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 $\{|\phi_n\rangle\}$ to another $\{|\chi_n\rangle\}$. A state $|\psi\rangle$ in the Hilbert space can be expanded in these two bases as $$|\psi\rangle=\sum\limits_{n}C_m|\phi_m\rangle=\sum\limits_{i}D_i|\chi_i\rangle$$ where $\langle\phi_m|\phi_n\rangle=\delta_{mn}$ and $\langle\chi_i|\chi_j\rangle=\delta_{ij}$. The change of basis is a unitary transformation i.e., $$|\chi_n\rangle=U|\phi_n\rangle.$$

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

Answer 1

Some comments probably related to your confusion:

1. 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.

2. Given two different bases $\{\lvert \psi_i\rangle\}$ and $\{\lvert \phi_i\rangle\}$, the map $$ U: H\to H, \lvert \psi_i \rangle\mapsto \lvert \phi_i\rangle$$ is a unitary operator with matrix components $U_{ij} = \langle \psi_i \vert \phi_j\rangle$ in the $\psi$-basis (compute this explicitly if you do not see it).

3. 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.