Simple question concerning unitary transformation

1. Apr 6, 2014

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?

2. Apr 6, 2014

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.

3. Apr 6, 2014

M. next

So it doesn't matter?

4. Apr 6, 2014

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.