Question

Is there a better way to visualize the complex hilbert spaces used in qm than euclidean vectors (arrows)?

Answer 1

The analogy with R^2 works fine. There is not a lot deeper than that. In particular, the thinking that complex multiplication involves a rotation is very misleading, because now the complex numbers are your *scalars*, not your vectors. When you do i(1,2) to obtain (i,2i) what you get is parallel to the original vector.

About adjoints: I have worked with adjoints daily for the last few decades, and I have absolutely no geometric intuition about them. Over time one develops an algebraic intuition, and there are lots of times where I know which formulas/tricks work. But I couldn't possibly express anything geometric about adjoints.

Answer 2

> multiplication of a (real) euclidean vector with a real scalar λ can be visualised as stretching the vector by a factor of λ [...] thinking of scalar multiplication in a complex vector field only as "stretching" is misleading.

In the real case, stretching is not all that happens: if the scalar is negative then the vector rotates by 180 degrees. So even in the real case you have experience with scaling doing more than just stretching.

Don't expect that every single aspect of higher-dimensional geometry must have a visualization. Learn to rely on *theorems* about linear algebra as much as pictures. We can't visualize why all complex linear operators on C^(n) have an eigenvalue, but we have the fundamental theorem of algebra available over C, so we know the existence of eigenvectors despite the lack of an accompanying picture.

Answer 3

Well, when it comes to such things the interpretation of the meaning of the vector is always linked to the basis of the space, since they define it knowing what your basis is dirrectly affects the kind of image will be most helpful to visualize what's happening in the system you're studying .For example, in most cases in QM the basis is the eigen-functions of the Hamiltonian operator, which provides us, at least in common cases, with a countable basis, and that for physical/statistical reasons are integrable and square intregrable, which are nice enough properties for them to have a Fourier transform/ Fourier series form. So in this case the complex components of the vector represent the amplitude of the wave an it's phase, so when you multiply this vector with a complex  number what you're doing is altering the amplitude and the phase of each wavepacket that builds your quantum state. If your vector space is the spin of the electron a different image would be most helpful.

EDIT: in short what i tried to say is this, the basis will give you a hint on what the best visualization for your problems is.