I Dirac Notation: Bra & Ket Conjugation Rules

[Somali_Physicist]

hey guys just a quick question , within the Dirac notation I we have bras and kets.Is it allowable to simply hermitianly conjugate everything , e.g:

<w|c> = <b|c> - <d|c>
Can we then:
<c|w> = <c|b> -<c|d>

Or is there some subtly hidden rule.

 

[Mentz114]

Try expanding (<w|c> = <b|c> - <d|c>)^*

 

[stevendaryl]

hey guys just a quick question , within the Dirac notation I we have bras and kets.Is it allowable to simply hermitianly conjugate everything , e.g:

<w|c> = <b|c> - <d|c>
Can we then:
<c|w> = <c|b> -<c|d>

Or is there some subtly hidden rule.

The quantity $\langle w | c \rangle$ is just a complex number, and it has the property that $(\langle w | c \rangle^* = \langle c | w \rangle$. So it's perfectly fine to apply the $^*$ operation to both sides of an equality.

 

[Somali_Physicist]

The quantity $\langle w | c \rangle$ is just a complex number, and it has the property that $(\langle w | c \rangle^* = \langle c | w \rangle$. So it's perfectly fine to apply the $^*$ operation to both sides of an equality.

Okay well that leads to my real conundrum:

<w|c><c|w> = α = P
conjugation of both sides
(<w|c><c|w>)* = α* = P*
<c|w><w|c> =α*
<c|w><w|c><w|c><c|w> = α²
=(<w|c><c|w>)² = <w|c><c|w><w|c><c|w>

but does not this imply

<c|w><w|c> = <w|c><c|w> which means <w|c><c|w> real?

i don't understand why that would be the case as the operator P should act differently when conjugated.

 

[PaleMoon]

yes the number alpha is real
what is the définition of your operator P?

 

[protonsarecool]

Okay well that leads to my real conundrum:

but does not this imply

<c|w><w|c> = <w|c><c|w> which means <w|c><c|w> real?

Yes, <c|w><w|c> = |<c|w>|^2 is a real number.