# Why is the "expansion postulate" a postulate of quantum mechanics?

I'm currently reading the following set of lecture notes on quantum chemistry, which includes the so-called "expansion postulate" as a fundamental postulate of quantum mechanics: "The eigenfunctions of a linear and hermitian operator form a complete basis set."

How is this a postulate of quantum mechanics, rather than a provable mathematical property of linear hermitian operators? Isn't the expansion postulate just a consequence of the spectral theorem for an infinite-dimensional Hilbert Space?

If the eigenfunctions of a linear, hermitian operator forming an orthogonal basis is not a mathematically provable fact about wavefunctions, wouldn't that necessarily imply that there exist hermitian, linear operators on wavefunctions whose eigenfunctions do not form an orthogonal basis?

• Perhaps these notes are written for an audience that is not expected to understand functional analysis (like the spectral theorem): In such a situation, it may be better to simply say "we postulate"...
– Danu
Jan 26, 2016 at 17:23
• It's just the spectral theorem, although care must be taken since it holds only for self-adjoint, not merely Hermitian operators (there is a difference on infinite-dimensional spaces). This question is basically unanswerable since the premise that the "expansion postulate" is a postulate is just false. Jan 26, 2016 at 17:39
• I never heard of this didactic strategy before. It seems to be a way of declaring "we won't be looking at cases where this isn't true". physics.stackexchange.com/questions/68822/… Jan 26, 2016 at 20:08

It's a postulate, not because it's a self-consistent mathematical property, but rather because it is an assumption about how the physical world may, or may not, be described.

There are many many different types of linear vector spaces other than a Hilbert space, so to make things easy they make an assumption and run with it until proven otherwise.

• Your argument doesn't seem consistent to me. A position vector in classical mechanics can be written as a linear combination of any set of orthogonal basis vectors, but this isn't a postulate of classical mechanics and it would be ludicrous to make it one. Further, the statement of the expansion postulate is entirely mathematical in nature: it claims that the eigenfunctions of any linear, hermitian operator form a basis, not just operators which correspond to an observable. Jan 27, 2016 at 21:11
• "The eigenfunctions of a linear and hermitian operator form a complete basis set." Just because the rules of mathematics allows for a linear hermitian operator to form a complete basis set, doesn't mean that the rules of nature do. Hence, it's an assumption about the relationship between the math rules and the observable.
– JQK
Jan 27, 2016 at 21:23
• This is wrong. This is not axiomatic; it can be proven mathematically. Your distinction between the rules of mathematics and nature is completely irrelevant. Physical correspondence is based on the assumption of (a) some quantum-mechanical axioms, which are not the "expansion postulate", and (b) some mathematical and meta-logical axioms, from which all QM follows, like the OP's question. Following your argument, all QM statements should therefore be "postulates" or "assumptions", as one cannot positively prove them. Science follows falsifiability. Dec 20, 2020 at 13:56
• @juanrocamonde your comment doesn’t make sense. The proof of connection between a mathematical framework and physics is an experiment. Moreover only an experiment can falsify a theory. The original post concerns itself with a certain mathematical property. It’s easy to think of examples from physics where this property wouldn’t be applicable.
– JQK
Dec 20, 2020 at 21:13
• The completeness "postulate" is not a postulate or an assumption. The assumptions are the axioms, and that is not one of them. Experiments test the applicability of the axioms. Dec 20, 2020 at 22:35