Rough Description of State Space

[chessforce]

I have done only a small bit of reading/studying of quantum mechanics. So, from what I have gathered thus far, I have the following rough semi-graphical description of state space:

Imagine a space in which you have a lot, but a definite number of discrete points. Each of those points (state vectors) can be described using a combination of base states with appropriate coefficients, provided that the dot/inner product between any two base states (base vectors) i, j is orthogonal (i.e. defined by the Kronecker delta function).

So, what I am wondering is if this is correct by any means. As aforementioned, my exposure to QM is minimal and therefore I may have this completely wrong. Thanks in advance.

 

[StatusX]

You're not that far off. The state space is just an ordinary (complex) vector space. It has an infinite number (in fact, a continuum) of points, just like an ordinary vector space, and often has in fact an infinite number of dimensions.

Like any vector space, it has many bases, not all of which are orthonormal, and while its not necessary to take an orthonormal basis, this is almost always done in practice because we usually take as a basis the eigenstates of some collection of observables, and such states are always orthogonal. Of course, since this space doesn't have an obvious geometric interpretation, you should be wondering where the inner product comes from, and a rough answer is that its define so that the last sentence I said is true.

 

[chessforce]

Thank you for the clarification.