Difference between symmetry transformation and basis transformation

Question

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 ${|phi_nrangle}$ to another ${|chi_nrangle}$. A state $|psirangle$ in the Hilbert space can be expanded in these two bases as $$|psirangle=sumlimits_{n}C_m|phi_mrangle=sumlimits_{i}D_i|chi_irangle$$ where $langlephi_m|phi_nrangle=delta_{mn}$ and $langlechi_i|chi_jrangle=delta_{ij}$. The change of basis is a unitary transformation i.e., $$|chi_nrangle=U|phi_nrangle.$$

By a symmetry transformation, I understand a rotation (for example). How is that different from a basis transformation?

ACuriousMind · Accepted 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 ${lvert psi_irangle}$ and ${lvert phi_irangle}$, the map
$$ U: Hto H, lvert psi_i ranglemapsto lvert phi_irangle$$
is a unitary operator with matrix components $U_{ij} = langle psi_i vert phi_jrangle$ in the $psi$-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 probabilities 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.

Difference between symmetry transformation and basis transformation

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