Well, the reason I bring up the two torus is because one of the sample problems in the book I'm using is to "prove that the two torus is a manifold by explicitly constructing an appropriate atlas." Well, I did fail to mention it need not be a maximal one.
I'm trying to get a better understanding of some topology for a GR class I'm taking...I'm wondering if someone can help me understand how to go about constructing atlases or just charts in general. I understand the concept but I am trying to get a better handle on the math.
Homework Statement
To relate E to the total energy of the expanding sphere, we need to integrate over
the sphere to determine its total energy. These integrals are most easily carried out
by dividing the sphere into shells of radius r, and thickness dr, so that each shell
has a volume dV...
Homework Statement
Consider a rigid rotor with moment of inertia Ix = Iy = I, Iz = (1 + ε)I
Classically, his energy is given by, E = L^2/2I.
(c) What are the new energy eigenstates and eigenvalues?
(d) Sketch the spectrum of energy eigenvalues as a function of ε. For what sign of do the...
Homework Statement
We only briefly mentioned this in class and now its on our problem set...
Show that all eigenvalues i of a Unitary operator are pure phases.
Suppose M is a Hermitian operator. Show that e^iM is a Unitary operator.
Homework Equations
The Attempt at a Solution...
Consider a one-dimensional system described by a particle of mass m in the presence
of a pair of delta function wells of strength Wo > 0 located at x = L, i.e.
V(x) = -Wo (x + L) - Wo(x - L) This is a rough but illuminating toy model of an electron in the presence of two positive.
charges...
Homework Statement
Are the momentum eigenfunctions also eigenfunctions of e free particle energy. Operator?
Are momentum eigenfunctions also eigenfunctions of the harmonic oscillator energy operator?
An misplayed system evolves with time according to the shrodinger equation with potential...
Homework Statement
I've never had an exam before in this format. It's 50-min long for Quantum I. It will be mostly short answer with small computations. It's supposed to test our conceptual understanding.
I've been reviewing my notes, making sure I harp on the central ideas of quantum...
Homework Statement
Compute the mass of volatile material (specifically just H and He) not accreted by the Earth. Assume the mass accreted by the Earth is 7e23 kg. How does the mass of the missing volatiles compare with the actual mass of the Earth? Use the following table:
Principal...
Homework Statement
Give a physicist's proof of the following statements regarding energy eigenfunctions:
(a) We can always choose the energy eigenstates E(x) we work with to be purely real
functions (unlike the physical wavefunction, which is necessarily complex). Note: This does not mean...
Homework Statement
For my homework assignment, I'm supposed to find a basis for the space of 3x3 matrices that have zero row sums and separately for zero row columns. I am having a hard time with this as it seems to me that there are a lot of combinations I have to consider. For the first...
Homework Statement
Using <\hat{p}n> = ∫dxψ*(x)(\hat{p})nψ(x) and \hat{p} = -ihbar∂x and the definition of the fourier transform
show that <\hat{p}> = ∫dk|\tilde{ψ}(k)|2hbar*k
2. The attempt at a solution
Let n = 1 and substitute the expression for the momentum operator. Transform the...
hmm, you've given me something to think about. So I can see how B is not invertible because well Gaussian elimination on it fails, but what about C? I don't think you can immediately see it by trying to do the more familiar elimination steps, so is it enough to say that in a way it a dependent...
Homework Statement
The 3x3 matrix A is given as the sum of two other 3x3 matrices B and C satisfying:1) all rows of B are the same vector u and 2) all columns of C are the same vector v.
Show that A is not invertible. One possible approach is to explain why there is a nonzero vector x...
Homework Statement
Carry out the integration ψ = ∫[M(dr/r2)] / √(2m(E-U(r)) - (M2/r2))
E = energy, U = potential, M = angular momentum
using the substitution: u = 1/r for U = -α/r
Homework Equations
The Attempt at a Solution
This is as far as I've gotten: -∫ (Mdu) /...
Homework Statement
We are given L = 1/2mv2 - mgz.
a) Find the equations of motion.
b) Take x(0) [vector] = 0; v(0) [vector] = v0 [vector] ; v0z > 0 and find x(τ) [vector] and v(τ) [vector], such that z(τ) = 0; τ≠0.
c) Find S.
Homework Equations
Euler-Lagrange equation and...
Well if I do this (∂/∂t + ∂/∂x...)* each component of uβ there's still the uα I have to worry about. aα = duα/dτ so this seems to imply that uβ∂β = d/dτ which doesn't seem right...?
And I'm not seeing how I can get that?
Oh wait never mind...I think I got it! If say I transform into a rest...
Homework Statement
As part of a problem I am doing I am asked to show uβ∂βuα = aα where u is 4 velocity and a refers to 4 acceleration. The way to do this is not immediately obvious to me, especially since the problem implies there should be a chain rule step involved which I am not seeing. I...
Homework Statement
"Texbooks that describe perfect fluids are often a little unclear about what is being assumed. It may not be immediately obvious why can't the pressures be different in different directions? Let's examine this. Suppose Tαβ = diag(ρ,(1+ε)P,P,P) . Show that if one performs a...
Homework Statement
I have an exam on Monday and I'm trying to get a few concepts straight before then. I have a few questions...
1) p = E/c for a photon. Is this also true for a wave?
2) Assuming complete absorption, radiation pressure is 1/c x Poynting vector. What happens when there's...
Homework Statement
Define a sequence of numbers Ai by: A0 = 2, An+1 = An/2 + 1/An (for n greater than or equal to 1). Prove that An less than or equal to √(2) + 1/2n for all n greater than or equal to 0. I think it's a safe bet that induction should be used here. I'm having trouble finding the...