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1. ### Show the formula which connects the adjoint representations

Well, this is one exercise from my quantum mechanics class...
2. ### Show the formula which connects the adjoint representations

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...
3. ### Net force acting on a charged particle ##+Q##

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?
4. ### Net force acting on a charged particle ##+Q##

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.
5. ### Net force acting on a charged particle ##+Q##

They're still exerting force over ##+Q##, but as they're are diametrically opposed to each other, so they cancel out. Right?
6. ### Net force acting on a charged particle ##+Q##

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...
7. ### Find the eigenvalues and eigenvectors

Yes, the values are ##\left( t-2\right) \left(t^2-5t+2\right)##, with ##\lambda_1 = 2, \lambda_2 = \frac {5} {2} - \frac {\sqrt {17}} {2}## and ##\lambda_3 = \frac {5} {2} + \frac {\sqrt {17}} {2}##.
8. ### Find the eigenvalues and eigenvectors

The eigenvector associated to these eigenvalues are ##\vec v_1 = (0,0,0) , \vec v_2 = (0,0,0)##... That's what I've found out.
9. ### Find the eigenvalues and eigenvectors

Yes, ##\left( t-2\right) \left(t^2-5t+2\right)##, with ## \lambda_2 = \frac {5} {2} - \frac {\sqrt {17}} {2}## and ##\lambda_3 = \frac {5} {2} + \frac {\sqrt {17}} {2}##.

13. ### Find the electric field at an arbitrary point

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...

15. ### Find the eigenvalues and eigenvectors

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...
16. ### Find the eigenvalues and eigenvectors

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.
17. ### Find the eigenvalues and eigenvectors

Homework Statement Find the eigenvalues and eigenvectors of the following matrix: $$A = \begin{bmatrix} 3 & 0 & 0 \\ 0 & 3 & 2 \\ 0 & -1 & 0 \end{bmatrix}$$ Homework Equations Characteristic polynomial: $$\Delta (t) = t^3 - Tr(A) t^2 + (A_{11}+A_{22} +A_{33})t - det(A) .$$ The Attempt at...
18. ### Find the electric field at an arbitrary point

Which one is wrong? Yes, that's what I did at the answer of a)... Is that answer wrong? I'm not sure if b) answer is right.
19. ### Find the electric field at an arbitrary point

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...
20. ### Find the normalization constant ##A##

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.
21. ### Find the normalization constant ##A##

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...

28. ### Find the eigenvalues and eigenvectors

Oh!!! Guys I'm sorry I wrote the values wrong! Now I understand what I was doing wrong. Thank you very much guys!
29. ### Find the eigenvalues and eigenvectors

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...
30. ### Write ##5-3i## in the polar form ##re^\left(i\theta\right)##

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...