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...
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...
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...
A problem on my homework:
We learn early on that "the shortest distance between two points is a straight line." Let's prove it...Using the Euler equation, compute the extrema of
∫sqrt(1 + (dy/dx)2)dx from x1 to x2 ...show that this corresponds to lines "y = mx +b".
Euler had a lot of...
Homework Statement
I am looking at the classic resistor cube problem-the one in which there is a cube with resistors all of the same value on each side. Particularily I am having trouble understanding the symmetry argument for the case in which the current enters and exits across the...
Homework Statement
I am supposed to derive this equation: tan(phi) = γtan(θ) by performing an integration to find the flux of E through each of 2 spherical caps; the flux through each of these caps should be equal. The first cap spans the angle θ; the element of surface area may be taken as...
Homework Statement
The coils which first produced a slight but detectable kick in Faraday's galvanometer he describes as made of 203 feet of copper wire each, wound around a large block of wood. The turns of the second spiral (that is, single layer coil) were interposed between those of the...
Homework Statement
The sphere has a radius a and is filled with a charge of uniform. They way I am asked to do this is by building the sphere up layer by layer. And I know that the field outside the sphere is the same as if it were a point charge.
Homework Equations
U = ε/2∫E^2 dv
The...
Homework Statement
Two point charges Q and -Q are separated by a distance d. The point p forms an equilateral triangle with the two charges of side length d. Find the magnitude and direction of the electric field at point p. What is the electric potential at P? Then, a charge of 2Q is placed...
Homework Statement
cupric bromide is heated. I know I should get cuprous bromide and bromide vapor
is this right? CuBr2 --> CuBr + Br2 ?
then, nitric acid is added to the cuprous bromide
CuBr + HNO3 --> ?
This is heated...I know I should get CuO...
then the CuO + CH4 --> Cu + H2O...
Homework Statement
a9t) = 1.2t. If v(1) = 5m/s, v(2) = ?
If x(1) = 6m, x(2) = ?
Homework Equations
The Attempt at a Solution
I know (anti-deriv) that is something like v(t) = .6t^2, but plug in 1 and that's not 5m/s?
Homework Statement
An alert hiker sees a boulder fall from the top of a distant cliff and notes that it takes 1.3s for the boulder to fall the last 3rd of the way to the ground. Ignore air resistance.a) what is the height of the cliff in meters? b) If in part a) you get 2 solutions what does...
Homework Statement
Has anyone ever been on one of these. (I admit I have always been too chicken.) Can anyone explain to me what was felt at the bottom of the first drop and the physics why that was so. Also, estimate the time when the car began to slow down as a fraction of the total...
Homework Statement
A 171 kg roller coaster car is on a track that forms a circular loop of radius 45.9 m in the vertical plane. If the car is to just maintain contact with track at the top of the loop, what is the minimum value for its net inward acceleration at this point?
On the Round...