Quantum Mechanics: Degenerate Perturbation Theory on square well

Evil Harry
Messages
2
Reaction score
0

Homework Statement


Hi I am trying to apply degenerate perturbation theory to a three dimensional square well v= 0 for x, y,z interval 0 to a, perturbed by H' = xyz (product) from 0 to a, otherwise infinite. I need to find the correction to energy of the first excited state which I know is triply degenerate. I am using Griffith's textbook.


Homework Equations





The Attempt at a Solution



The problem I am having is in constructing the matrix. the one that looks like
Waa Wab
Wba Wbb

Or H11 H12
H21 H22
as some other textbooks calls it. I know for this problem I need to use a three dimensional matrix the above is just to clarify. Specifically my problem arises when i need to solve the values of the non diagonal elements. say Wab because this leads to an integral that I can't seem to solve the integral of say:
integrate x sin(Ax) sin(Bx) between 0 and a, where A and B are different because the wave functions are different say that n=1 for the argument in A and n=2 for the argument in B. so that the wave functions are \psi 112 and \psi211. Now I am trying to avoid solving the integral having been unsucsesful in numerous attempts and instead use another approach of finding an operator that commutes with H' and H (the unpeturbed system). So my new problem is an operator that I can use. I have never before used this method and there isn't really a worked example in the book. So any help with this integral (see bottom) or in using this method will be much appreciated. How do I know which operator to use and since they have simultaneous eigenfunctions wouldn't the diagonal elements be the same as using non degenerate perturbation theory <ψ | H'|ψ > or am I now using different eigenfunctions than those of the square well?
E=∫_0^a▒〖xsin(xπ/a) sin⁡(2πx/a)dx〗
 
Physics news on Phys.org
Your integral is of the form x sin(ax) sin(bx). There are many ways to do this. You could first use sin(ax)sin(bx) = (1/2)cos((a-b)x) - (1/2)cos((a+b)x), then you have to integrate x cos(cx), which can be done by integration by parts.
 
Thanx for the help, I forgot about that identity, haven't used it in a while.
 
Hi, I had an exam and I completely messed up a problem. Especially one part which was necessary for the rest of the problem. Basically, I have a wormhole metric: $$(ds)^2 = -(dt)^2 + (dr)^2 + (r^2 + b^2)( (d\theta)^2 + sin^2 \theta (d\phi)^2 )$$ Where ##b=1## with an orbit only in the equatorial plane. We also know from the question that the orbit must satisfy this relationship: $$\varepsilon = \frac{1}{2} (\frac{dr}{d\tau})^2 + V_{eff}(r)$$ Ultimately, I was tasked to find the initial...
The value of H equals ## 10^{3}## in natural units, According to : https://en.wikipedia.org/wiki/Natural_units, ## t \sim 10^{-21} sec = 10^{21} Hz ##, and since ## \text{GeV} \sim 10^{24} \text{Hz } ##, ## GeV \sim 10^{24} \times 10^{-21} = 10^3 ## in natural units. So is this conversion correct? Also in the above formula, can I convert H to that natural units , since it’s a constant, while keeping k in Hz ?
Back
Top