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This is solving a coupled mass problem using the techniques used in QM, it's in the mathematical introduction of Shankar.

This equation was obtained by using the solution for ##x(t) = x_i(0)cos(\omega _i t)## and plugging it into ##\left| x(t) \right \rangle = \left| I \right \rangle x_i (t)+\left| II \right \rangle x_{II} (t)##

$$\left| x(t) \right \rangle = \left| I \right \rangle x_1 (0)cos(\omega _1 t) + \left| II \right \rangle x_{II} (0) cos(\omega _{II} t)$$

Where the kets of I and II are an orthogonal basis, and this turns into:

$$\left| I \right \rangle \langle I \left| x(0) \right \rangle cos(\omega _I t) + \left| II \right \rangle \langle II \left| x(0) \right \rangle cos(\omega _{II} t) $$

Edit: ##\langle I \left| x(0) \right \rangle## is just the projection of ##x(0)## onto the basis of ##\left|I \right \rangle##, right? So this is just reworking the equation in terms of ##x(0)##?

Since I think I've figured out my original question, I'd like to pose a new one. My linear algebra isn't very strong, and I'm having problems with the following.

##\left| \ddot{x}(t) \right \rangle = \Omega \left| x(t) \right \rangle##, $$\Omega = \begin{bmatrix} -\frac{2k}{m} & \frac{k}{m} \\ \frac{k}{m} & -\frac{2k}{m} \end{bmatrix} $$

We want to use the basis that diagnolizes ##\Omega##, we need to find it's eigenvectors.

##\Omega \left| I \right \rangle = - \omega ^2 \left| I \right \rangle ##

How does one go about finding the eigenvalues and eigenvectors? the general formula is ##\det(\Omega - \omega I)=0##.

This equation was obtained by using the solution for ##x(t) = x_i(0)cos(\omega _i t)## and plugging it into ##\left| x(t) \right \rangle = \left| I \right \rangle x_i (t)+\left| II \right \rangle x_{II} (t)##

$$\left| x(t) \right \rangle = \left| I \right \rangle x_1 (0)cos(\omega _1 t) + \left| II \right \rangle x_{II} (0) cos(\omega _{II} t)$$

Where the kets of I and II are an orthogonal basis, and this turns into:

$$\left| I \right \rangle \langle I \left| x(0) \right \rangle cos(\omega _I t) + \left| II \right \rangle \langle II \left| x(0) \right \rangle cos(\omega _{II} t) $$

**Where did these inner products come from?**Edit: ##\langle I \left| x(0) \right \rangle## is just the projection of ##x(0)## onto the basis of ##\left|I \right \rangle##, right? So this is just reworking the equation in terms of ##x(0)##?

Since I think I've figured out my original question, I'd like to pose a new one. My linear algebra isn't very strong, and I'm having problems with the following.

##\left| \ddot{x}(t) \right \rangle = \Omega \left| x(t) \right \rangle##, $$\Omega = \begin{bmatrix} -\frac{2k}{m} & \frac{k}{m} \\ \frac{k}{m} & -\frac{2k}{m} \end{bmatrix} $$

We want to use the basis that diagnolizes ##\Omega##, we need to find it's eigenvectors.

##\Omega \left| I \right \rangle = - \omega ^2 \left| I \right \rangle ##

How does one go about finding the eigenvalues and eigenvectors? the general formula is ##\det(\Omega - \omega I)=0##.

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