# Is this Dirac notation Bra-Ket manipulation correct?

In my introductory condensed matter course, my teacher wrote the following for a 2-band model with eigenstates $$|m\rangle,|n\rangle$$. The dots indicate derivatives with respect to some arbitrary Hamiltonian parameter. However, I think there's a mistake. $$\langle\dot{m}|\dot{n}\rangle=\Sigma_p \langle\dot{m}|p\rangle\langle p|\dot{n}\rangle \text{ (insert identity)} = \langle\dot{m}|m\rangle\langle m|\dot{n}\rangle + \langle\dot{m}|n\rangle\langle n|\dot{n}\rangle \text{ (2 bands)} =\langle m|\dot{n}\rangle[\langle n|\dot{n}\rangle -\langle m|\dot{m}\rangle ],$$ where he used the product rule: $$d\langle n|n\rangle = d(1) = 0 \implies \langle n|\dot{n}\rangle +\langle \dot{n}|{n}\rangle = 0 \implies \langle n|\dot{n}\rangle = -\langle \dot{n}|{n}\rangle$$.

But, shouldn't the $$\langle n|\dot{n}\rangle$$ term in his equation also have a minus sign, because $$d\langle m|n\rangle = d(0) = 0 \implies \langle m|\dot{n}\rangle +\langle \dot{m}|{n}\rangle = 0 \implies \langle m|\dot{n}\rangle = -\langle \dot{m}|{n}\rangle$$? That is, shouldn't it be: $$\langle m|\dot{n}\rangle[-\langle n|\dot{n}\rangle -\langle m|\dot{m}\rangle ]=-\langle m|\dot{n}\rangle[\langle n|\dot{n}\rangle +\langle m|\dot{m}\rangle]?$$

• Your answer seems right to me, but I could be missing something too.
– Guy
Commented Aug 18, 2020 at 2:14
• I agree with David, something is off in your teacher's derivation. Commented Aug 18, 2020 at 21:46
• Your argument is correct. Commented Feb 15, 2023 at 21:17