Hilbert space algebra

Question

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?

To clarify: (1) Does ONS mean orthonormal system? (2) Did you mean "mathematical rules" instead of "mathematica rules"? (They're both meaningful, but the meanings are different. Mathematica is a symbolic calculation software package.) — Chiral Anomaly, Apr 17 '21 at 21:03
What do you mean by 'solving' this equation? Do you mean how to simplify this equation? — Tobias Fünke, Apr 17 '21 at 21:05
When $V$ and $W$ are inner product spaces, then for pure tensors we have $\langle v_1\otimes w_1, v_2\otimes w_2\rangle = \langle v_1,v_2\rangle\langle w_1,w_2\rangle$. Extending by bilinearity gives you the inner product on $V\otimes W$. — doetoe, Apr 17 '21 at 22:04
KroneckerProduct[] is the function you want to perform tensor products in Mathematica. — AccidentalTaylorExpansion, Apr 18 '21 at 09:02

Cosmas Zachos · Accepted Answer · 2021-04-17T22:26:03.477

2

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.

edited Apr 17 '21 at 22:26

answered Apr 17 '21 at 22:08

Cosmas Zachos

62,595

and how do you know this rule ? Can you expand it, if possible, like are there in between steps ? – imbAF Apr 17 '21 at 22:13
WP or your freshman year notes are not enough? Your teacher surely reviewed the tensor product for you? – Cosmas Zachos Apr 17 '21 at 22:16
ofc no! We talked about reducible tensors of order n for about 20 min. And in my bachelor we have talked about tensor about 1 hour max. I wasn't aware of clifford algebra until recently. And that's the tip of the iceberg when it comes to the gap knowledge in math (physics related) – imbAF Apr 17 '21 at 22:38
I hope the above example suffices. Sometimes mathematicians get carried away by bifunctors and pomposities, instead of illustrating the bland, practical object. I could tweak to address specific questions. – Cosmas Zachos Apr 17 '21 at 22:40
It did help and the link also, i will be studying that.Thank you – imbAF Apr 17 '21 at 22:44
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My previous answer, at the bottom, attaches two pages summarizing "addition" of angular momentum in this Kronecker product formalism... – Cosmas Zachos Apr 17 '21 at 22:46
I really appreciate the help! It looks overwhelming but i will give it my best – imbAF Apr 17 '21 at 22:48

Hilbert space algebra

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