(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Not relevant, but I have some work that reaches incorrect conclusions, and I can't see the mistake "in the middle".

2. Relevant equations

3. The attempt at a solution

[tex]\begin{array}{l}

{(XY)^\dag } = {\left( {\left\langle {a'} \right|X\left| {a''} \right\rangle \left\langle {a''} \right|Y\left| {a'''} \right\rangle } \right)^\dag } \\

= \left( {\left\langle {a'} \right|X{{\left| {a''} \right\rangle }^\dag }} \right)\left( {\left\langle {a''} \right|Y{{\left| {a'''} \right\rangle }^\dag }} \right) \\

= \left( {\left\langle {a''} \right|Y{{\left| {a'''} \right\rangle }^\dag }\left\langle {a'} \right|X{{\left| {a''} \right\rangle }^\dag }} \right) = {Y^\dag }{X^\dag } \\

\end{array}[/tex]

...Somewhere, here, is a mistake. I distributed the adjoint operation "+" (dagger) across the two scalars, "<||>", but if I did that, this would imply:

[tex]{\left( {XY} \right)^\dag } = \left\langle {a'} \right|X{\left| {a''} \right\rangle ^\dag }\left\langle {a''} \right|Y{\left| {a'''} \right\rangle ^\dag } = {X^\dag }{Y^\dag } = {\rm{uh - oh}}{\rm{.}}[/tex]

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# Working with Hermitian-Adjoint - Sakurai Problem 1.4b

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