Simple question concerning unitary transformation

[M. next]

Is the transformation of an operator under INFINITESIMAL unitary transformation, U^-1AU or UAU^-1?? I saw that two books defined it differently?

 

[Matterwave]

Remember that the unitary matrices form a group. So if U is a unitary matrix, then U^-1 is also a unitary matrix. Therefore, the distinction is superficial depending on which transformation you want to define as your U.

 

[M. next]

So it doesn't matter?

 

[Matterwave]

Well, it does matter a little bit. If you define U to be the unitary transformation that transforms kets |a>, then U^-1 will be the unitary transformation that makes the same transformation on the bras <b|. Using this, one should see that the matrix element <b|S|a> under a transformation goes to <b|U^-1SU|a> which means that the matrix S'=U^-1SU is the matrix that represents the operator in this new basis. If I defined U the opposite way, as U is the transformation on bras, then I get the other definition.