What is the difference between a basis transformation and a symmetry transformation in the Hilbert space of a quantum system?
By a basis transformation, I mean transforming from one orthonormal basis {|ϕn⟩} to another {|χn⟩}. A state |ψ⟩ in the Hilbert space can be expanded in these two bases as |ψ⟩=∑nCm|ϕm⟩=∑iDi|χi⟩
By a symmetry transformation, I understand a rotation (for example). How is that different from a basis transformation?
Answer
Some comments probably related to your confusion:
Just writing a state in two different bases is not a transformation, you aren't doing anything to the state. A transformation is a non-trivial map from the Hilbert space to itself.
Given two different bases {|ψi⟩} and {|ϕi⟩}, the map U:H→H,|ψi⟩↦|ϕi⟩
is a unitary operator with matrix components Uij=⟨ψi|ϕj⟩ in the ψ-basis (compute this explicitly if you do not see it).There are two different notions of symmetry in this context (see also this answer of mine:
The weaker one is that a symmetry is a transformation on states that leaves all quantum mechanical amplitudes invariant, this is a symmetry in the sense of Wigner's theorem which tells us that such transformations are represented by unitary operators.
The stronger one is that a symmetry is a symmetry in Wigner's sense that additionally commutes with time evolution, i.e. whose unitary operator commutes with the Hamiltonian.
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