Multiplication of Operators in Quantum Mechanics

[Penny57]

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>

=

1	0
0	0

However, that was because I was following this image;

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.

 

[anuttarasammyak]

Why don't you investigate it with the expression of the states
$$|1>= \begin{pmatrix} 1 \\ 0 \\ \end{pmatrix}$$
$$|2>= \begin{pmatrix} 0 \\ 1 \\ \end{pmatrix}$$

 

[TSny]

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 $\langle i| j \rangle = \delta_{ij}$.

 

[Penny57]

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!