Why identical particle states are multiplied? In case of identical particles we multiply the individual wave functions of the particles to get the system wave funtion. But why are we not adding? Or performing any other operation to get the system wave function. Can anyone show the math that what physics will be violated if i simply add the individual particle's wave function?
 A: The wave function for any composite system consisting of two uncorrelated (unentangled) subsystems is the (tensor) product 
$$ |\psi_A\rangle |\psi_B\rangle \equiv |\psi_A\rangle \otimes |\psi_B\rangle  $$
This multiplicative behavior of states isn't a specific feature of identical particles – or any particles. It holds for any physical system that may be divided to two (or several) parts.
The reason why the "combination" of several particles or objects is given by the multiplication and not addition is that the numerical values of wave functions in quantum mechanics don't represent "properties of objects" themselves but the probabilities (probability amplitudes, more precisely) of different properties of the objects.
We have the Born rule that says e.g.
$$ \rho (x) = |\psi(x)|^2 $$
The probability or probability density is given by the squared absolute value of some value of the wave function (or a linear combination of these values).
If we describe properties of objects $A,B$ that are independent, they may have various probabilities. The probability that the system $A$ has the $i$-th property is $P_{A,i}$. The probability that $B$ has the $j$-th property is $P_{B,j}$. If the objects $A,B$ are independent of each other, the probability that $A$ has the $i$-th property and $B$ has the $j$-th property is the product of probabilities
$$ P_{AB,ij} = P_{A,i} P_{B,j} $$
This is the usual multiplicative rule for probabilities of independent things. For example, if die $A$ has $P=1/6$ to land as 6 and $B$ has $1/6$ to land as $6$, the probability that we get $6+6$ is $1/6\times 1/6 = 1/36$.
The multiplicative formula for the wave functions $\psi_A$ and $\psi_B$ is basically just a "square root" of the formula for the probabilities. The calculus of the wave functions must reproduce and does reproduce some rules from the probability calculus – because the probabilities are linked to the wave functions by the relatively simple rule.
In particular, all Hilbert spaces of allowed states of composite systems $A+B$ are unavoidably given by tensor products ${\mathcal H}_A \otimes {\mathcal H}_B$ and unentangled states of two particles are tensor products of the two pure states.
The addition is something completely different. The state
$$|\uparrow\rangle + |\downarrow\rangle $$
describes a particle whose spin (that's what my arrow referred to) is either up or down. The addition translates to the word "or", not the word "and", because the addition of the wave functions is sort of analogous to the addition of probabilities, and the probability of "U or V" is
$$ P(U\text{ or }V) = P(U) + P(V) - P(U\text{ and } V) $$
If $U,V$ are mutually exclusive, the last term is zero. But we still see that "or" refers to the addition of probabilities, and a similar statement about the wave function says that their sums refer to "either one term" or "the other terms". Except that the relative phase between the complex probability amplitudes also matters in quantum mechanics – while the classical probabilities never come with any phases.
A: Consider this example:  
Imagine that the state vectors are represented by real-space wave functions.  For any particular value of the possible location of particle A, the entire space of values for B are available for the position of B.   For another location of A, again, the entire space is available for B.  
In order to account for all possible combinations of the location of A and B, one multiplies the probability amplitude for finding the particles in their respective locations.
A: To simplify things--it's just what you do with probabilities.
Consider 2 non interacting particles uniformly distributed in a box of length 1. Their wave functions are:
$ \psi_1(x) = \Pi(x_1) $
$ \psi_2(x) = \Pi(x_2) $
where $\Pi$ is the unit box function equal to 1 on [0, 1] and 0 elsewhere. If we were to add probabilities (or amplitudes), the probability that both particles are in the box is 2. (Unitarity anyone?). If we multiple probabilities, the it is 1--they are in the box.
