Using log identities:
##log((\alpha - 1)!^2) = 2(log(\alpha - 1)!)##
Then apply Stirling's Approximation
##(2[(\alpha - 1)log(\alpha - 1) - (\alpha - 1)##
## = 2(\alpha -1)log(\alpha -1) - 2\alpha+2##
Is this correct? I can't find a way to check this computationally.
I know that the ##F_{AonB} = -F_{BonA}##, but I just wanted to check something. If object A is the truck, then the x-direction should have a vector coming from the force of the engine driving the truck forward, a vector pointing in the negative direction for friction, and a vector in the...
Thanks, I see what the problem is, also I was using bad representation of the spins, this is from chapter 7 problem 7 in Sakurai 2nd ed. and the way my professor told me to do the problem made it a lot easier.
I am having trouble with the normalization part.
To get a spin ##|32>## state I could have the following possibilities
##C_1|111110> + C_2|111011> + C_3|101111>##
This should be equivalent to
##C_1|11>|21> + C_2|11>|21> + C_3|10>|22>##
That is a spin 1 particle and a spin 2 particle that need...
I found a paper that does this in 1-dimensions and I can kind of expand that to 3-dimensions, but they integrate between ##\pm \sqrt{\mu}##. Is this because at ##\sqrt{\mu}## you have a density that drops below the level where you can still be in the Thomas-Fermi regime and the kinetic energy...
1. Since N is large, ignore the kinetic energy term.
##[-\mu + V(r) + U|\Psi (r)|^2]\Psi (r) = 0##
2. Solve for the density ##|\Psi (r)|^2##
##|\Psi (r)|^2 = \frac{\mu - V(r)}{U}##
3. Integrate density times volume to get number of bosons
##\int|\Psi (r)|^2 d\tau = \int \frac{\mu -...
Ok, thanks, I was unclear on whether the permutations were degenerate or if the spin states were degenerate. However, if the spin states are not degenerate then there are like 27 of them for identical spin 1 boson and I'm not sure how many of those would be symmetric.
S and A are the operators that project a ket from ##V^{\otimes 3}## into the subspaces of symmetric (##Sym^NV##) and antisymmetric (##Anti^NV##) configurations by way of the permutation operators. I wrote ##|33>## because it's the simplest bosonic spin configuration I could think of that would...
My question is really about the degeneracies. I know that the symmetrization postulate says that there is only 1 unique ket in the subspace ##Sym^{N}V##, but does this mean that if I know one unique spin configuration that is symmetric, say ##|33>## then is it correct to say the ground state...
Ok, I see the mistake now. It looks like I wrote the solution to the integral as ##\frac{1}{4}(u-cos(kx)sin(kx))## instead of ##\frac{u}{2} - \frac{1}{4}cos(kx)sin(kx)##
Everything else that I have written down is correct those were just typos in the latex. Thank you so much for your help.
The ##sin^2(kx)## term came from multiplying from the left the complex conjugate, am I wrong there? And I got a ##\frac{2}{4w}## from the first term because I did a trig sub of ##sin(kx)cos(kx) = 2sin(2kx)## that changed my bounds of integration from ##0 \rightarrow w## to ##0 \rightarrow2n\pi##...
I took the w derivative of the wave function and got the following. Also w is a function of time, I just didn't notate it for brevity:
$$-\frac{\sqrt{2}n\pi x}{w^{3/2}}cos(\frac{n\pi}{w}x) - \frac{1}{\sqrt{2w^3}}sin^2(\frac{n\pi}{w}x)$$
Then I multiplied the complex conjugate of the wave...
Split the integral
$$\frac{Aa}{\sqrt{2\pi}}\int^{\infty}_{-\infty}e^{ikx}dk - \frac{A}{\sqrt{2\pi}}\int^{\infty}_{-\infty}|k|e^{ikx}dk$$
Apply the boundary conditions, this is where my biggest source of uncertainty comes from I doubled the integral and integrated from 0 to a instead of from -a...
I just graduated with a bachelor's in physics and I am having trouble getting accepted to a PhD program. I transferred to the university from a community college where my GPA was something like a 2.1 (I went back and took some classes and it is not a 2.5) and I spent my first year in chemistry...