In quantum mechanics, the state space is a separable complex Hilbert space.
A Hilbert space is separable if and only if it has a countable orthonormal basis [1, 2].
Why the quantum mechanical state space must be separable?
In [3], we read that separability is a mathematically convenient hypothesis, with the physical interpretation that countably many observations are enough to uniquely determine the state of a quantum system.
In Merzbacher’s quantum mechanics (3d ed.), page 185, we read that “The infinite-dimensional vector spaces that are important in quantum mechanics are analogous to finite-dimensional vector spaces and can be spanned by a countable basis. They are called separable Hilbert spaces.”
From a historical point of view, the two descriptions (or versions) of quantum mechanics that were initially developed in 1920’s, namely the Schrödinger’s wave mechanics and the Heisenberg’s matrix mechanics, were respectively based on the Hilbert spaces of square integral functions and square summable sequences of complex numbers, which are both separable, and physically equivalent (mathematically isomorphic). Thus, the invariant (or representation-free) description of quantum mechanics, through the abstract Hilbert space of Dirac kets, that was followed, had to be based on a separable Hilbert space too, otherwise it would not be equivalent to the two existing descriptions.
As it happens with the property of completeness [4], the property of separability of the quantum mechanical state space is also overlooked or mentioned very briefly in standard textbooks and the reader, especially the physics-oriented one, is left with the impression that it is rather a mathematical “decoration” of minor physical importance that can be forgotten.
From my own experience, it is also worth noting that the expression “a Hilbert space is separable if and only if it has a countable basis”, which is often given as definition of separability, is tricky and, to some extent, misleading. A reader with some background in functional analysis is rather easy to understand that, here, “it has a countable basis” actually means “ALL bases are countable”, as two basis sets are related by a one-to-one and onto mapping, thus they have the same cardinality, and then if one is countable, the other is countable too. But, a physics student may be confused and left with the impression that separable Hilbert spaces have also uncountable bases, which is the wrong picture, especially in connection with the uncountable (continuous) sets of the position and momentum eigenstates that although span the state space, they are not actually bases, because they are not belong to the state space, and this point is not highlighted in literature either.
[1] https://en.wikipedia.org/wiki/Separable_space
[2] http://www.math.ucsd.edu/~bdriver/231-02-03/Lecture_Notes/Hilbert-Spaces.pdf
[3] https://en.wikipedia.org/wiki/Mathematical_formulation_of_quantum_mechanics
[4] https://www.researchgate.net/post/Is_the_property_of_completeness_of_the_quantum_mechanical_state_space_presented_and_analyzed_adequately_in_standard_quantum-mechanics_textbooks
There aren't, however, any issues of either principle, or practice, that depend on whether the basis is countable or not. It only makes sense bringing the subject up, if such issues can be presented.
There's no problem in describing the properties of the free particle in Rd , despite the, apparent, fact that the eigenfunctions of the position operator and of the momentum operator aren't countable. It is, however, known, how deal with this issue: introduce appropriate cutoffs, work with a countable basis and prove that the limits exist.
For potentials that are bounded from below and confine at infinity no such problem arises, since the spectrum of the Hamiltonian is discrete and, thus, countable, anyway.
Issues that might arise regarding scattering states are discussed in detail in all textbooks.
And there's been a huge amount of work on the singular continuous spectrum of Schrödinger operators, relevant for describing localization.
Dear Stam Nicolis, thank you. Then, you would say that it is not necessary to include separability into the definition of the state space?
Of course not-only that it isn't correct to claim that it hasn't been taken into account. It may not be mentioned explicitly, that's all-but that's just because its consequences have, in fact, been taken into account.
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