Multiplication of Operators in Quantum Mechanics

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The discussion focuses on the multiplication of operators in quantum mechanics, specifically how to form a matrix representation of operators using state vectors. The initial matrix was constructed correctly, but the user expressed uncertainty about the process and how to proceed with part e of the problem. Guidance was provided on using the inner product rule, which simplifies the computation of operator products. The user successfully resolved their confusion after applying the advice regarding state expressions. Overall, the interaction highlights the importance of understanding the mathematical framework behind operator multiplication in quantum mechanics.
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Homework Statement
Operators can also be multiplied just like matrices. Physically, this represents applying two operations in succession. To compute in the abstract setting, we just need the rule ⟨i|j⟩ = δij .

(d) Compute the operator product of |1⟩⟨1| and |1⟩⟨1| + |2⟩⟨2|.
(e) Compute the operator product of |1⟩⟨2| + |2⟩⟨1| and |2⟩⟨2|.
Relevant Equations
O = O[SUB]ij[/SUB] |i⟩⟨j| = O[SUB]11[/SUB] * |1⟩⟨1| + O[SUB]12[/SUB] * |1⟩⟨2| + O[SUB]21[/SUB]|2⟩⟨1| + O[SUB]22[/SUB]|2⟩⟨2|.
For the first part of the problem, I managed to form this matrix;

<1|O|1><1|O|2>
<2|O|1><2|O|2>

=
10
00

However, that was because I was following this image;

MUuW2cj.png


I'm not entirely sure how this was obtained, and I'm not really sure what to do to continue forward with part e. I apologize for my lack of knowledge - I've attempted to search for any youtube videos to help and go through online textbooks, but I'm unable to find what I am looking for.
 
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Why don't you investigate it with the expression of the states
<br /> |1&gt;=<br /> \begin{pmatrix}<br /> 1 \\<br /> 0 \\<br /> \end{pmatrix}<br />
<br /> |2&gt;=<br /> \begin{pmatrix}<br /> 0 \\<br /> 1 \\<br /> \end{pmatrix}<br />
 
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penny_ss said:
To compute in the abstract setting, we just need the rule ⟨i|j⟩ = δij .
For example, the operator ##| 2 \rangle \langle 1|## multiplied by the operator ##| 1 \rangle \langle 1|## would be $$| 2 \rangle \langle 1| \cdot | 1 \rangle \langle 1| = | 2 \rangle \langle 1|| 1 \rangle \langle 1| = | 2 \rangle \langle 1| 1 \rangle \langle 1|$$ The middle part ##\langle 1| 1 \rangle## of the expression on the far right can be evaluated using the rule ##\langle i| j \rangle = \delta_{ij}##.
 
Thank you so much for the help! I've managed to work through it with the advice given. I forgot to account for the expression of the states - once I did, the problem became much easier. Thank you for the help! And that rule makes a lot more sense now. Thank you again!
 
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