Simple question concerning unitary transformation

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Discussion Overview

The discussion revolves around the transformation of operators under infinitesimal unitary transformations, specifically the expressions U^-1AU and UAU^-1. Participants explore the implications of different definitions of the unitary transformation U as it relates to kets and bras in quantum mechanics.

Discussion Character

  • Debate/contested
  • Conceptual clarification

Main Points Raised

  • One participant questions whether the transformation of an operator should be expressed as U^-1AU or UAU^-1, noting that different texts provide conflicting definitions.
  • Another participant mentions that since unitary matrices form a group, the distinction between U and U^-1 is superficial and depends on the chosen definition of U.
  • A subsequent reply suggests that the choice of U does matter, particularly in how it affects the transformation of kets and bras, leading to different representations of the operator in a new basis.
  • It is noted that defining U as the transformation on kets leads to one expression, while defining it as the transformation on bras leads to the alternative expression.

Areas of Agreement / Disagreement

Participants express differing views on the significance of the choice between U^-1AU and UAU^-1, indicating that the discussion remains unresolved with multiple competing perspectives.

Contextual Notes

The discussion highlights the dependence on definitions of unitary transformations and the implications for operator representation, but does not resolve the mathematical nuances involved.

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Is the transformation of an operator under INFINITESIMAL unitary transformation, U^-1AU or UAU^-1?? I saw that two books defined it differently?
 
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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.
 
So it doesn't matter?
 
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.
 
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