0
$\begingroup$

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?

$\endgroup$
2
  • $\begingroup$ You wrote down how a state can be expanded in two bases, but you didn't write down what you consider the actual transformation. $\endgroup$
    – ACuriousMind
    Apr 25 '18 at 17:27
  • $\begingroup$ Yes. But isn't rotation just one such basis transformation? I think I'm confusing. $\endgroup$
    – SRS
    Apr 25 '18 at 17:29
2
$\begingroup$

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.

$\endgroup$
0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.