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here's the problem:

"In a two dimensional vector space, consider the operator whose matrix is written as

[tex] \sigma_x = \left(\begin{array}{cc}0&1\\1&0\end{array}\right) [/tex].

in an orthonormal basis {|1>, |2>}.

Calculate the eigenvalues and normalised eigenvectors for the operator in this basis."

I'm uncertain over what I'm being asked to do here. This operator clearly has not been expressed in the basis of its own eigenstates, otherwise it would have non-zero elements (ie. its eigenvalues) in the (1,1) and (2,2) elements, and zeros elsewhere, right?

I know how to find these eigenvalues and eigenstates, and have done this by calling the eigenstates [tex]{|\phi_j>} [/tex] and doing [tex] \sigma_x |\phi_j> = \lambda |\phi_j> [/tex], where [tex]|\phi_j> = \left(\begin{array}{c}\phi_1\\\phi_2\end{array}\right) [/tex].

I do this and out pops the eigenvalues [tex] \lambda = +1 , -1 [/tex] and the eigenvectors [tex] |\phi_a> = \frac{1}{\sqrt{2}} \left(\begin{array}{c}1\\1\end{array}\right) [/tex] and [tex] |\phi_b> = \frac{1}{\sqrt{2}} \left(\begin{array}{c}1\\-1\end{array}\right) [/tex].

This is where my confusion arises, because the problem asks me to "calculate normalised eigenvectors for the operator

If so, how do I manage this, given that I am not told what [tex] |1>, |2> [/tex] are?

Or am I mistaken - are they actually what I have already calculated? If so, I am very confused about how these problems work.

Thanks.

"In a two dimensional vector space, consider the operator whose matrix is written as

[tex] \sigma_x = \left(\begin{array}{cc}0&1\\1&0\end{array}\right) [/tex].

in an orthonormal basis {|1>, |2>}.

Calculate the eigenvalues and normalised eigenvectors for the operator in this basis."

I'm uncertain over what I'm being asked to do here. This operator clearly has not been expressed in the basis of its own eigenstates, otherwise it would have non-zero elements (ie. its eigenvalues) in the (1,1) and (2,2) elements, and zeros elsewhere, right?

I know how to find these eigenvalues and eigenstates, and have done this by calling the eigenstates [tex]{|\phi_j>} [/tex] and doing [tex] \sigma_x |\phi_j> = \lambda |\phi_j> [/tex], where [tex]|\phi_j> = \left(\begin{array}{c}\phi_1\\\phi_2\end{array}\right) [/tex].

I do this and out pops the eigenvalues [tex] \lambda = +1 , -1 [/tex] and the eigenvectors [tex] |\phi_a> = \frac{1}{\sqrt{2}} \left(\begin{array}{c}1\\1\end{array}\right) [/tex] and [tex] |\phi_b> = \frac{1}{\sqrt{2}} \left(\begin{array}{c}1\\-1\end{array}\right) [/tex].

This is where my confusion arises, because the problem asks me to "calculate normalised eigenvectors for the operator

**in this basis**". Does this mean I am somehow supposed to write the eigenvectors [tex] |\phi_a_,_b> [/tex] in terms of the basis vectors [tex] |1>, |2> [/tex] that [tex] \sigma_x [/tex] was originally expressed in?If so, how do I manage this, given that I am not told what [tex] |1>, |2> [/tex] are?

Or am I mistaken - are they actually what I have already calculated? If so, I am very confused about how these problems work.

Thanks.

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