Quantum Mechanics problem: Determine the value of the constant

Ineedhelpimbadatphys
Messages
9
Reaction score
2
Homework Statement
The problem states work for word.

Using canonical quantization relation, prove that
sum operator ((E_n -E_0)) |< E_n | X | E_0 >|^2) = constant

Where E_0 is the energy corresponding to the eigenstate | E_0 >. Determine the value of the constant. Assume the hamiltonian had a general form H = P/2m +V(X)

Hint: One way to proof this is to think how [H, X], X] is connected to the obove identity.
Relevant Equations
all equations i have are in the statement.
I have no idea where to start with this problem. I am interested in any hints, or ways to proof this. But i would especially like to know how the commutator is connected to the identity.
 

Attachments

  • 385A8420-90F2-4204-8439-15C0224B4160.jpeg
    385A8420-90F2-4204-8439-15C0224B4160.jpeg
    25.8 KB · Views: 131
Physics news on Phys.org
Please, everyone, be respectful of poster asking for a hint about one specific aspect of this problem.
 
  • Like
Likes topsquark, Lord Jestocost and berkeman
Ineedhelpimbadatphys said:
Homework Statement:: The problem states work for word.

Using canonical quantization relation, prove that
sum operator ((E_n -E_0)) |< E_n | X | E_0 >|^2) = constant

Where E_0 is the energy corresponding to the eigenstate | E_0 >. Determine the value of the constant. Assume the hamiltonian had a general form H = P/2m +V(X)

Hint: One way to proof this is to think how [H, X], X] is connected to the obove identity.
Relevant Equations:: all equations i have are in the statement.

I have no idea where to start with this problem. I am interested in any hints, or ways to proof this. But i would especially like to know how the commutator is connected to the identity.
What is ##< E_0 \mid [H,X],X]] \mid E_0 >##?

-Dan
 
Thank you so much. I did actually manage to figure it out. I had tried calculatibg that, and got stuck at < E_0 | XHX | E_n > and assumed I was wrong.

After seeing this, I just kept trying and got it. thank you.
 
  • Like
Likes vanhees71 and topsquark
Thread 'Need help understanding this figure on energy levels'
This figure is from "Introduction to Quantum Mechanics" by Griffiths (3rd edition). It is available to download. It is from page 142. I am hoping the usual people on this site will give me a hand understanding what is going on in the figure. After the equation (4.50) it says "It is customary to introduce the principal quantum number, ##n##, which simply orders the allowed energies, starting with 1 for the ground state. (see the figure)" I still don't understand the figure :( Here is...
Thread 'Understanding how to "tack on" the time wiggle factor'
The last problem I posted on QM made it into advanced homework help, that is why I am putting it here. I am sorry for any hassle imposed on the moderators by myself. Part (a) is quite easy. We get $$\sigma_1 = 2\lambda, \mathbf{v}_1 = \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix} \sigma_2 = \lambda, \mathbf{v}_2 = \begin{pmatrix} 1/\sqrt{2} \\ 1/\sqrt{2} \\ 0 \end{pmatrix} \sigma_3 = -\lambda, \mathbf{v}_3 = \begin{pmatrix} 1/\sqrt{2} \\ -1/\sqrt{2} \\ 0 \end{pmatrix} $$ There are two ways...
Back
Top