That's my attempting: first I've wrote ##e## in terms of the power series, but then I don't how to get further than this $$ \sum_{n=0}^\infty (-1)^n \frac {Â^n} {n!} \hat B \sum_{n=0}^\infty \frac {Â^n} {n!} = \sum_{n=0}^\infty (-1)^n \frac {Â^2n} {\left( n! \right) ^2} $$. I've alread tried to...
I'll try to write up this when I get home. This exercise have got my brain confused. At the first question that I've posted, would you, if you were my physics teacher, consider it right?
In the first case, the net force is going to be a sum of the individual contributions of each charge acting over ##+Q##, superposition principle. And then if I was left with 10 equally spaced charges the system is going to equilibrium state.
Homework Statement
Twelve equal particles of charge ##+q## are equally spaced over a circumference (like the hours in a watch) of radius R. At the center of the circumference is a particle with charge ##+Q##.
a) Describe the net force acting over ##+Q##.
b) If the charge located at...
Hey guys, I'm here again... I don't know why I'm still having troubles with this kind of subject. I did my readings and so on, but I'm still struggling to get it right... Look, I got another exercise here, I need to find the eigenvalues and eigenvectors of: $$ \begin{bmatrix}
4 & -2 & 0 \\
-1 &...
Oh thank you, I was doing my calculation wrong. So I've tried to do the integrals over r and I got a different answer this time (it doesn't match with the solutionary): $$ E \int_0^r \, da = \frac {4 \pi \rho_0} {\varepsilon_0} \int_0^r e^{\frac {-r} {a}} \, dr \\ 4 \pi r^2 E = \frac {4 \pi...
First I've used the Gauss law, with the information I got from a): $$ E r^2 4 \pi = \frac {8 \pi a^3 \rho_0} {\varepsilon_0 r^2} \\ \vec E = \frac {4 \pi a^3 \rho_0} {\varepsilon_0 r^2} \vec r .$$
The integral of the left side I did under spherical cordinates and the right side I've used the...
Oh my God... I've done wrong again. The right answer for the eigenvalues is ##\lambda_1= 2, \lambda_2 = 1## and ##\lambda_3 = 3##!
Thank you! I'm going to check my calculations before freaking out. I'm so impulsive...
Yes, I did my calculation wrong. I'd computed ##3+3=9## instead of ##3+3=6##. Now I got it right, my eigenvalues are ##\lambda_1 = 2, \lambda_2 = -1## and ##\lambda_3 = -3##.
Sorry for that, I'll post it at the right place next time. Thank all of you, by the way.
Homework Statement
A distribution of charge with spherical symmetry has volumetric density given by: $$ \rho(r) = \rho_0 e^{ \frac {-r} {a} }, \left( 0 \leq r < \infty \right); $$
where ##\rho_0## and ##a## is constant.
a) Find the total charge
b) Find ##\vec E## in an arbitrary point...
Thank you guys! I got this solved. My problem this time is to find ##<x^2>##. I did some calculation and it leads me to ##<x^2> = \frac {1} {8} ## and it doesn't seems the right answer.
Now I've changed it to ##cos(\alpha)##. But it doesn't change the final result in terms of ##w## like I wrote above.
To write the exponentials in terms of cossine I'd divided ## \left( 2 + e^{i \alpha} + e^{-i\alpha} \right) ## for ##2##. Is this an aceptable answer? Because what I've got to...
This question is from the master degree qualification test that I'm intended to do here in my city (northeast of Brazil). I've obtained ##A## doing what you've told me to. First I got the exponentials in terms of ##cosh##, then I've turned it into a constant ##w## so I got: $$ \frac {\left| A...
So, I'm here again... I've done my calculations right this time, but I still can't get ##A## though... I got the probability density: $$ \left| \psi(x) \right|^2 = \left| A \right|^2 \left[ 2 e^\left(-2 x^2\right) +e^\left(-2x^2 +i\alpha \right) + e^\left(-2x^2 -i\alpha \right) \right] . $$
And...
Homework Statement
Find the noralization constant ##A## of the function bellow: $$ \psi(x) = A e^\left(i k x -x^2 \right) \left[ 1 + e^\left(-i \alpha \right) \right], $$ ##\alpha## is also a constant.
Homework Equations
##\int_{-\infty}^{\infty} e^\left(-\lambda x^2 \right) \, dx = \sqrt...
I did it the way you told me to. I'd wrote in terms of tg: $$ \tan \left( \frac {-3} 5 \right) \\ \theta = \operatorname {arctg} \left( \frac {-3} 5 \right) \\ z =\sqrt {34} \left[ \cos \left( \operatorname{arctg} \left( \frac {-3} 5 \right) \right) + i\sin \left( \operatorname {arctg} \left(...
I did what you've said ## y=1## then ##x=1/i=-i##, so I got ## v_1=(1, -i)##. When I put this vector in the matrix to verify ##Mv_1=0## it leads me to a non-zero value...
Homework Statement
Write ##5-3i## in the polar form ##re^\left(i\theta\right)##.
Homework Equations
$$
|z|=\sqrt {a^2+b^2}
$$
The Attempt at a Solution
First I've found the absolute value of ##z##:
$$ |z|=\sqrt {5^2+3^2}=\sqrt {34} $$.
Next, I've found $$ \sin(\theta) = \frac {-3} {\sqrt...
Homework Statement
Find the eigenvalues and eigenvectors fro the matrix: $$
A=\begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix} $$.
Homework Equations
Characteristic polynomial: ## \nabla \left( t \right) = t^2 - tr\left( A \right)t + \left| A \right|## .
The Attempt at a Solution
I've found...
Homework Statement
I've got to integrate the following $$ \int dx =\int \frac {d\phi} {\phi \sqrt {1 - \phi²}}. $$
Homework Equations [/B]
I already know the answer but not how to get it. The answer that I got from solution is ## x = \operatorname {arcsech}{\phi} ##.
The Attempt at a...
Yes, that is the full question. I know what a matrix is. This question is really confusing me. I don't understand how to representate this 3 state system as a matrix. It is a superposition of 3 states, so ## |x⟩=|0⟩+|1⟩+|2⟩ ## can be represented as a column matrix? $$ \begin{pmatrix}
\langle 0 |...
Homework Statement
Hi, guys. The question is: For a 3-state system, |0⟩, |1⟩ and |2⟩, write the matrix representation of the raising operators ## \hat A, \hat A^\dagger ##, ## \hat x ## and ##\hat p ##.
Homework Equations
I know how to use all the above operators projecting them on...
The wave equation is $$ U(x,t) = a_1 \cos \left( \frac {2 \pi x a t} {L} \right) \sin \left( \frac {2 \pi x} {L} \right) + a_2 \cos \left( \frac {4 \pi x a t} {L} \right) \sin \left( \frac {4 \pi x} {L} \right) $$ with ##n = 2,4##. Do you have some tip to find ##a_1## and ##a_2##?
I've tried it. But that's what is getting me confuse. ##B_n = a_1, a_2## then? Thank you the help. It's the first exercise that I'm really stuck about vibrating waves.
Ok, that was how I did. The ##U(x,t)## solution must me a sum of sine and cosine functions, then $$ U\left(u,t\right) = \sum_{n=0}^\infty B_n \cos\left( \frac {n \pi a t} {L} \right) \sin\left( \frac {n \pi a t} {L} \right)$$ and ##B_n## can be found using $$ B_n = \frac 2 L \int_0^L...
I did, but the answer is a sum of sine and cosine times Bn term which is still in terms of a1 and a2. I'm very confused. I did the calculus throught Maple.
Find the wave equation U(x,t) of a vibrating string with linear density d, tension p, initial velocity zero, weight L and initial displacement
U0(x) = a1*sin(2*pi*x/L)+a2*sin(4*pi*x/L).
Guys, please help me with this task. I did the following procedure:
The U(x,t) solution must me a sum of...