I've just started on QM and I'm puzzled with a lot of new ideas in it.

1.On a recent lecture I've attended, there is an equation says: $\langle q'|\sum q|q\rangle \langle q|q' \rangle =\sum q \delta(q,q')$

I don't understand why $\langle q'|q\rangle \langle q|q' \rangle =\delta (q,q')$

Can you explain this equation for me?

2.Actually, I'm still not clear about the bra-ket notation. I've learnt the bra and the ket could be considered as vectors. Then what are the elements of the vectors?

Thank you very much!

  1. The equation is true, if $|q\rangle$,$|q'\rangle$ are chosen from an orthonormal set of vectors, such as an eigenbasis of an operator. Then, by definition, $\langle q|q' \rangle = \delta_{q,q'}$

  2. $| q \rangle$ just denotes some vector labeled $q$ in some Hilbert space. The dimension equals the number of distinct classical states that your system can be in.


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