Very Basic questions on bra-ket notation

[Alexis21]

Hello,

i am a beginner in quantum mechanics and i have those basic questions on the bra-ket notation:

Which dimension has a ket $| \phi >$ describing a state normally? Maybe$\quad C ^n$?

Which dimension has a bra-ket $<\psi | \phi >$then? Maybe $\quad C$?

How do you get the matrix representation of an operator in general? I have been reading something like this: $<\phi_n |\hat{A}| \phi_m >$? I think i have to figure out then, how A works on phi_m but what to do next?

Thanks for helping

 

[Fredrik]

See post #3 here for the relationship between matrices and linear operators. This old post explains bra-ket notation.

What do you mean by "dimension"? Do you mean in the sense of units? I have never seen a reason to assign them a unit, so I would consider them dimensionless. Note that $|\langle\alpha|\beta\rangle|^2$ must be dimensionless since it's supposed to be interpreted as a probability.

 

[dextercioby]

A vector or a scalar has no dimension, a space of vectors has. Yes, if phi_n is a set of vectors which form a basis in a (pre-)Hilbert space, then $\langle \phi_n, A \phi_n\rangle$ is a matrix element, a complex number.