces have no “holes”.
If the state space is infinite-dimensional, we implicitly invoke its completeness every time we expand a state in terms of a complete set of eigenstates, such as the energy eigenstates or the eigenstates of another observable, since an infinite series of eigenstates is meant as the limit of the sequence of the respective partial sums when the number of terms tends to infinity. The sequence of partial sums is then a Cauchy sequence converging to the initial state, which must belong to the space.
Qualitatively, considering a convergent sequence of physical states, we expect that it converges to a physical state too, because it would be unphysical, by means of such a sequence, to end up at an unphysical state. For instance, assume that we perform a series of small changes to the state of a quantum system and suddenly we reach an unphysical state. This would be physically unacceptable. Thus, from a physical perspective, the completeness of the state space seems unavoidable.
However, looking in some of the so-called standard textbooks of quantum mechanics, particularly in Sakurai’s, Merzbacher’s, Gasiorowicz’s, and Griffiths’s, this essential property is either overlooked or just mentioned, and it is not highlighted properly.
It is an interesting point. I suppose one may look at the dual space. Perhaps convergence properties of mathematical objects like vector product, scalar product etc. can be easily proved in the dual space.
I believe that these questions it's better that you ask to your professor of quantum theory, because fundamental is not to get lost in the boundless bibliography these topics….this is the real danger. However, is well known that the Hilbert space formalism has some flaws (e.g. the eigenstates of position operators are not vectors of the Hilbert space, then we introduced the notion of rigged Hilbert spaces or we introduced the notion of affiliated operators for to manage unbounded operators in von Neumann formalism etc…) If you want to have a more mathematics rigorous context (no the good physics textbooks which you mentioned ) I suggest you the book of Blank et al “Hilbert space operators in quantum physics" it is a intermediate book between maths and physics, while for a quick overview on the quantum formalism see Emch's book “mathematical and conceptual foundations of 20th-century physics (part III)
Dear Carlo Pandiscia, thank you for your answer and suggestions. Actually, I'm not a student. I wish to discuss the question on a general basis, beyond students, professors, etc. I think that the property of completeness is so fundamental that every standard textbook of quantum mechanics, even if it is introductory, should present and analyse it properly to readers, even to physics undergraduates. The rigged Hilbert space you mention has to do with continuous sets, such as the position and momentum eigenstates, but not with the completeness of the space.
Dear Spiros, I'm sorry, probabily I did not understand your question, are you wondering why you use Hilbert spaces and not pre-Hilbert spaces?
Well the answer is simple...... the linear operator on the Hilbert space are easy to manage, while on the pre-hilbert space we lose many fundamental results e.g. the adjoint operator is not well-defined.
However it is not forbidden to consider pre-hilbertian spaces
(technical difficulties aside). Indeed with prof. accardi we use pre-hilbert spaces in mathematics formalism of the bose-einstein condensation, work that you can easily find on the web (i hope so)
Dear Carlo Pandiscia, no worries, thank you, I'll look at it.
Mathematical subtleties that are of relevance in quantum theory are indeed discussed in many books, though they are not usually suggested as references to undergrads. For instance, see the two volumes by Galindo and Pascual. They discuss finer points such as the deficiency index, self-adjoint extension, etc, in addition to a neat introduction to Hilbert spaces. Another accessible introduction is the book by Capri. There are, of course, the well known mathematical treatises by Reed-Simon, Preguvecki, etc.
Dear Colleagues,
IMHO, Spiros Konstantogiannis' question is part of a much larger question: whether equations of quantum mechanics in general, and of the quantum field theory in particular, are defined mathematically. The answer is surely "no", as witnessed by divergences in QFT (to think of one example). Again IMHO, a well-defined Hilbert-space formulation exists only for simplest problems of little practical interest. No one has ever calculated Lamb shift in Hilbert space.
All practical approaches use the Hilbert-space formulation in a purely symbolic manner, in order to "derive" approximate equations for physical quantities of interest. I put "derive" in quotation marks because such derivations are purely symbolic. One example is the Feynman-Dyson approach in QFT. It implies that Heisenberg equations are mathematically sound, which is simply not the case (cf. Plimak et al 2018 Phys. Scr. https://doi.org/10.1088/1402-4896/aad5ff). There is also growing evidence that we need Krein spaces rather than Hilbert spaces, et cetera, et cetera.
Dear L. I. Plimak, thank you for your answer. Then, would you say that the
inconsistencies in the Fock space formulation of QFT are originated from - or, more generally, reflect - similar inconsistencies in the Hilbert space formulation of quantum mechanics?
Dear Spiros,
Let's be specific. Choose a physical problem you are interested in, and we'll discuss its mathematics.
Best
Lev
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