failexam said:
I see! So, to reiterate your point, a ket/state vector/wavefunction |ψ> = a1|1> + a2|2> + a3|3> + ... + an|n> = (a1, a2, a3, ... , an), where the an's are eigenvalues (possible outcomes of a measurement on the state of a system with wavefunction ψ) and the |n>'s are the eigenvectors (possible states that a system can collapse into after a measurement performed on the system).
That sum would usually have infinitely many terms, not stop at some integer n. The notation ##(a_1,a_2,\dots)## should be avoided. Even if the number of components had been finite, it would be more appropriate to represent the vector as a column matrix
$$\begin{pmatrix}a_1\\ a_2\\ \vdots\end{pmatrix}$$ because then a linear operator acting on the vector can be represented as a matrix that multiplies this matrix of components from the left.
failexam said:
The set (a1, a2, a3, ... , an) is the 'complete set of amplitudes', isn't it?
Yes, I think that's a valid way of looking at it.
failexam said:
I think you mention this point because my knowledge of vector spaces is rudimentary?
Yes, you didn't seem to be familiar with the definition of "vector space".
But there's also the fact that linear algebra is so useful in QM. There's a complex Hilbert space (a special kind of inner product space) associated with each physical system. Preparation procedures (ways the system can be prepared before the measurement begins) are represented by vectors. Measuring devices are represented by self-adjoint linear operators. The possible results correspond to eigenvalues and eigenvectors. Probabilities of measurement results are computed using the inner product. Etc.
failexam said:
Anyway, thanks for the titles. I'll try to read the books of Axler and Anton from cover to cover as they will help me with the mathematics of quantum mechanics and relativity.
I actually didn't mean to recommend Anton, so let me explain what I think about it. It's an excellent book. But, in the 6th edition at least, which is the only one I'm familiar with, the things we really need are postponed to the very end. So the
order of the topics is bad for a physics student. That's why I prefer Axler. Axler starts with complex vector spaces right away, and introduces linear operators (linear transformations) as soon as possible. However, since Axler doesn't start with a bunch of "how to calculate" stuff like Anton, and starts proving theorems right away, people who have no experience with proofs seem to find it difficult.
There are many books that are good enough, and it doesn't matter much which one(s) you study. The one by Friedberg, Insel & Spence seems to be good as well, and there are several free books available online.
Here's something I said in another thread, about which topics are important:
Fredrik said:
Complex vector spaces, linear independence, bases, inner products, inner product spaces, orthonormal bases, linear operators, matrices, matrix multiplication, a theorem about which matrices are invertible, the relationship between linear operators and matrices, the adjoint operation, self-adjoint linear operators, eigenvectors and eigenvalues, and the spectral theorem.
Since the relationship between linear operators and matrices is very important, I recommend that you use a book that presents those things early in the book, like Axler or Friedberg, Insel & Spence. (I have only read the former, but I've heard good things about the latter).
You may not need all of those things for an introductory course. It may be enough to understand complex inner product spaces, orthonormal bases and self-adjoint linear operators. But you will need the rest if you want to get good at QM.
