Homework Statement
As the title says, I'm trying to calculate the fundamental noise limit for an ideal photodetector, by specifically looking at the rate of incidence of annihilation of photons (and subsequent excitation of conducting electrons) on the detection surface. Since I'm looking for...
Well, I can only think of two ways to do the dot product, and they must both be wrong. The way I thought was correct is
S^{2} = (S_{1}+S_{2})\cdot (S_{1}+S_{2})=S_{1}^{2}+S_{2}^{2}+2S_{1}\cdot S_{2}
2S_{1}\cdot S_{2}=(S^{2}-S_{1}^{2}-S_{2}^{2})\frac{1}{2}
The other way would be S_{1}\cdot...
Also, a question on why we need to prove those facts. I understand P_{\pm}^{2}=P_{\pm} as that's the definition of the projection operator. I'm unsure how the first and third requirements tie into things though.
Could you elaborate further on what you mean in the first sentence? I can apply the found dot product into P to get
P_{\pm}=((\frac{1}{2}) \pm (\frac{1}{4}) \pm (\frac{(S^{2}-S_{1}^{2}-S_{2}^{2})}{\hbar^{2}}))|\psi>
but this gives me the eigenvalues of P. To find the eigenvalues of S^2, do I...
Homework Statement
Consider the center of mass system of two interacting fermions with spin 1/2.
a) What is the consequence of the Pauli exclusion principle on the two-particle wave function?
b)Let S1 and S2 be the spin operators of the two individual fermions. Show that the operators...
Okay so P is uniform in the z direction. z hat dotted with r hat is equal to the cosine of the angle between the two, which I call phi. This angle is the top angle in the triangle that i utilized with the law of cosines to find r^2. Using the law of cosines again, I can relate phi to the rest of...
Homework Statement
Calculate the potential of a uniformly polarized sphere directly from eq. 9
Homework Equations
V(r)=k \int \frac {P(r') \cdot \hat{r}} {r^2} d\tau
The Attempt at a Solution
P is a constant and can be factored out. Since r is taken, call the radius of the sphere R and and...
k, that makes sense. I'm not sure why I had that backwards..it definitely makes sense to have it the other way around because theta as defined should always rotate through more than phi if b is larger.
a\theta=b\phi
\theta = \frac {b} {a} \phi
but that's just the rotation, we add in another...
Theres rotation occurring too, I know that now. You can tell this by keeping the point of contact fixed and just sliding along the edge of the coin. If they are of the same size, it will rotate once just by doing this, and this holds regardless of the relative size of the two coins. one full...
well, when considering two spheres rolling around each other of the same size, they both have the same circumference so they rotate through the same angle. if the lower one is four times larger then it takes the outer sphere four full rotations to get all the way around, so inner angle = 4 *...
yeah, i think the other angle is the angle between the distance from the point of contact to the center of curvature of the bottom hemisphere and the distance between the initial contact and the center of curvature. is that correct?
sorry, had a project to take care of for a couple days there...
Homework Statement
A solid homogeneous hemisphere of radius a rests on top of a rough hemispherical cap of radius b, the curved faces being in contact. Show that the equilibrium is stable if a is less than 3b/5.
Homework Equations
V = mgh
The Attempt at a Solution
So the center of mass of a...
Okay. I have to go into work now but the second part should just involve taking the derivative of the angular momentum, right? Not sure how that shakes out to both sin and cos in the term.