A lot of textbooks give the definition of an S-matrix element as:(adsbygoogle = window.adsbygoogle || []).push({});

[tex]\langle \beta_{out}| \alpha_{in}\rangle = \langle \beta_{in}| S| \alpha_{in} \rangle=\langle \beta_{out}| S| \alpha_{out} \rangle=S_{\beta \alpha}[/tex]

and that [itex]S|\alpha_{out} \rangle =|\alpha_{in} \rangle [/itex]

I don't understand that definition. Shouldn't the S-matrix take an in-state, and map it to the corresponding out-state:

[tex]S|\alpha_{in} \rangle=|\alpha_{out} \rangle [/tex]

Moreover, shouldn't the amplitude for a state prepared in [itex]\alpha [/itex] to be detected as β be:

[tex]\langle \beta_{out} | S|\alpha_{in} \rangle [/tex]?

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# S-Matrix definition and terminology

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