Physics Asked by Wheels on January 22, 2021
I am new to QM and have a question regarding bra-ket notation and dual vectors.
I have a vector $|Psirangle = |arangle + k|brangle$, where $k$ is some complex number k = $x + iy$. I want to find the dual vector $langlePsi|$. I have some ideas about this but I’m not sure which (if any) is right.
$langlePsi|$ = ($|Psirangle)^{dagger}$ = $(|arangle + k|brangle)^{*}$ = $|arangle^{*} + k^{*}|brangle^{*}$ = $langle a|^{*} + k^{*}langle b|^{*}$.
Is this correct?
If we have an equation among kets such as $$a|Vrangle=b|Wrangle+c|Zrangle+cdots$$
this implies another one among the corresponding bras: $$langle V|a^*=langle W|b^*+langle Z|c^*+cdots$$
In your case $$|Psirangle=|arangle+k|brangle$$ the corresponding relation in bras:
$$langle Psi|=langle a|+langle b|k^*$$
Note: Given a ket $alpha|Vrangleequiv |alpha Vrangle$ the corresponding bra is $$langle alpha V|=langle V|alpha^*$$
In the same way, given a ket $$Omega|Vrangle=|Omega Vrangle$$ the corresponding bra is $$langleOmega V|=langle V|Omega^dagger$$
where $Omega $ is operator.
Answered by Young Kindaichi on January 22, 2021
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