Proof of |g> = A|f> implies <g| = <f|B

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Homework Help Overview

The discussion revolves around proving the relationship between the states |g> and |f> in the context of quantum mechanics, specifically involving operators and their Hermitian conjugates. The original poster is attempting to show that |g> = A|f> implies

Discussion Character

  • Conceptual clarification, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • Participants discuss the notation involved in the proof and explore the implications of the definition of the Hermitian conjugate. Some suggest using matrix representation and properties of complex conjugation to approach the problem.

Discussion Status

The discussion is ongoing, with participants providing insights into the relationship between the states and the properties of operators. There is acknowledgment of the challenges posed by the notation and the newness of the concepts to some participants. Some guidance has been offered regarding the use of matrix operations and the definition of the Hermitian conjugate.

Contextual Notes

Participants note that they have recently been introduced to Dirac notation and may lack experience with linear analysis and dual vector spaces, which could be contributing to their difficulties in understanding the problem.

rayveldkamp
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Hi,
I am trying to show that |g> = A|f> implies
<g| = <f|B

where A is an operator and B is its Hermitian conjugate.
I think my problem is with notation, but i have not been able to show this as yet.
Thanks

Ray
 
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rayveldkamp said:
Hi,
I am trying to show that |g> = A|f> implies
<g| = <f|B

where A is an operator and B is its Hermitian conjugate.
I think my problem is with notation, but i have not been able to show this as yet.
Thanks

Ray
In a matrix representation, you can write the original equation as a sum of products using matrix multiplication rules. Take the complex conjugate, and replace the conjugates of the elements of A with elements of B. Then from the relationship between elements of <g| and |g>, <f| and |f> you have all you need.
 
rayveldkamp said:
Hi,
I am trying to show that |g> = A|f> implies
<g| = <f|B

where A is an operator and B is its Hermitian conjugate.
I think my problem is with notation, but i have not been able to show this as yet.
Thanks

Ray

That's pretty much the definition of Hermitian conjugate, isn't it?
 
HallsofIvy said:
That's pretty much the definition of Hermitian conjugate, isn't it?
As far as I'm aware, that is how the hermitian conjugate (or adjoint) is defined - though the dual correspondence for A|a> !

I guess you could take the matrix operation of finding the adjoint as definition, and "derive" this result as Older Dan suggests.
 
Hi,
We have only just been introduced to Dirac notation, and have not had a lot of experience in linear analysis and dual vector spaces etc... I have figured out how to do it, i just contract |g> with an arbitrary bra <h|, then do the same with <g| and an arbitrary bra |h>, show the two are equal and hence the expression for <g| must be correct.
Thanks for the help guys, much appreciated
 

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