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MisterX

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**x^2 - y^2 in the [itex]\mid n\ell m \rangle[/itex] basis - tensor op.**

## Homework Statement

I must determine the matrix elements of [itex]x^2 - y^2[/itex] in the [itex]\mid n\ell m \rangle[/itex] basis.

"...use the fact that [itex]x^2 - y^2[/itex] is a sum of spherical components of a rank two tensor, together with the explicit form of the Winger-Eckart theorem."

"Show that [exp. proportional to [itex]x^2 - y^2[/itex]] is a sum of ITOs with [itex]\ell =2[/itex]."

## Homework Equations

Winger Eckart Theorem:

[itex]\langle \ell_1 m_1|T^\ell_m|\ell_2 m_2\rangle = \underbrace{\langle \ell_2m_2;\ell m|\ell_1 m_1 \rangle}_{\text{C.G. coeff.}} \langle j||T^\ell||j'\rangle[/itex]

The operators must satisfy these properties:

[itex]\left[L_z, T^{\ell}_m\right] =mT^{\ell}_m[/itex]

[itex]\sum_{i=1}^3\left[ L_i,\left[L_i, T^{\ell}_m\right]\right] =\ell\left(\ell + 1 \right) T^{\ell}_m[/itex]

[itex]L_{\pm} = L_{1} \pm iL_{2}\;\;\;\;\;\;\;\; \left[L_\pm, T^{\ell}_m\right] =\sqrt{\ell\left( \ell + 1\right)-m\left(m\pm 1\right)}\,T^{\ell}_{m+1} [/itex]

I have some operators with a lower value of [itex]\ell[/itex]:

[itex]T^1_{1} = -x -iy[/itex] is an operator with [itex]\ell = 1, m = 1[/itex].

[itex]T^1_{0} = z[/itex] is an operator with [itex]\ell = 1, m = 0[/itex].

[itex]T^1_{-1} = x -iy[/itex] is an operator with [itex]\ell = 1, m = -1[/itex].

[itex]\left[L_i, x_j\right] = \sum_k \epsilon_{ijk}x_k[/itex]

## The Attempt at a Solution

What I need to do is find the family of five operators with [itex]\ell = 2[/itex], and express [itex]x^2-y^2[/itex] as superposition of those operators. All I need is one of these operators and I can construct the rest. I guessed that [itex]z^2[/itex] might be an operator with [itex]\ell = 2, m = 0[/itex] but I found that [itex]\sum_{i=1}^3\left[ L_i,\left[L_i, z^2\right]\right] =4z^2 - 2x^2 - 2y^2 \neq \ell \left(\ell + 1 \right)z^2 = 6z^2 [/itex].

Is there a general procedure for constructing [itex]\ell = 2[/itex] operators from [itex]\ell = 1[/itex] operators?

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