The philosophy behind the mathematics of quantum mechanics

My field of study is computer science, and I recently had some readings on quantum physics and computation.

This is surely a basic question for the physics researcher, but the answer helps me a lot to get a better understanding of the formulas, rather than regarding them "as is."

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Whenever I read an introductory text on quantum mechanics, it says that the states are demonstrated by vectors, and the operators are Hermitian matrices. It then describes the algebra of vector and matrix spaces, and proceeds.

I don't have any problem with the mathematics of quantum mechanics, but I don't understand the philosophy behind this math. To be more clear, I have the following questions (and the like) in my mind (all related to quantum mechanics):

• Why vector/Hilbert spaces?


• Why Hermitian matrices?



• Why tensor products?


• Why complex numbers?

(and a different question):

Is the answer just "because the nature behaves this way," or there's a more profound explanation?

Answer

Vector spaces because we need superposition. Tensor product because this is how one combines smaller systems to obtain a bigger system when the systems are represented by vector space. Hermitation operator because this allows for the possibility of having discrete-valued observables. Hilbert space because we need scalar products to get probability amplitudes. Complex numbers because we need interference (look up double slit experiment).

The dimension of the vector space corresponds to the size of the phase space, so to speak. Spin of an electron can be either up or down and these are all the possibilities there are, therefore the dimension is 2. If you have $k$ electrons then each of them can be up or down and consequently the phase space is $2^k$-dimensional (this relates to the fact that the space of the total system is obtained as a tensor product of the subsystems). If one is instead dealing with particle with position that can be any $x \in \mathbb R^3$ then the vector space must be infinite-dimensional to encode all the independent possibilities.

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Edit concerning Hermitation operators and eigenvalues.

This is actually where the term quantum comes from: classically all observables are commutative functions on the phase space, so there is no way to get purely discrete energy levels (i.e. with gaps in-between the neighboring values) that are required to produce e.g. atomic absorption/emission lines. To get this kind of behavior, some kind of generalization of observable is required and it turns out that representing the energy levels of a system with a spectrum of an operator is the right way to do it. This also falls in neatly with rest of the story, e.g. the Heisenberg's uncertainty principle more or less forces one to have non-commutative observables and for this again operator algebra is required. This procedure of replacing commutative algebra of classical continuous functions with the non-commutative algebra of quantum operators is called quantization. [Note that even on quantum level operators can still have continuous spectrum, which is e.g. required for an operator representing position. So the word "quantum" doesn't really imply that everything is discrete. It just refers the fact that the quantum theory is able to incorportate this possibility.]