The Schrodinger representation
Compatible Dynamical Variables
Let us begin with the uncertainty relation which states that a very accurate simultaneous measurement of a component of position and corresponding
momentum component is not possible. So at any given time we can talk about the accurate simultaneous measurements of one of two variables only; we may take either position or the momentum. This raises the question,`` How do we know which quantities can be measured simultaneously?'' The answer is that the
operators corresponding to the all quantities which can be measured simultaneously must commute pairwise. These are known as compatible
variables.
Examples: Let us look at examples of a complete set of compatible dynamical variables.

1. For a system of one or more particles, the operators  corresponding to the generalized coordinates form a complete set of commuting operators. The set of all canonical momentum operators also form a complete set of commuting operators.
2. For a particle with spin, the spin operators do not form a commuting set. But $\vec{S}^2$ and $S_z$ from a commuting set. The set  $S^2, S_z$ and all position operators will be a complete commuting set.
3. Thus if we measure all the components of position, experimentally we can only get probabilities of position being in a range $x$ to $x+dx$. Theoretically, we should be able to compute this probability. In order to account for wave like  properties satisfactorily, one has to work with probability amplitudes for describing outcomes of all possible measurement.

 The quantum description of states consists of just these probability  amplitudes. The state vector describing a system consists of this information about (joint) probability amplitudes for a chosen  complete set of  commuting dynamical variables.

Specifying the state vector is simply a way of supplying a set of probability amplitudes for all compatible dynamical variables. The linear span of set of all such vectors is the vector space needed for a quantum system.

Measurable Quantities for a Point Particle
We will now describe a way of replacing abstract quantities for a point particle with more familiar objects. Technically this is known as introducing a  representation. The basic dynamical variables of a point particle are coordinates and momenta  $(x,y,z), (p_x,p_y,p_z)$. And not all pairs formed of these variables commute and therefore such pairs of variables can be simultaneously measured very accurately.

We know that position and momentum can not be measured simultaneously very precisely. Only position ( or only momentum ) can be measured.  In other words {\it one set} of compatible variable can be taken to  be the three position coordinates $(x,y,z)$.  We have started with a complete set of compatible variables consisting  of the three position coordinates, experimentally observable quantity  is the probability density of finding the particle  with position in a range $(\vec{r}, \vec{r} + \vec{dr})$. (It is implicitly assumed that the position is not quantized)

This probability density and hence the corresponding amplitude  will be a function of $(x,y,z)$. The quantum state of a point  particle will be described by corresponding probability amplitudes. The amplitudes will be some function $\psi(\vec{r})$ of position.
In terms of $\psi(\vec{r})$, the probability of position being in a  small volume $\Delta V$ at location $\vec{r}$, will
                be $|\psi(\vec{r})|^2  \Delta V$.
Since the sum of probabilities of all possible outcomes should be  one, we must have
        $$                   \iiint _{-\infty}^ \infty |\psi(x,y,z)|^2 dx\, dy\, dz =1.     (1)  $$
The set of all functions satisfying the above condition is not a  vector space. Recall that the first postulate tells you that the functions $\psi(\vec{r})$ and $c \psi(\vec{r})$ represent  the same state, where $c$ is a complex number. So we relax the requirement (1) and take the set of all functions which are square integrable but need not be normalized. These are the functions, which satisfy the  condition       
              $$  \iiint _{-\infty}^ \infty |\psi(x,y,z)|^2 dx\, dy\, dz  <   \infty,   $$

         and they do form a vector space.

Thus  required vector space is linear span of the set of all possible position probability amplitudes. The wave function, representing the state vector, when normalized, gives the position probability density. This representation of  states is known as the {\bf coordinate representation} and is also called the {\bf Schr\"{o}dinger representation}.

Dynamical Variables of a  Point Particle

Use of the coordinate representation makes the position as a privileged variable and the action of corresponding operator on a square
integrable function of position $\psi(\vec{r})$ becomes very simple. The action of position operators is  given by is given by multiplication :
$$ \hat{x} \psi(\vec{r}) = x  \psi(\vec{r}),~~\hat{y} \psi(\vec{r}) = y \psi(\vec{r}), ~~~ \text{and}~~~ \hat{z} \psi(\vec{r}) = z  \psi(\vec{r}).$$
Any function of position, for example potential energy $V(\vec{r})$, is also a dynamical variable and the corresponding operator $\widehat{V(\vec{r})}$ acting
on a state vector is multiplication
  $$      \widehat{V(\vec{r})}\, \psi(\vec{r}) = V(\vec{r})\, \psi(\vec{r})  $$.
Another set of dynamical variables of a point particle are momenta. They satisfy the canonical commutation relations
$$     [\hat{x}, \hat{p}_x] = i\hbar, \quad  [\hat{y}, \hat{p}_y] = i\hbar, \quad
 [\hat{y}, \hat{p}_y] = i\hbar.         (3)$$
These commutation relations imply that, in the coordinate representation, the action of momentum operator is
$$     \hat{\vec{p}} \psi(x) = -i \hbar \psi(x).     (4) $$
This action of momentum operator is directly related to the uncertainty principle. It is not very difficult to show that the position and momentum operators as introduced above satisfy the canonical commutation relations as  given in the postulates.
 Several type of representations are possible. This means that the  states and dynamical variables  can be described in several different  equivalent ways.  These different representations may in general appear very different.
 Every other dynamical variable, such as energy (the Hamiltonian), will be a function of position and momenta. So, for the Hamiltonian operator we have
    $$ H = \text{K.E.} + \text{P.E.} = \frac{p^2}{2m} + V(\vec{r}) \text{and} $$

\[ \quad \hat{H} \psi(\vec{r}) = -\frac{\hbar^2}{2m} \nabla^2 \psi(\vec{r}) + V(\vec{r}) \psi(\vec{r}).\]

Alternative representations

In the coordinate representation the vectors describing the states of a point particle are just the probability amplitudes for position. The position
operator then has a simple action of multiplication by position. Alternatively we could describe the possible states by the probability
amplitudes of {\bf momentum}. The vector space of possible states would then be the space of all square integrable functions of momentum, denoting them by
$\phi(\vec{p})$, with
     $$ \iiint |\phi(\vec{p})|^2 dp_x\,dp_y\, dp_z < \infty. $$
The momentum operators $\hat{p}_x$ etc. will then have a simple action of multiplying by momentum $p_x$ etc. In this case
             $$ \hat{\vec{p}} \phi(\vec{p}) = \vec{p} \phi(p), \qquad
               \hat{\vec{r}}\phi(\vec{p}) = + i\hbar \nabla_p \phi(\vec{p}). $$
Note the plus sign in the action of position operator. The probability amplitude for momentum is called the {\bf momentum space wave function} and
this representation is called the {\bf momentum space representation}.
One can have other representations, for example, energy representation. Of course any two representations will be related and one can transform
equations in one representation to corresponding equations in another representation. We will not go into details of this {\it change of
representation}  here. The coordinate representation is most useful and you will see more of it.