Dirac Notation: Bra & Ket Conjugation Rules

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Discussion Overview

The discussion revolves around the rules of hermitian conjugation in Dirac notation, specifically regarding the manipulation of bras and kets. Participants explore the implications of conjugating expressions and whether certain equalities lead to conclusions about the nature of the quantities involved.

Discussion Character

  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant questions whether it is valid to hermitianly conjugate expressions in Dirac notation, providing an example with = - .
  • Another participant suggests expanding the conjugation of the initial expression to explore its implications.
  • A participant asserts that the quantity is a complex number and that its conjugate can be applied to both sides of an equality, leading to further exploration of the implications of this property.
  • Concerns are raised about whether the equality = implies that is real, questioning the behavior of the operator P under conjugation.
  • Another participant confirms that = ||^2 is indeed a real number, suggesting a resolution to the concern raised.

Areas of Agreement / Disagreement

Participants express differing views on the implications of hermitian conjugation and whether certain expressions can be equated. While some agree on the real nature of specific quantities, the broader implications of the conjugation rules remain contested.

Contextual Notes

Participants do not fully resolve the implications of the operator P when conjugated, leaving open questions about its behavior in different contexts.

Somali_Physicist
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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.
 
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Try expanding (<w|c> = <b|c> - <d|c>)*
 
Somali_Physicist said:
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.
 
stevendaryl said:
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.
 
Last edited:
yes the number alpha is real
what is the définition of your operator P?
 
Somali_Physicist said:
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.
 

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