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.
This is probably a straight forward question, but can someone show me how to solve this problem:
\frac {d^2} {d \phi^2} f(\phi) = q f(\phi)
I need to solve for f, and the solution indicates the answer is:
f_{\substack{+\\-}} (\phi) = A e^{\substack{+\\-} \sqrt{q} \phi}
I know...
I think Hawking estimates it would take a particle accelerator larger than the size of our solar system to probe the Planck length. That is, assuming we don't find a more effective way of creating high energy collisions.
However, some are optimisitic that CERN's energies will be high enough...
SamCJ:
You may find some insight into what you are looking for from this book:
Wholeness and the Implicate Order by David Bohm
David Bohm advanced a lot of research in quantum mechanics, and was even sought after by Einstein himself to be an assistant for him. However, the book...
a few quick notes:
(A) string theory imposes minimums and maximums on size. A minimum size implies no exact singularity, but rather something very small istead. A few in the field have advanced theories of black holes that look more like places for strings to get caught and entangled...
a lot of what I was going to mention was already said, but what really killed what could have been my willingness to give the Discovery Channel a second chance was the 3 "Science of Star Wars" specials...
somehow, helicopters were equated with "star wars spy droids," monks "use the force...
Okay, I figured out how to get the answer via the couple ratios.
(25)(4.4)/(1.1) = By
(By)(1.6)/(1.2) = FE = 133.3 lb
However, I'm still not understanding how I could arrive at the via the equilibrium equations... attached is my free body diagrams for the top pieces. Any help?
Here's the problem:
"The bone rongeur shown [refer to attachment] is used in surgical procedures to cut small bones. Determine the magnitude of the forces exerted on the bone at E when two 25-lb forces are applied as shown."
I understand that this "machine" can be broken into 4 free-body...
Good grief....
That's totally correct Gale17 - thanks. I was trying to use partial fractions on just this part:
\frac{ 1 }{ \left( s+1 \right) \left( s+2 \right) }
and then mutliply the answer by s+3. Obviously that was not getting me anywhere, because I still had an s term on the...
I am asked to find the inverse laplace transform of the following function:
\frac{ \left( s+3 \right) }{ \left( s+1 \right) \left( s+2 \right) }
Using tables, can anyone help me understand why the answer is:
2e^{-t} - e^{-2t}
I'm completely loss on this one, and yet the book...
Note:
Here's the equations and coordinates I'm using (notice that I've moved the coordinate system down to the bottom, so D is at (0,0,0):
A: (-0.045 m, 0, 0)
B: (0.035 m, 0, 0.038 m)
C: (0.035 m, 0, -0.038 m)
D: (0, 0, 0)
F: ( \alpha , 0, \beta )
where F is where the line of...
Here's the question:
"A camera of mass 240g is mounted on a small tripod of mass 200g. Assuming that the mass of the camera is uniformly distributed and that the line of action of the weight of the tripod passes through D, determine (a) the vertical components of the reactions at A, B, and C...
More generally, is there some approach that can be used to calculate the interatomic spacing using the ideal gas law outside of the quantum mechanics context?
Here's the question:
For what temperatures are the atoms in an ideal gas at pressure P quantum mechanical?
Hint: Use the idea gas law
PV = N k_B T
to deduce the interatomic spacing.
Answer:
T < \left( \frac{1}_{k_B} \right) \left( \frac{h^2}_{3m} \right)^{\left(...
x(\theta) = r cos \theta
\frac{dx}{d \theta} = - r sin \theta
therefore,
d\theta = \frac{dx}_{-r sin \theta}
This is where I get my negative from. Should I then just throw out the negative?
I guess the part I'm not understanding is this:
the needle could technically go either direction - it could move right-to-left or left-to-right. One corresponds to positive changes in x, and the other negative.
Likewise, one probability density is always positive, and the other is always...