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I do not know the rules (mathematica rules) in order to perform the following calculation:

Lets say we have a $2$ particle system. Each particle has its own eigenbasis:

$|\phi_r\rangle$ is an ONS of System $S$ (particle 1).

$|\phi_a^\prime\rangle$ is an ONS of System $S'$ (particle 2).

Then if the whole system is in an arbitrary state $|\psi\rangle$, we can write:

$|\psi\rangle = \sum\limits_{r,a} c_{ra}\, |\phi_r\rangle \bigotimes |\phi_a^\prime\rangle $

Now if I want to find the expectation value of an observable in the system $S$ (and ignore $S'$) I would do the following:

$\langle \psi|A\bigotimes I|\psi\rangle = ( \sum\limits_{r,a} c_{ra}^*\, \langle \phi_r| \bigotimes \langle \phi_a^\prime|) A\bigotimes I( \sum\limits_{s,b} c_{sb}\, |\phi_s\rangle \bigotimes |\phi_b^\prime\rangle$

How do I solve this equation? Like what are the rules in this case?

Tobias Fünke
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imbAF
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1 Answers1

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The rule for tensor products is $$ (\langle a| \otimes \langle \alpha|) (A\otimes I)(|b\rangle\otimes |\beta\rangle) = \langle a|A|b\rangle ~ \langle \alpha|\beta\rangle. $$ Try to understand it by using 2x2 matrices, tensored to a 4x4 matrix, taking, e.g., $A=\sigma_1$. Do you see the block matrix and what it does to the tensor products? Thinking about particles and Hilbert spaces is a bit counterproductive here. This is bland, basic linear algebra.

OK, as per request, here is an example, $$ |b\rangle \otimes |\beta\rangle =\begin{pmatrix} b_1\\b_2\end{pmatrix} \otimes \begin{pmatrix} \beta_1\\\beta_2\end{pmatrix} = \begin{pmatrix} b_1 \beta_1\\b_1\beta_2\\ b_2\beta_1\\ b_2\beta_2 \end{pmatrix}, $$ and $$ \sigma_1\otimes I_2 = \begin{pmatrix} 0&0&1&0\\ 0&0&0&1\\ 1&0&0&0\\ 0&1&0&0\end{pmatrix} . $$ See how the blocks multiply and why.

Cosmas Zachos
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