I Definition of a symmetry transformations in quantum mechanics

  • Thread starter Lebnm
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By the Wigner theorem, symmetries transformations are implemented by operators ##\hat{U}## that are unitary or antiunitary. This is what is written in most books. But I have read somewhere that, to ##\hat{U}## represent a symmetrie, it's necessary that ##\hat{U}^{\dagger} \hat{H} \hat{U} = \hat{H}##, that is, the hamiltonian has to be invariant under the transformation. Is it true? Are these definitions equivalents?
 
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They are different. Wigner proved that the most general transformation that preserves the total probability has to be linear and unitary or anti-unitary. On the other hand, such ##U## is a symmetry of ##H## (i.e. of a system with Hamiltonian ##H##) if ##U^{\dagger}HU = H##.
 

haushofer

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Or: not every transformation (described by U) is necessarily a symmetry transformation (described by [H,U]=0) ;)
 

haushofer

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Or even different: being unitary is necessary but not sufficient to be a symmetry transformation.
 
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Ok, thank you!
 

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