Can you plz solve this Given two bases |𝐵𝑖 ⟩ and |𝐶𝑖 ⟩, show that the change of basis matrix 𝑃𝐵←𝐶 is the inverse of the change of basis matrix in the other direction, 𝑃𝐶←B
I am actually working on a video covering precisely this: changing from one basis to another. It should come out in the next few weeks, I'll let you know when it does!
An operator can indeed be written down as a matrix in the matrix formulation of quantum mechanics. We cover this topic in this video: th-cam.com/video/wIwnb1ldYTI/w-d-xo.html
The adjoint operator is very useful because its action in the dual space is equivalent to the action of the "normal" operator in state space. And as we very often need to use the dual space in quantum mechanics, then we need the adjoint operator. There are many examples, but just to give you a quick one: if you have a state A|psi> and you want to normalize it, then you need to calculate , where A^{dagger} is the adjoint operator. To learn more about dual space, you can check out these two videos: th-cam.com/video/hJoWM9jf0gU/w-d-xo.html th-cam.com/video/pNFna7zZbgE/w-d-xo.html
To show that an operator A is Hermitian, we need to show that, for two arbitrary states |psi> and |phi>, we have: =* When A is the position operator, it is easiest to show that this holds working in the position representation, where the operator then just acts as a multiplicative factor. I hope this helps you get started in the proof!
@@ProfessorMdoesScience Let M be a multiplication operator (1.5) and show that ker M = (0) if and only if f.L({x:cp(x) = O}) = O. G'lVe necessary and sufficient conditions on cp that ran M is closed ............ Can the question be solved?
We want to keep the comments about the content of the videos; for your questions I recommend you try Stack Exchange or similar. Having said this, we plan to publish a video on unitary operators in the next few weeks which should help with your question. Shortly, an operator U is unitary if U^(dagger)=U^(-1).
Hi please, Guide me how to prove that ihcut d/dt is also a Hermitian operator? since it represents Hamiltonian it must be Hermitian, to have this we must have (d/dt)^+ =-d/dt,, But in case of deriving probability density we come across, terms like d¥/dt and d¥*/dt,,, But we see , ihcut d¥/dt = H¥, and thus d¥/dt=-i/h ¥ But to have d¥*/dt,, we have to take Hermitian to the previous, but then,, If (ihd/dt)^+ =ihd/dt, then we again have the same expression, for d¥*/dt =i/h ¥*, but it is non consistent with books, in Nouredine Zettili book there is d¥*/dt=-i/h ¥*,, there is minus sign bothering me , if ihcut d/dt is Hermitian then there shouldn't be minus sign at all, but books can't be wrong,,, so what is my mistake here ???
Your starting point looks incorrect, as the Hamiltonian is not equal to i*hbar*d/dt. The Schrödinger equation tells us how a quantum state evolves in time (just like Newton's second law tells us how a classical state evolves in time), but does not define the Hamiltonian. The Hamiltonian is defined separately (e.g. for a simple harmonic oscillator it would have a kinetic energy term and a potential that depends quadratically on the displacement). I hope this helps!
@@ProfessorMdoesScience Alright Professor, I understand that ihcut d/dt is not equal to the Hamiltonian,,, but Isn't ihcut d/dt a Hermitian operator?,, And If it is Hermitian then how to prove this ? For space operators like momentum operator -ih d/dx we can prove it by integrating by parts but how to prove ihcut d/dt as it contains time derivative??
@@vivekpanchal3338 Given our disucssion above, I guess here you are thinking of i*hbar*d/dt as a purely mathematical differential operator, right? I haven't thought about this, but not sure there is a reason why it should be Hermitian. I hope this helps!
@@ProfessorMdoesScience , Yes professor, I am thinking about ihbar d/dt as an operater, usually we write in time dependent case as ihbar d¥/dt =H¥,,, Then I thought that, is ihabar d¥/dt is the same as Hamiltonian H !! And we know that H is a Hermitian then ihbar d/dt should must be a Hermitian operator. 😅 Alright I understand, I was wrong, H is not same as ihbard/dt it is mere a differential operator,, for some function ¥ it gives equality for H¥ 😊, Thank you for giving your precious time to explain 😊😊
This is for a general operator A, which is not necessarily Hermitian. If we did have a Hermitian operator, A=A† by definition. So, the equality still holds.
Everyone should watch these videos
Thanks for your support!
Amazing explanation, thanks!
Glad you enjoyed it!
Can you plz solve this
Given two bases |𝐵𝑖 ⟩ and |𝐶𝑖 ⟩, show that the change of basis matrix 𝑃𝐵←𝐶 is the inverse of the change of basis matrix in the other direction, 𝑃𝐶←B
I am actually working on a video covering precisely this: changing from one basis to another. It should come out in the next few weeks, I'll let you know when it does!
As promised, here is the video on changing basis:
th-cam.com/video/CDmXvPDMIFs/w-d-xo.html
Can I conceptually treat an operator as a matrix?
An operator can indeed be written down as a matrix in the matrix formulation of quantum mechanics. We cover this topic in this video: th-cam.com/video/wIwnb1ldYTI/w-d-xo.html
hello professor, a small question, does taking conjugation removes a dagger?
Yes, for example: = *. Or perhaps what you are asking is: (A^dagger)^dagger = A, which is also true. I hope this helps!
@@ProfessorMdoesScience Thanks a lot sir . (-:
HI, A QUICK QUESTION ,what is the need for an adjoint operator. what is the significance of using the adjoint operator?
The adjoint operator is very useful because its action in the dual space is equivalent to the action of the "normal" operator in state space. And as we very often need to use the dual space in quantum mechanics, then we need the adjoint operator. There are many examples, but just to give you a quick one: if you have a state A|psi> and you want to normalize it, then you need to calculate , where A^{dagger} is the adjoint operator.
To learn more about dual space, you can check out these two videos:
th-cam.com/video/hJoWM9jf0gU/w-d-xo.html
th-cam.com/video/pNFna7zZbgE/w-d-xo.html
@@ProfessorMdoesScience thank you so much.ur explanations are so good ,that i subbed ur channel for more
Thanks for your support! :)
How can we show position operator is hermitian?
To show that an operator A is Hermitian, we need to show that, for two arbitrary states |psi> and |phi>, we have:
=*
When A is the position operator, it is easiest to show that this holds working in the position representation, where the operator then just acts as a multiplicative factor. I hope this helps you get started in the proof!
@@ProfessorMdoesScience Done professor! Thank you :)
Very nice sir 😊🙏🙏
Thanks, glad you like it!
Hello
How can I contact you?
You can find our contact information in the "About" page of the channel.
@@ProfessorMdoesScience
Let M be a multiplication operator (1.5) and show that ker M = (0) if and
only if f.L({x:cp(x) = O}) = O. G'lVe necessary and sufficient conditions on cp
that ran M is closed ............ Can the question be solved?
@@ProfessorMdoesScience
If fez) = expz = L~~oZ"jn! and A is hermitian, show that f(iA) is unitary .......... Can the question be solved?
We want to keep the comments about the content of the videos; for your questions I recommend you try Stack Exchange or similar. Having said this, we plan to publish a video on unitary operators in the next few weeks which should help with your question. Shortly, an operator U is unitary if U^(dagger)=U^(-1).
Hi please, Guide me how to prove that ihcut d/dt is also a Hermitian operator? since it represents Hamiltonian it must be Hermitian, to have this we must have (d/dt)^+ =-d/dt,,
But in case of deriving probability density we come across, terms like d¥/dt and d¥*/dt,,,
But we see ,
ihcut d¥/dt = H¥, and thus d¥/dt=-i/h ¥
But to have d¥*/dt,, we have to take Hermitian to the previous, but then,, If (ihd/dt)^+ =ihd/dt, then we again have the same expression, for d¥*/dt =i/h ¥*, but it is non consistent with books, in Nouredine Zettili book there is d¥*/dt=-i/h ¥*,, there is minus sign bothering me , if ihcut d/dt is Hermitian then there shouldn't be minus sign at all, but books can't be wrong,,, so what is my mistake here ???
Your starting point looks incorrect, as the Hamiltonian is not equal to i*hbar*d/dt. The Schrödinger equation tells us how a quantum state evolves in time (just like Newton's second law tells us how a classical state evolves in time), but does not define the Hamiltonian. The Hamiltonian is defined separately (e.g. for a simple harmonic oscillator it would have a kinetic energy term and a potential that depends quadratically on the displacement). I hope this helps!
@@ProfessorMdoesScience Alright Professor, I understand that ihcut d/dt is not equal to the Hamiltonian,,,
but Isn't ihcut d/dt a Hermitian operator?,, And If it is Hermitian then how to prove this ? For space operators like momentum operator -ih d/dx we can prove it by integrating by parts but how to prove ihcut d/dt as it contains time derivative??
@@vivekpanchal3338 Given our disucssion above, I guess here you are thinking of i*hbar*d/dt as a purely mathematical differential operator, right? I haven't thought about this, but not sure there is a reason why it should be Hermitian. I hope this helps!
@@ProfessorMdoesScience , Yes professor, I am thinking about ihbar d/dt as an operater, usually we write in time dependent case as ihbar d¥/dt =H¥,,,
Then I thought that, is ihabar d¥/dt is the same as Hamiltonian H !! And we know that H is a Hermitian then ihbar d/dt should must be a Hermitian operator. 😅
Alright I understand, I was wrong, H is not same as ihbard/dt it is mere a differential operator,, for some function ¥ it gives equality for H¥ 😊, Thank you for giving your precious time to explain 😊😊
thx i love you
Glad you like this! :)
Why you write the condition of hermitian operator
= ( )*
Must be write
= ( )*
Not A+ between the bra ket
This is for a general operator A, which is not necessarily Hermitian. If we did have a Hermitian operator, A=A† by definition. So, the equality still holds.
❤