Hermitian conjugate of an operator

In summary, the conversation discusses the concept of Hermitian conjugate operators, specifically focusing on the relationship between bra and ket vectors and how they relate to the adjoint of an operator. The fourth line explains that the ket corresponding to <v|A is A†|v>, and this follows from the previous lines and the properties of the inner product. The fifth line introduces notation and provides an example using the inner product to illustrate the relationship between the bra and ket vectors and their adjoints.
  • #1
spaghetti3451
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Hey guys, I'm doing a third year course called 'Foundations of Quantum Mechanics' and there's this thing in my notes I don't quite get. I was hoping to get your help on this, if you don't mind. It's about Hermitian conjugate operators. The sentences go

(v, Au) = (A†v|u)

<v|A|u> = <v|(A|u>)
<v|A|u> = (<v|A)|u>
(<v|A)† = A†|v>
<v|A|u>*=<u|A†|v>

I am wondering how the fourth line might follow from the third and if the fifth line belongs to this chunk or if it's a separate piece of info on its own.
I would appreciate any form of help!
 
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  • #2
The right-hand side of the first line looks weird. Did you mean (A†v,u)?

I don't think it's very useful to define the adjoint of a bra ket, but if we define it, it's natural to define it as the bra that's assigned to it by the isomorphism from the Hilbert space into its dual space that's guaranteed to exist by the Riesz representation theorem for Hilbert spaces. (The adjoint of a bra is defined similarly, using the inverse of the same isomorphism, to ensure that |u>††=|u>).

So the fourth line says that ket that corresponds to the bra <v|A is A†|v>. This follows from the previous lines and the properties of the inner product.

If we denote the ket that corresponds to <v|A by |w>, and use the notation ##\big(|\alpha\rangle,|\beta\rangle\big)## for the inner product of ##|\alpha\rangle## and ##|\beta\rangle##, we have
$$\big(|w\rangle,|u\rangle\big) =\langle w|u\rangle =\big(\langle v|A\big)|u\rangle =\langle v|\big(A|u\rangle\big) =\big(|v\rangle,A|u\rangle\big) =\big(A^\dagger|v\rangle,|u\rangle\big).$$
 
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What is the Hermitian conjugate of an operator?

The Hermitian conjugate of an operator is a mathematical operation that involves taking the complex conjugate of the operator and then transposing it.

Why is the Hermitian conjugate important in quantum mechanics?

In quantum mechanics, the Hermitian conjugate is important because it allows us to calculate the expectation values of physical quantities, such as energy or momentum, which are represented by operators.

How is the Hermitian conjugate different from the adjoint of an operator?

The Hermitian conjugate and the adjoint of an operator are two different mathematical operations. While the Hermitian conjugate involves taking the complex conjugate and transposing the operator, the adjoint involves only transposing the operator without taking the complex conjugate.

What is the physical significance of the Hermitian conjugate?

The Hermitian conjugate has physical significance because it allows us to determine whether an operator is self-adjoint, which means it is equal to its own Hermitian conjugate. This is important because self-adjoint operators correspond to physical observables in quantum mechanics.

How is the Hermitian conjugate related to the inner product of two vectors?

The Hermitian conjugate is related to the inner product of two vectors in the sense that it is used to calculate the inner product of two vectors in a complex vector space. This is done by taking the Hermitian conjugate of one vector and then multiplying it with the other vector.

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