Homework Statement
Hi, I'm trying to familiarize with the bra-ket notation and quantum mechanics. I have to find the hamiltonian's eigenvalues and eigenstates.
##H=(S_{1z}+S_{2z})+S_{1x}S_{2x}##
Homework Equations
##S_{z} \vert+\rangle =\hbar/2\vert+\rangle##
##S_{z}\vert-\rangle...
Homework Statement
We have an infinite cylinder that, from radius 0 to a, has a volume current density ##\vec{J(r)}=J_{0}(r/a) \hat{z}## , then from a to 2a, it has a material with uniform linear magnetic permeability ##\mu=(3/2)\mu_0##
, and at the surface, it has surface current...
Thanks pasmith, indeed that is a counterexample, but I have no idea how to define that function (except for the x=0, when x<0, part, of course). Nevertheless, describing it as you did would not count as correct answer (whatever the explicit expression for its formula might be) in replying that...
The b) is a different problem, that says if the integral from zero to infinity converges then its integrand must go to zero when x extends to infinity. I was refering to that problem when I brought up the series.
For the a) problem, following your suggestion: suppose that for some ##a \subset...
Thank you for your patience, I misread you, I see what you mean now, I'll work on that.
Meanwhile I thought something for the b) part. If could treat the problem (since f is continous) from a series point of view (maybe justifying this with the archimedean property of real numbers that states...
Thanks for your reply Hallsoflvy, but I don't quite understand what you are saying. The statement has the condition that the integral is zero on every interval ##[a,b] \subset [0,1]##, therefore how could I suppose that for some ##a## the conclusion is that its integral is not zero?
I thought of...
Homework Statement
a) If ##f: [0,1] \rightarrow \mathbb{R}## is continous and ##\int^{b}_{a} f(x)dx = 0## for every interval ##[a,b] \subset [0,1]##, then ##f(x)=0 \forall x \in [0,1]##
b) Let ##f: [0,\infty) \rightarrow [0,\infty)## be continous. If ##\int^{\infty}_{0} f(x)dx## converges...
Hi Ray Vickson, I was trying to use the theorem that states that if I can decompose the function like this ##F(r,\theta)=H(r)G(\theta)## and ##\lim_{(r) \rightarrow (0)} H(r) = 0## and ##G(\theta)## is bounded in [0,2pi] then the original function is differentiable, but I see now that since the...
I'm sorry, that happens for hurrying. I forgot to divide by ##||(u, v)||## in the definition, which gives me an extra ##r## in the denominator:
##\frac {r^3 cos(\theta) sin(\theta)^2} {r^3} = cos(\theta) sin(\theta)^2##
which means it doesn't exist, therefore is not differentiable in (0,0)...
You're right, Samy_A, it doesn't, and with the polar coordinates I have a function which is the product of two functions ##H(r,\theta)=F(r)G(\theta)## and while ##\lim_{(r) \rightarrow (0)} F(r) = 0##, ##G(\theta)## is bounded in [0,2pi], therefore the original function is differentiable in...
Homework Statement
I need to see if the function defined as
##f(x,y) = \left\{
\begin{array}{lr}
\frac{xy^2}{x^2 + y^2} & (x,y)\neq{}(0,0)\\
0 & (x,y)=(0,0)
\end{array}
\right.##
is differentiable at (0,0)
Homework Equations
[/B]
A function is differentiable at a...
Hi OldEngr63, thanks for your reply, I tried your approach with the angles, so I got
##\hat{θ}) 2m(R+a) \ddot{θ} + mg sinθ=0 ##
which is nice for small angle approximation and describing the oscillatory movement.
And Fr meant friction force. Sorry it took long for me to answer back. Thanks...
Hi everybody, I know this problem has been posted before, but it envolved Lagrangian methods which I haven't seen yet. I would appreciate any help.
1. Homework Statement
A small hoop is rolling without slipping on a bigger cylinder which is stationary. I need to write Newton's Laws and...
Ok, using your hint, I thought that since a is the only component of the distance ## \vec{r} ## that's involved in the cross product, I get for the first case:
## \vec{L} = 2a m V0 ##
and using the same reasoning in the second case I get
## \vec{L}= 2bmV1 ##
Since they should be equal, I...
Homework Statement
Two atoms of equal mass m, that move with the same speed but opposite direction, interact when they're in some region R of space, as in fig.1. After the interaction, one of the atoms moves with velocity ## \vec{V1} ## as in fig.2.
a) Are the linear and angular momentum of...
Hi. There are two masses connected by a massless bar, and from the unstable equilibrium position shown in the figure is slightly inclined so it falls down, being the final state of the system that both masses are in contact with the surface. There is no friction between the floor and m2. The...
Homework Statement
A particle of mass m is free to slide on a thin rod. The rod rotates in a plane about one end at a constant angular velocity w. Show that the motion is given by r=Ae^(-γt)+Be^(γt), where γ is a constant which you must find and A and B are arbitrary constants. Neglect...
Homework Statement
A wheel of radius R rolls along the ground with velocity V.
A pebble is carefully released on top of the wheel so that it is instantaneously at rest on the wheel.
Show that the pebble will immediately fly off the wheel if V> sqrt(Rg)
The Attempt at a Solution
Hi...
Thanks, I thought a little bit more:
T=Ma*Aa; Aa= (r_a''-Ra*w^2)êr
T=Mb*Ab; Ab= (r_b''-Rb*w^2)êr
(I eliminated the êθ part of the vector, since it hasn't started yet to move -"immediately after the catch is removed"- therefore r'=0, and of course θ''=0).
Now, both centripetal accelerations...
Hi TSny! Thanks for your repply.
I've been thinking again, and saw that I was equating the centripetal acceleration of Mb to the radial acceleration of Ma.
Now I've reasoned that (centripetal force on of Mb)-(centripetal force on Ma) should result in the radial force on Ma. In this case...
Homework Statement
A disk rotates with angular velocity w. Two masses, Ma and Mb, slide without friction in a groove passing through the cnter of the disk. They are connected by a light string of length L, and are initially held in position by a catch, with mass Ma at distance Ra from the...