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  1. Mutatis

    Show the formula which connects the adjoint representations

    Well, this is one exercise from my quantum mechanics class...
  2. Mutatis

    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. Mutatis

    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. Mutatis

    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. Mutatis

    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. Mutatis

    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. Mutatis

    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. Mutatis

    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. Mutatis

    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}##.
  10. Mutatis

    Find the eigenvalues and eigenvectors

    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 &...
  11. Mutatis

    Find the electric field at an arbitrary point

    I've took out ##\rho_0## of the integral because it's a constant...
  12. Mutatis

    Find the electric field at an arbitrary point

    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...
  13. Mutatis

    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...
  14. Mutatis

    Find the electric field at an arbitrary point

    Would you please help me to get the right answer for ##\vec E##?
  15. Mutatis

    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. Mutatis

    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. Mutatis

    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. Mutatis

    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. Mutatis

    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. Mutatis

    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. Mutatis

    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...
  22. Mutatis

    Find the normalization constant ##A##

    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...
  23. Mutatis

    Find the normalization constant ##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...
  24. Mutatis

    Find the normalization constant ##A##

    Ok guys, thank you. I'm going to review my calculations and then I get back here.
  25. Mutatis

    Find the normalization constant ##A##

    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...
  26. Mutatis

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

    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(...
  27. Mutatis

    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!
  28. Mutatis

    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...
  29. Mutatis

    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...
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