Why does the description of a composite system involve a tensor product?

[prabin]

Can anyone answer me that why the description of composite system involve tensor product ? Is there any way to realize this intuitively ?

 

[andresB]

The tensor product is not only used for composite systems, even for a single particle moving in space the 3d base kets are built as a product of 1d kets

$$\left|x,y,z\right\rangle =\left|x\right\rangle \otimes\left|y\right\rangle \otimes\left|z\right\rangle $$

As I see it, the tensor product is an abstract formalization of the following fact: Let $\psi(x_{1,}x_{2})$ be a two-particle wave function and ${\phi_{n}(x)}$ an orthogonal basis of functions for the one particle Hilbert space $L^{2}$ (say, the Hermite polynomials). Then you can write $\psi(x_{1,}x_{2})$ as a linear combination of functions of the form $\phi_{n}(x_{1})\phi_{m}(x_{2})$. If we write this using an abstract vector, we are basically defining a tensor product.

 

[Structure seeker]

The tensor product is the mathematical respresentation of two independent spaces, two independent unit vectors/kets or two independent tensors to be viewed as one. The context is linearity, and the tensor product preserves the linearity of both factors in such a way that the restriction to one of both (via the trace over the other) returns the original vector/ket/space/tensor.