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I recently started learning about quantum physics. In the book, Quantum physics by H.C. Verma, the author explains that there are many ways to represent the state of a particle. The wave function $\psi(x)$ is only one way to represent the state. This particular representation is called position representation. One another way to represent the state of the particle is $|\psi(x)\rangle$ and it is called the abstract representation of the state. I searched up for it's meaning and found that the symbol $|\psi(x)\rangle$ is actually a column matrix. My questions are:

  1. What this matrix actually is?

  2. What are the elements of this matrix?

  3. Why is this matrix called the abstract representation of the state? (I know that the word "abstract" belongs to a branch of mathematics called Abstract Algebra but I do not know anything regarding this branch.)

  4. Why do I need this so called abstract representation $|\psi(x)\rangle$ when I already know about the position representation $\psi(x)$?

  5. When I see $\psi(x)$, I imagine a picture of a wave. When is see $|\psi(x)\rangle$, what picture is supposed to come to my mind? I will be glad if someone enlightens me.

P.S.: The book only says that it's a matrix and how to apply various mathematical rules to these matrices. It doesn't answer my above questions.

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    $\begingroup$ Regarding (1) and (2) only: In an abstract state, every position in space is a basis, and so the "column" vector has an uncountable infinite number of components in general. The ''$c$th" component is just $\psi(c)$. $\endgroup$ Jun 25, 2022 at 16:19

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Hello and welcome to the site. I will try to give an answer to some of your questions and hopefully the remaining ones will be dealt with, or if not, you can still ask in the comments.

First, some introductory remarks: The state $|\psi>$ (I prefer this notation rather than $|\psi(x)>$ at least in the braket) is a vector, living in a complex vector space. I like to think that the vector $|\psi>$ contains all sorts of information about the state of the quantum system under examination. The term abstract perhaps refers to the fact that the vector representation, despite the fact that contains all the information about the physical quantum system at hand, does not concretely allow you to access these pieces of information (at least in an immediate/actual/concrete way).

Let us take for example the position representation, denoted by $\psi(x)$. This representation is obtained by projecting the state onto an element of the posiiton basis $|x>$, a basis that belongs in the vector space mentioned earlier. This is hence a concrete way of obtaining information about the position of the quantum system at some certain point in time, since the projection $\psi(x)=<x|\psi>$ is indeed the probability amplitude for the system to be located at point $x$ in space. Note that the set of vectors that are in one-to-one correspondence to each possible position the quantum system might be located at, indeed comprises a basis: $$1=\int dx |x><x|,\hspace{2em} \delta(x-x')=<x|x'>$$

Equivalently, one can consider the momentum basis of the space. A representative of such a basis could be a vector of the form $|p>$ and hence the wave function in momentum space is given by $\tilde{\psi}(p)=<p|\psi>$ (tilde to distinguish from $\psi$). In this way, we obtain information about the momentum of the system by projecting onto the momentum basis, since $\tilde{\psi}(p)$ is the probability amplitude of the system having momentum $p$.

Now, to answer (1), $|\psi>$ is a column since you can represent vectors as columns. Similarly, its dual $<\psi|$ can be thought as a row. (2) is a more difficult question to answer. However, since $|\psi>$ is a vector inside a vector space and we have just discussed about two of the bases in that vector space, one can think of an analogue (finite dimensional): imagine the a 2D coordinate system and the basis {(1,0),(0,1)} and $\{(\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}}),(\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}})\}$. What I have done is to write down two different bases, according to which I can write any vector in the 2D plane as a linear combination of either of them (this would be our case if there were only two allowed values for both the momentum and the position of the quantum system). For more on that I refer you to (Are the position eigenkets $\lvert x \rangle$ really a basis for the space of states?, Ambiguity in number of basis vectors, Hilbert space of harmonic oscillator: Countable vs uncountable?, Conceptual difficulty in understanding Continuous Vector Space). For (4), according to my way of viewing things, I like to think that it is called abstract representation because it does not provide the information it (secretely) holds in an immediate way, unless you project it onto some basis. And as far as (5) is concerned, I do not suppose that you are to understand things by assigning some sort of picture to a vector representing the quantum state of your system.

I hope my answers do not confuse you and I hope I haven't told you anything wrong.

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  • $\begingroup$ What is this x space represented by |x>? Does it represent the real x axis? What is the momentum space, |p>? $\endgroup$
    – user231188
    Jun 26, 2022 at 2:47
  • $\begingroup$ The $|x>$ vectror space is the space with all the possible position vectors. Each different $|x>$ is simply a point in the real line. Similarly, each different $|p>$ is simply a vector assigned for a value of the momentum (again, the latter can take any real value) $\endgroup$
    – schris38
    Jun 26, 2022 at 6:22
  • $\begingroup$ For instance, if we had only two allowed values of the position of the quantum system, we would have $|x>=\{|x_1>,|x_2>\}$ and the vector space would be two dimensional. Instead, in the case in which it can have any position, then there exists one such vector for each and every one of the infinite possible positions of the system $\endgroup$
    – schris38
    Jun 26, 2022 at 6:24

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