- #1

CharlieCW

- 56

- 5

## Homework Statement

The SO(3) representation can be represented as ##3\times 3## matrices with the following form:

$$J_1=\frac{1}{\sqrt{2}}\left(\matrix{0&1&0\\1&0&1\\ 0&1&0}\right) \ \ ; \ \ J_2=\frac{1}{\sqrt{2}}\left(\matrix{0&-i&0\\i&0&-i\\ 0&i&0}\right) \ \ ; \ \ J_3=\left(\matrix{1&0&0\\0&0&0\\ 0&0&-1}\right)$$

On the other hand, the generators of rotations of SO(3) can also be expressed as ##3\times 3## matrices with the following form:

$$K_1=\left(\matrix{0&0&0\\0&0&-i\\ 0&i&0}\right) \ \ ; \ \ K_2=\left(\matrix{0&0&i\\0&0&0\\ -i&0&0}\right) \ \ ; \ \ K_3=\left(\matrix{0&-i&0\\i&0&0\\ 0&0&0}\right)$$

Since the representation for SO(3) is unique, these matrices should be the same but "disguised". In other words, there exists a similarity matrix ##M## that allow us to transform from the basis of ##J_i## to ##K_i##. Determine the explicit form of the matrix ##M##.

Hint: ##M## must be unitary to conserve hermicity.

## Homework Equations

Similarity transformation:

$$X'=M^{-1}XM$$

## The Attempt at a Solution

Given a basis of vectors ##v=(v_1,v_2,...,v_n)##, it is easy to transform it to another basis of vectors ##w=(w_1,w_2,..,w_n)## by finding an appropiate similarity transformation matrix ##M##. However, when I have a basis of matrices ##\{J\}##, I can't find a way to transform to another basis of matrices ##\{K\}##.

My first attempt was to find an ##M## such that ##K_i=M^{-1}J_iM##, I tried expressing the ##J_i## as a linear combination of the ##K_i##.

However, except for ##J_2=\frac{1}{\sqrt{2}}(K_1+K_2)##, we cannot do the same for the others (ex. there's no way to express ##J_3## in terms of the ##K_i##).

My second attempt was to try finding transformation matrices for each case, and so far I only have found 3 different matrices to transform from ##J_i## to ##K_i## like this:

$$M_1=\left(\matrix{-\frac{1}{2}&0&\frac{1}{2}\\0&-\frac{i}{2}&0\\ \frac{1}{2}&0&\frac{1}{2}}\right) \ \ \ so \ that \ K_1=(M_1)^{-1}J_1 M_1$$

$$M_2=\left(\matrix{0&-\frac{1}{\sqrt{2}}&0\\ \frac{1}{2}&0&\frac{1}{2}\\ -\frac{1}{2}&0&\frac{1}{2}}\right) \ \ \ so \ that \ K_2=(M_2)^{-1}J_2 M_2$$

$$M_3=\left(\matrix{-i&0&i\\1&0&1\\ 0&1&0}\right) \ \ \ so\ that\ K_3=(M_3)^{-1}J_3 M_3$$

However, the statement of the problem says that the matrix ##M## should be unique, and I can't find a way to combine the above matrices (I've tried every possible permutation over the last week).