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- Summary
- About writing the state vector of a composite system

Suppose I have a system of two (possibly interacting) spins of 1/2. Then the state of each separate spin can be written as a ##\mathbb{C}^2## vector, and the spin operators are made from Pauli matrices, for instance the matrices

##\sigma_z \otimes \hat{1}## and ##\hat{1} \otimes \sigma_z##,

which are tensor products of the z-direction spin matrix acting on one spin, and a unit matrix acting on the other spin, correspond to the spin-z operators for individual spins.

Now, in the bra-ket notation it is easy to write the product form states of the composite system as something like ##|s_1 \rangle |s_2 \rangle##. Is it also a correct notation to use the tensor product symbol ##\otimes## for constructing these from the ##\mathbb{C}^2## vectors:

##|s_1 \rangle |s_2 \rangle = \begin{bmatrix}a_1 \\ a_2 \end{bmatrix}\otimes\begin{bmatrix}b_1 \\ b_2 \end{bmatrix}##,

with ##a_1 ,a_2 ,b_1## and ##b_2## being the complex number components of ##|s_1 \rangle## and ##|s_2 \rangle## ?

##\sigma_z \otimes \hat{1}## and ##\hat{1} \otimes \sigma_z##,

which are tensor products of the z-direction spin matrix acting on one spin, and a unit matrix acting on the other spin, correspond to the spin-z operators for individual spins.

Now, in the bra-ket notation it is easy to write the product form states of the composite system as something like ##|s_1 \rangle |s_2 \rangle##. Is it also a correct notation to use the tensor product symbol ##\otimes## for constructing these from the ##\mathbb{C}^2## vectors:

##|s_1 \rangle |s_2 \rangle = \begin{bmatrix}a_1 \\ a_2 \end{bmatrix}\otimes\begin{bmatrix}b_1 \\ b_2 \end{bmatrix}##,

with ##a_1 ,a_2 ,b_1## and ##b_2## being the complex number components of ##|s_1 \rangle## and ##|s_2 \rangle## ?