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Suppose I have been given a ket vector $$|i\rangle = \frac{1}{\sqrt{2}}|u\rangle + \frac{i}{\sqrt{2}}|d\rangle$$ and I want to find the corresponding bra vector. How can I do that?

My attempt : My guess is that the corresponding bra vector would be something like : $$\langle i| = \frac{1}{\sqrt{2}}\langle u| + \frac{i}{\sqrt{2}} \langle d|$$ But I am not sure about it. Any help/hint is appreciated

Thanks in advance :)

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    $\begingroup$ @Jakob That is an answer. It's not appropriate to put answers in comments. It is possible that it will be deleted. $\endgroup$
    – Bill N
    Jul 29, 2021 at 14:36
  • $\begingroup$ Hint: have you tested your guess by computing $\langle i|i\rangle$? $\endgroup$
    – J.G.
    Jul 29, 2021 at 14:44
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    $\begingroup$ @J.G. ah I see, that's not correct. The correct one should be $$\langle i| = \frac{1}{\sqrt{2}}\langle u| - \frac{i}{\sqrt{2}} \langle d|$$? $\endgroup$
    – Om3ga
    Jul 29, 2021 at 14:47

1 Answer 1

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Bras are the Hermitian adjoint of kets; thus, as said by @Jacobs, you have to take the Hermitian adjoint of $|i\rangle$, which means taking the transpose of the complex conjugate. For numbers it's just equal to the complex conjugate. For example,

$$\frac{1}{\sqrt{2}}|u\rangle\to\langle u|\frac{1}{\sqrt{2}}$$ since the coefficient is real.

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