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.
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...
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...
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.
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.
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...
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...
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:
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...
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...