# Difference between symmetry transformation and basis transformation

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?

• You wrote down how a state can be expanded in two bases, but you didn't write down what you consider the actual transformation. Apr 25 '18 at 17:27
• Yes. But isn't rotation just one such basis transformation? I think I'm confusing.
– SRS
Apr 25 '18 at 17:29

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).