0
$\begingroup$

I'm trying to prove that unitary transformations preserve orthonormality for quantum chemistry course. There is a proof for this in many mathematical texbooks, but non of them is dealing with matrix elements and bracket notation. Books that introduce bracket notation don't deal with math that much so they just state this as a fact and don't prove it, which is what I need. This is how I tried but it seems that I can't manage.

Usual equations (definitions) regarding unitary transformations and their matrix elements:

$U^{\dagger}=U^{-1}$

$U^{\dagger}U=UU^{\dagger}=UU^{-1}=U^{-1}U=I$

$(U^{\dagger})_{ij}=U_{ji}^{*}$

$∑_{k}U_{ik}^{\dagger}U_{kl}=δ_{il}$

$∑_{k}U_{ki}^{*}U_{kl}=δ_{il}$

To prove that unitary transformation preserve orthonormality, I expressed two orthonormal vectors $|n⟩$ and $|m⟩$ ( of course $⟨m|n⟩=δ_{mn}$ ) in different basis where basis is changed by unitary transformation:

$|n⟩=∑_{k}|k⟩U_{nk}$

$|m⟩=∑_{l}|l⟩U_{ml}$

$⟨m|=∑_{l}⟨l|U_{lm}^*$

Taking inner product of these orthonormal vectors and then substituting their transformed form should yield Kronecker delta.

$⟨m|n⟩=∑_{l}∑_{k}U_{lm}^*⟨l|k⟩U_{nk}=∑_{l}∑_{k}U_{lm}^*δ_{lk}U_{nk}=∑_{k}U_{km}^*U_{nk}=∑_{k}U_{mk}^{\dagger}U_{nk}$

But, indices doesn't seem right comparing to definitons regarding matrix elements of unitary transformations.

Where did this go wrong? Should something here be done in a different manner?

$\endgroup$
  • $\begingroup$ Hint : Try to prove that in general $$ \langle\, \mathrm{A}\,m \,|\, n\,\rangle =\langle \,m \,|\,\mathrm{A}^{\dagger}\,\, n\,\rangle $$ so $$ \langle\, \mathrm{U}\,m \,|\,\mathrm{U}\, n\,\rangle =\cdots $$ $\endgroup$ – Frobenius Jul 19 '17 at 16:54
1
$\begingroup$

If $|{\psi}>=U|{\phi}>$ then $<{\psi}|=<{\phi}|U^\dagger$, from which the desired result directly follows.

$\endgroup$

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.