I see - so technically the eigenvalues are:
\hbar w \left( n + m + 1 \right)
and the eigenfunctions are:
\Psi_{nm} = C_{nm} (a_+^n \psi_0(x)) (b_+^m \psi_0(y))
with:
C_{nm} = A_n A_m
right?
Okay, I think I've got it. Does this look correct?
ANS:
I'm looking for first-order correction to the nth eigenvalue - so I need to solve this:
E_n^1 = \left< \psi_n^0 | H' | \psi_n^0 \right>
Where
\psi_n^0 (x) = \sqrt{ \frac{2}{a} } sin \left( \frac{n \pi x}{a} \right)
and
H'...
Okay, I think I've got it then. Is this correct:
\hat{H} = \frac{ (p_x)^2 }{2m} + \frac{ (p_y)^2 }{2m} + \frac{mw^2}{2} \left( x^2 + y^2 \right)
Which is broken up into components:
\hat{H} = \hat{H_x} + \hat{H_y}
Noting the 1-D harmonic oscillator gives:
E_x = \hbar w \left( n_x...
We have covered the 1D harmonic oscillator, but we haven't seen any other higher dimensional setups yet. We have also used the seperation of variables so far, just not in regards to higher-dimensions.
Just as a general question - once the equation is broken down into two 1D equations, how...
This might be another problem that our class hasn't covered material to answer yet - but I want to be sure.
The question is the following:
Find the eigenfunctions and eigenvalues of a two-dimensional isotropic harmonic oscillator.
Again, I need help simply starting.
Sorry for all the questions - I tend to save them till I'm done with assignments:
Here's the question:
Consider a particle of mass 'm' in a one-dimensional infinite potential well of width 'a'
V (x) = \left\{\begin{array}{c} 0 \ \ \ if \ \ \ 0 \leq x \leq a \\ \infty \ \ \ otherwise...
Thanks for the recommendations. I didn't realize C++ could be used in conjuction with Matlab/Mathematica - do you know of any sources out there that describe how to do it?
Nevermind, I got it now - didn't realize the relation between 1 and e^(i2n(pi))....
A e^{\sqrt{q} \phi} = A e^{\sqrt{q} \left( \phi + 2 \pi \right)}
e^{\sqrt{q} \phi} = e^{\sqrt{q} \phi} e^{\sqrt{q} 2 \pi}
1 = e^{\sqrt{q} 2 \pi}
e^{i 2 n \pi} = e^{\sqrt{q} 2 \pi}
i 2 n...
If I'm going to attempt creating computer programs for simulating theories as complicated as string theory, what language should I be looking at? I have some experience with C++, but if I'm going to devote a large part of my free time to learning a language, I'd like to learn something that...
An additional question, somewhat related:
When determining the eigenvalues, the problem indicates that
f (\phi + 2\pi) = f (\phi)
Given the answer already shown, why would this periodic function require:
2 \pi \sqrt{q} = 2 n \pi i
thanks - knew it was something simple. I actually remembered the other approach too, where you replace f'' with r^2, f' with r, and f with 1, and then solve for what r is. But either approach gives the same result.
Thanks again though.