Operators and vectors in infinite dimensional vector spaces

[fog37]

Hello Everyone. I am searching for some clarity on this points. Thanks for your help:

Based on Schrödinger wave mechanics formulation of quantum mechanics, the states of a system are represented by wavefunctions (normalizable or not) and operators (the observables) by instructions i.e. operations build from $x,y,z$ and the partial derivates like $\frac{\partial}{\partial x}$ etc. All (or almost) all operators can be built from the position operator and the linear momentum operator. All this is true regardless of the vector space being finite or infinite dimensional, discrete or continuous, correct?

According to Dirac (bra and ket) formulation instead, the system's state are viewed as vectors which can be described as row or column vectors, containing components, and the operators are matrices, with indexed elements, only if the vector space is finite/infinite dimensional and discrete. What if the vector space was finite/infinite dimensional and continuous? How would operators be represented in a continuous vector space? The matrix representation would not be possible. The vector would just be described as $|\Psi>$ using the ket notation but could not have a row or column representation, correct?

thanks,
fog37

 

[dRic2]

I may have misinterpreted what you're saying but, from what I know, the discrete/continuous problem refers to the eigenfunction of an operator, not to the space of the state. I think you are making some confusion. Last year I studied this stuff on "introduction to Quantum Mechanics" by Griffiths: the chapter on Dirac's formalism is quite short and easy to understand. I think you should look it up (or some other books on the matter). Not to be rude, but I think it will take quite long to answer your post and make some clarity.

Ric

Ps:

the states of a system are represented by wavefunctions (normalizable or not)

a non-normalizable wavefunction has no physical significance per se so it doesn't represent a "real" state , but a linear combination of non-normalizable wavefunctions could represent a state with physical significance.

 

[PeterDonis]

According to Dirac (bra and ket) formulation instead, the system's state are viewed as vectors which can be described as row or column vectors, containing components, and the operators are matrices, with indexed elements

The bra-ket formulation does not imply or require the existence of a matrix representation. Dirac used that representation for convenience, not because it was required. In fact the bra-ket formulation is equally compatible with the wave function representation you describe. The bra-ket formulation is a more abstract formulation that does not require any specific representation; that's a key reason why it's so useful.

 

[fog37]

Thank you to both.

I guess vector spaces can be finite or infinite dimensional but calling them discrete or continuous is probably incorrect. As dRic2 says, the bases in a vector space can be countable or uncountable (continuous) with countable meaning that the basis vectors in each basis can be put in one-one correspondence with the positive integer numbers.

I found the Griffith's chapter on Dirac notation which state on page 160 that if a state lives in a $N$-dimensional vector space it can be represented as column of $N$ components and operators take the form of ordinary $N\times N$ matrices. This can be extended to the case when N becomes infinite. But if the vector space is continuous, the matrix representation is not possible and I wondered if there was some other mathematical representation I am not aware of or we just describe the vectors and operators just with their abstract symbol...