I'm starting a series on "Quantum Mechanics Basics". Feel free to comment with any questions about the posted material. Be sure to stay tuned and enjoy!pic.twitter.com/NHmeD3kR4H
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Good question. To answer your question I would have to follow up with "which Hilbert space?". That is, every particle has different degrees of freedom and these are reflected in the nature of the Hilbert space.
For example, the Hilbert space of an electron is C^2, the set of all 2 component complex vectors. To answer your question directly, the answer is *sometimes* (e.g. C^2).
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Though I should mention that the vectors in this space shouldn't be thought of as pointing somewhere in 3D or 2D space. Rather, they are "abstract" vector spaces that represents the set of all states of the system.
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Thanks! So, can I use the same Hilbert space for an electron and a proton? Or I should worry about electron being a fundamental particle and proton being made up of quarks?
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It so happens that they have the same Hilbert space, but that's because the proton is also a spin 1/2 particle. When we only consider the spin degrees of freedom, the spin of the particle is what determines the Hilbert space.
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