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Viona
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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
Looks good. Can you say anything more if ##\hat A## is Hermitian?Viona said:
Can you prove it?Viona said:
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/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
If A is Hermitian then the eigenvalue (a) is a real number.PeroK said:Looks good. Can you say anything more if ##\hat A## is Hermitian?
I am not sure if this is true or not.PeroK said:Can you prove it?
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.Viona said:I am not sure if this is true or not.
Can we say that if the operator is Hermitian then: <ψ| A |Φ> =<Φ| A |ψ>*= a <Φ | ψ>* = a <ψ | Φ> ?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.
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!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.
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.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!
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.
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.
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.
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.
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.