Operator acts on a ket and a bra using Dirac Notation

In summary, operator acts on a ket and a bra using Dirac Notation. If the operator is Hermitian then: <ψ| A |Φ> =<Φ| A |ψ>*= a <Φ | ψ>* = a <ψ | Φ>
  • #1
Viona
49
12
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​
asas.png
 
Physics news on Phys.org
  • #2
We should treat this as a homework problem. Can you make an attempt at answering it?
 
  • #3
and.png
 
  • #4
and2.png
 
  • #7
Viona said:
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​
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/
 
  • Like
Likes Viona
  • #8
PeroK said:
Looks good. Can you say anything more if ##\hat A## is Hermitian?
If A is Hermitian then the eigenvalue (a) is a real number.
 
  • Like
Likes PeroK
  • #9
PeroK said:
Can you prove it?
I am not sure if this is true or not.
 
  • #10
Viona said:
I am not sure if this is true or not.
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.
 
  • Like
Likes Viona
  • #11
Switching to normal linear algebra notation. If ##A## commutes with ##A^{\dagger}## and ##v## is an eigenvector of ##A## with eigenvalue ##a##, then:
$$A(A^{\dagger}v) = A^{\dagger}(Av) = A^{\dagger}(av) = a(A^{\dagger}v)$$ 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.
 
  • Like
Likes docnet and Viona
  • #12
PeroK said:
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.
Can we say that if the operator is Hermitian then: <ψ| A |Φ> =<Φ| A |ψ>*= a <Φ | ψ>* = a <ψ | Φ> ?
 
  • Like
Likes PeroK
  • #13
Yes, and if we assume that ##A## and ##A^{\dagger}## share eigenvectors with cc eigenvalues, then: $$\langle \psi |A| \phi \rangle = \langle \phi |A^{\dagger}| \psi \rangle^* = \langle \phi |a^*| \psi \rangle^* = a\langle \psi |\phi \rangle$$ So, that's slightly more general than Hermitian, with ##A## normal and non-degenerate.
 
  • Like
Likes Viona
  • #14
PeroK said:
Yes, and if we assume that ##A## and ##A^{\dagger}## share eigenvectors with cc eigenvalues, then: $$\langle \psi |A| \phi \rangle = \langle \phi |A^{\dagger}| \psi \rangle^* = \langle \phi |a^*| \psi \rangle^* = a\langle \psi |\phi \rangle$$ So, that's slightly more general than Hermitian, with ##A## normal and non-degenerate.
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
Viona said:
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!
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.
 

1. What is Dirac Notation?

Dirac Notation, also known as bra-ket notation, is a mathematical notation used in quantum mechanics to represent vectors and operators. It was developed by physicist Paul Dirac and is based on the concept of a vector space.

2. What is a ket and a bra in Dirac Notation?

In Dirac Notation, a ket (|ψ⟩) represents a quantum state or vector, while a bra (⟨ψ|) represents the conjugate transpose of a ket. Together, they form a bracket called a bra-ket, which is used to represent the inner product between two vectors.

3. How does an operator act on a ket and a bra using Dirac Notation?

An operator, represented by a symbol such as Ĥ, acts on a ket and a bra by multiplying them from the left and right, respectively. For example, the action of an operator on a ket |ψ⟩ can be written as Ĥ|ψ⟩, while the action on a bra ⟨ψ| can be written as ⟨ψ|Ĥ.

4. What is the significance of using Dirac Notation in quantum mechanics?

Dirac Notation is a powerful and concise way to represent and manipulate quantum states and operators. It allows for easy calculation of inner products, expectation values, and other important quantities in quantum mechanics. It also simplifies the notation for mathematical operations such as addition, multiplication, and differentiation.

5. Are there any limitations to using Dirac Notation?

One limitation of Dirac Notation is that it only applies to finite-dimensional vector spaces. It also does not take into account the physical interpretation of the quantities being represented, so it should be used in conjunction with other physical concepts and principles in quantum mechanics.

Similar threads

  • Advanced Physics Homework Help
Replies
1
Views
1K
  • Advanced Physics Homework Help
Replies
15
Views
2K
  • Advanced Physics Homework Help
Replies
22
Views
2K
  • Advanced Physics Homework Help
Replies
3
Views
829
  • Advanced Physics Homework Help
Replies
8
Views
2K
  • Advanced Physics Homework Help
Replies
2
Views
1K
  • Quantum Physics
Replies
3
Views
918
  • Advanced Physics Homework Help
Replies
7
Views
2K
  • Advanced Physics Homework Help
Replies
0
Views
488
Replies
14
Views
949
Back
Top