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Many expositions of Quantum Mechanics begin by stating axioms of the following type:

"The state of a quantum mechanical system is given by a nonzero vector in a complex vector (or Hilbert space) with an Hermitian Inner Product"

These types of expositions then go on to define observables as self-adjoint linear operators etc.

Landau and Lifshitz build up their edifice in a seemingly different way, beginning with an assertion (or perhaps, for them, a kind of axiom) that probabilities are bilinear on $\Psi$ and $\Psi^{*}$ (I think most of us would now instead say that it was a sesquilinear form rather than conjugate-linear). They then deduce conclusions throughout the exposition. In particular they never explicitly assert that states form a vector space (which then would have a basis).

My difficulty stems from discerning whether, at various points, certain statements are to be taken as "axiomatic" or derived. Specifically:

Landau and Lifshitz, in section 3 of Quantum Mechanics, state that (I will refer to this as statement 1):

"In accordance with the principle of superposition, we can assert that the wave function $\Psi$ must be a linear combination of those eigenfunctions $\Psi_n$ which correspond to the values $f_n$ that can be obtained, with probability different from zero, when a measurement is made on the system and it is in the state considered."

Their previous statement of the superposition principle (in section 2) is as follows:

Suppose that, in a state with wave function $\Psi_1 (q)$, some measurement leads with certainty to a definite result (result 1), while in a state $\Psi_2 (q)$ it leads to result 2. Then it is assumed every linear combination of $\Psi_1 (q)$ and $\Psi_2 (q)$ ... gives a state in which measurement leads to either result 1 or result 2.

In this superposition statement (sect 2) L&L have not explicitly stated the converse; that every state in which a measurement leads to either result 1 or result two can be expressed as a linear combination of the eigenfunctions $\Psi_1 (q)$ and $\Psi_2 (q)$. (And this would not be unique - for example an electron state in which there was equal chance of being z up or z down could potentially be in x up or x down etc.)

The Question I am trying to ask is:

"Is L&L's statement 1 to be taken as axiomatic? Or is it derived?"

I presently see it as probably axiomatic because I cannot yet derive it from previous statements (as explained in the paragraph below), but I am not sure, hence the question. It seems to me to be perhaps equivalent to saying that quantum states must live in a vector space.

I can see why $\Psi$ could be written as a linear combination of those eigenfunctions (from their statement of the principle of superposition), but it also seems to me that that when L&L assert that the wave function $\Psi$ must be a linear combination of those eigenfunctions $\Psi_n$ they may be introducing a new physical principle of perhaps greater depth than simply deriving from the principle of superposition. In particular it seems to be equivalent to saying that these eigenfunctions form the basis for a vector space.

Further clarification

The assertion I am referring to is the "assert" in bold in the first quotation.

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  • $\begingroup$ Can you please tell specifically what is your "this assertion" and "previous content"? As a general rule, the eigenfunctions of a Hermitian operator produce a basis set for the Hilbert space, and any vectors of the Hilbert space (which is what you refer to as a general $\Psi$ can be expressed as a linear combination of these basis vectors. I can elucidate more if I know what is the assertion you're talking about, but all the statements from L&L follow from the fact that any element of a Linear vector space is a linear superposition of basis vectors. $\endgroup$ – Feynfan Jun 12 '18 at 14:54
  • $\begingroup$ The previous content I am referring to is in fact in the second quotation block of the original question and I have edited the question to clarify what the "assertion" is. In the language of the above comment I am asking then if the top quotation block is in fact L&L's actual statement of "the fact that any element of a Linear vector space is a linear superposition of basis vectors" and secondly, that this is a new statement in their reasoning and not deduced from their previous chain of reasoning. $\endgroup$ – Nathan C Jun 12 '18 at 15:09
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It is making a new claim using the previous. Since all the values the system can take upon measurement are possible, then one can capture the state by writing it as an expansion in terms of all of the individual states which correspond to those measurement values respectively.

V.2 I'm saying if you're OK with the second quote you made from earlier, then the first quote later on follows from it. If you insist you give it a name, it seems to me the second quote earlier has all the flavor of a vector space by itself.

I wouldn't think too deeply about the math naming, just what is actually being represented, otherwise you'll get bogged down in clarifications like this imo.

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  • $\begingroup$ Having somewhat updated the wording and language of the question, would the author to the above posted answer go so far as to say "It is making an "axiomatically" new claim? It seems to me that without at some point axiomatically stating that state space is a vector space, one could never be sure of "capturing the state" by writing it in terms of an expansion. It could be a strange type of space. Once we know it is a vector space, then I agree. Hence I am asking whether this is indeed the point at which L&L have sneaked in the vector space requirement. $\endgroup$ – Nathan C Jun 15 '18 at 13:56
  • $\begingroup$ If one looks at the QM history (E. Schroedinger), one can understand that once one gets into mathematical physics realm, one deals with vector spaces of different kinds. From mathematical physics it is clearly seen which states can form superpositions and which cannot. For example, one never makes a superposition of eigenvectors of $l_z$ with eigenvectors of $L^2$ because they belong to different vector spaces of separated variables. $\endgroup$ – Vladimir Kalitvianski Jun 16 '18 at 17:01

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