Dirac Notation and Hermitian operators

[Nickweynmann]

Homework Statement

Using Dirac Notation prove for the Hermitian operator B acting on a state vector |ψ>, which represents a bound particle in a 1-d potential well - that the expectation value is <C^2> = <Cψ|Cψ>.
Include each step in your reasoning. Finally use the result to show the expectation value of the kinetic energy of the particle in the state |ψ> is <Ekin> = (1/2m)<Pψ|Pψ> ?

Can someone enlighten my thought via a diagram exactly what I am looking at pls

Homework Equations

I have to find the right eqns, which I guess are the Hermitian related eqns: stating what is Hermitian
<a|Cb> = <Ca|b> , where C is operator, a and b are normalisable functions.
<a|λCb> = λ<a|Cb> , where λ is a real constant.
any power of C is Hermitian

The Attempt at a Solution

My attempt has been quite measely as I cannot get any visual sense of it. (ideas?)

So> To prove the <C^2> = <Cψ|Cψ>, all I can think to say is...
The RHS with C being Hermitian.

<Cψ|Cψ> = <ψ|CCψ> = <ψ|C^2ψ>
As we are dealing with the same operator there is no difference if the order in which they so therefore obeys the requirement for a product to commute CC - CC = 0.

therfore that will give the expectation value of <C^2> = <Cψ|Cψ>.

Is that it or is there more reasoning?

Turning to <Ekin> I have no idea what to do because it seems like the (1/2m) is just stuck in there because it previously known. Is there another reasoning?

 

[strangerep]

Using Dirac Notation prove for the Hermitian operator B acting on a state vector |ψ>, which represents a bound particle in a 1-d potential well - that the expectation value is <C^2> = <Cψ|Cψ>.

What is B here? Was that a typo? Where did the original question come from?

For a diagram, try searching Wikipedia for "potential well". Then follow its "see also" link to "finite potential well".

Regarding the kinetic energy, do you know the formula for classical kinetic energy in terms of momentum?

 

[Nickweynmann]

Yeah oops It should be C, I am aware of the relation for Ekin - square of momentum and the factor 1 / 2m. How does the factor come about in the instance of <Px Psi | Px Psi>??

 

[strangerep]

Yeah oops It should be C, I am aware of the relation for Ekin - square of momentum and the factor 1 / 2m. How does the factor come about in the instance of <Px Psi | Px Psi>??

Oh, we're both online. Did you see my edits?

Also, check the 2nd equation in your original "relevant equations" section?

 

[Nickweynmann]

Thanks, I have seen your recommendations, not sure how to see your edits exactly! I am quite new to this forum and it is my first post. So I am basing my working off of these rules for hermitian operators /BUT/ do they still work when I principely have a sandwich int...? Because when the momentum comes together from <P Psi | P Psi> it gives <Psi | P^2 Psi>, so I am held back by the factor which cannot be in the sandwich integral but just attached from prior 'classical' knowledge and the fact that 'm' is real and so obeys the hermitian rules.

Am I a little barking?

 

[strangerep]

Am I a little barking?

Everyone is a little barking (imho), just in different ways.

Thanks, I have seen your recommendations, not sure how to see your edits exactly!

Just reload the page on your browser.

So I am basing my working off of these rules for hermitian operators /BUT/ do they still work when I principely have a sandwich int...? Because when the momentum comes together from <P Psi | P Psi> it gives <Psi | P^2 Psi>, so I am held back by the factor which cannot be in the sandwich integral but just attached from prior 'classical' knowledge and the fact that 'm' is real and so obeys the hermitian rules.

m is a constant here. So let's start again... you know the classical kinetic energy expression, so you can write the quantum kinetic energy operator as $T: = P^2/2m$, right? Now use the result from the first part of the question, and the rules in your "relevant equations", to re-express
$$
\langle\psi| T |\psi\rangle ~=~ ?
$$

 

[Nickweynmann]

Cheers strangerep...

 

[strangerep]

Vector spaces, are destroying me I cannot understand them at all...Bring back wave mechanics...

Nooo... wave mechanics are actually a special case of vector spaces -- infinite-dimensional vector spaces, and one must study Functional Analysis to understand them thoroughly. But for now, have you tried just reading some Wiki pages? E.g., for "vector space", "inner product space", and "linear operator".

seemingly this question is really simple and I am able to show that it all fits for the expectation value but it means naf all to me.

You didn't answer one of my earlier questions: where did your original question come from? Are you doing a course? Studying a book? Or... what?

Do you know of any other resources that can train me up on this material?

My standard reference for QM is Ballentine's "QM -- A Modern Development". The 1st chapter summarizes the necessary math, but I don't whether that would be too advanced at this stage. You can probably look at some of it using Amazon's "look inside" feature, or else on Google Books, to get a feel for whether it would be right for you at this time.

 

[Nickweynmann]

Hi, sorry for replying late, I am a 3rd year undergrad of Physics, and I am struggling to make sense of the vector spaces and usiing dirac notation as before I covered wave mechanics using Schrodingers eqn. I have looked at harmonic oscillators, expectation value, uncertainty etc from the aspect of the wavefunction.

Now I am sweating for the deadline as I cannot make sense of this plus other parts of the same question.

Nick

 

[strangerep]

Now I am sweating for the deadline as I cannot make sense of this plus other parts of the same question.

Can you get a copy of Ballentine from your library? I just looked through ch1 again, and it covers that stuff quite efficiently, and from a physicist-friendly perspective.

(Aside: I'm rather surprised that linear algebra and Dirac notation haven't been covered in your coursework already.)

 

[Nickweynmann]

Hi, I have the dilema that my course is a distance learning degree (open university) so I am balancing learning with my work and being in the sticks reduces the time I can get to any library. Currently my local uni library is closed to outsiders until the exam period finishes (conveniently after my deadline) the e-library is a little useless. So I am stuck to asking questions on this forum...Which I appreciate are to be elicited from me, but with time running low and my work during the day is a little difficultSo you (from what I hear) are a true physicist by nature, resolving problems after midnight..

 

[strangerep]

Hi, I have the dilema that my course is a distance learning degree (open university) so I am balancing learning with my work and being in the sticks reduces the time I can get to any library. Currently my local uni library is closed to outsiders until the exam period finishes (conveniently after my deadline) the e-library is a little useless. So I am stuck to asking questions on this forum...Which I appreciate are to be elicited from me, but with time running low and my work during the day is a little difficult

Urgle! So... these are actual exam questions? And your coursework hasn't covered bra-ket stuff??

[...] resolving problems after midnight..

No. Think: what shape is Earth?

P.S: Check your private messages in a few minutes time.

 

[Nickweynmann]

Oh, now I have it! your wisdom has prevailed on me thanks, wink, nod nod ;) I am very much appreciative for your time.