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d3nat
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Homework Statement
Consider a particle confined in a cubical box with the sides of length L each.
Obtain the general solution to the eigenvalues and the corresponding eigenfunctions.
Compute the degeneracy of the first excited state.
A perturbation is applied having the form
H' = V from 0 <x < ##\frac{L}{2}## and 0<y<##\frac{L}{2}##
= 0 elsewhere
Compute the first order shifts in the energy levels of the first excited state due to the perturbation.
Construct the corresponding eigenstates.
Homework Equations
The Attempt at a Solution
I'm struggling with perturbation theory.I solved the first part of the problem.
the first excited state is triply degenerate
and has Energy equal to:
## E = \frac{\pi^2\hbar^2}{2mL^2}(n_1^2+n_2^2+n3^2) = \frac{6\pi^2\hbar^2}{3mL^2}####\psi = (\frac{2}{L})^\frac{3}{2}sin(k_1x)sin(k_2y)sin(k_3z)##
the quantum numbers have to start at n = 1
so for the excited state the degeneracy is
##n_1 = 1, n_2 = 1, n_3 = 2 ## corresponds to ##\psi_1## and ##E_1##
##n_1 = 1, n_2 = 2, n_3 = 1 ## corresponds to ##\psi_2## and ##E_2##
##n_1 = 2, n_2 = 1, n_3 = 1 ## corresponds to ##\psi_3## and ##E_3##
Now, I've used projection operators on the Hamiltonian and have gotten through a series of steps the following matrix:
\begin{array}{cc}
E_1 + <1|V|1> & <1|V|2> & <1|V|3> \\
<2|V|1> & E_2 + <2|V|2> & <2|V|3>\\
<3|V|1> & <3|V|2> & E_3+<3|V|3>
\end{array}
which is equal to the matrix:
\begin{array}{cc}
<1|i>\\
<2|i>\\
<3|i>
\end{array}
with ##E_i## multiplied in front (sorry, I don't know how to write that here)
Now, I think I should bring that over and solve like a normal
## (A-\lambda I)x = 0## problem
but what do I do with the subscript i?
Do I drop it?
I also don't know what to do from here.
I tried asking my professor, but he just told me to "solve the integrals."
What integrals??
I'm so lost.
Any help would really be appreciated.
I know that matrix with the bras and kets should have lots of zeros, but I don't know how to get it like that...