# Dirac Notation: Bra & Ket Conjugation Rules

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• Somali_Physicist
In summary: This is because the Hermitian conjugate of a complex number is its complex conjugate, so the Hermitian conjugate of <c|w> is <w|c>. Therefore, <c|w><w|c> = <w|c><c|w> = |<c|w>|^2, which is a real number. The operator P does not change this fact. In summary, when applying the Hermitian conjugate to both sides of an equality in Dirac notation, the resulting expression will be a real number.

#### 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.

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.

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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.

## 1. What is Dirac notation?

Dirac notation, also known as bra-ket notation, is a mathematical notation used in quantum mechanics to represent state vectors and operators. It was developed by physicist Paul Dirac to simplify and unify the mathematical framework of quantum mechanics.

## 2. What is the purpose of using bra and ket in Dirac notation?

The bra and ket notation is used to represent vectors and dual vectors in a vector space. In quantum mechanics, the bra represents the dual vector or "conjugate" of a ket, and vice versa. This allows for a more compact and elegant way of writing mathematical expressions involving vectors and operators.

## 3. What are the rules for conjugating a ket?

Conjugating a ket involves taking the complex conjugate of each element in the ket vector. This means changing the sign of the imaginary component, if present, in each element. The resulting vector is then written as a bra, with the notation $\langle a|$ instead of $|a\rangle.$

## 4. How do you conjugate a bra?

Conjugating a bra is similar to conjugating a ket, but in the opposite direction. This means taking the complex conjugate of each element in the bra vector and writing it as a ket, with the notation $|a\rangle$ instead of $\langle a|.$

## 5. Why is bra-ket notation useful in quantum mechanics?

Bra-ket notation provides a concise and intuitive way of representing mathematical expressions in quantum mechanics. It also allows for the easy manipulation and calculation of inner products, expectation values, and other operations involving vectors and operators. Additionally, bra-ket notation is essential for understanding and solving problems in quantum mechanics, making it a fundamental tool for scientists in this field.