- #1

Viona

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Thread moved from the technical forums to the schoolwork forums

**Summary::**Operator acts on a ket and a bra using Dirac Notation

Please see the attached equations and help, I Think I am confused about this

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- Thread starter Viona
- Start date

- #1

Viona

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Thread moved from the technical forums to the schoolwork forums

Please see the attached equations and help, I Think I am confused about this

- #2

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We should treat this as a homework problem. Can you make an attempt at answering it?

- #3

Viona

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- #4

Viona

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- #5

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Looks good. Can you say anything more if ##\hat A## is Hermitian?

- #6

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Can you prove it?

- #7

Nugatory

Mentor

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Please take a few minutes to learn how to use this site's Latex feature... There's a guide in the help section at https://www.physicsforums.com/help/latexhelp/Summary::Operator acts on a ket and a bra using Dirac Notation

Please see the attached equations and help, I Think I am confused about this

- #8

Viona

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If A is Hermitian then the eigenvalue (a) is a real number.Looks good. Can you say anything more if ##\hat A## is Hermitian?

- #9

Viona

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I am not sure if this is true or not.Can you prove it?

- #10

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I don't think it is true in general for any operator. Certainly for Hermitian operators and also for normal operators (these are operators that commute with their Hermitian conjugate), but not in general.I am not sure if this is true or not.

- #11

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$$A(A^{\dagger}v) = A^{\dagger}(Av) = A^{\dagger}(av) = a(A^{\dagger}v)$$ and we see that ##A^{\dagger}v## is an eigenvector of ##A## with eigenvalue ##a##. Which means that ##A## and ##A^{\dagger}## share eigenspaces.

And it's easy to show that the eigenvalues are complex conjugates.

You can use that to prove the identity in your question, but I think you need that condition.

- #12

Viona

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Can we say that if the operator is Hermitian then: <ψ| A |Φ> =<Φ| A |ψ>I don't think it is true in general for any operator. Certainly for Hermitian operators and also for normal operators (these are operators that commute with their Hermitian conjugate), but not in general.

- #13

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- #14

Viona

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It is clear now. It seems to me that I need to educate myself and study more in linear algebra. Thank you for your help!

- #15

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PS ultimately it's simply this equality you need: $$A^{\dagger}| \psi \rangle = a^*| \psi \rangle$$ And that holds for Hermitian operators, some other operators, but not all operators.It is clear now. It seems to me that I need to educate myself and study more in linear algebra. Thank you for your help!

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