Y^2 - x^2 in the [itex]\mid n\ell m \rangle[/itex] basis - tensor Op.

In summary, the homework statement is asking for the matrix elements of x^2-y^2 in the \mid n\ell m \rangle basis. The student is trying to solve for x^2-y^2 as superposition of five operators with \ell = 2.
  • #1
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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  • #2
Try T±22=(x±iy)2...
 
  • #3
Yeah I found out - apparently it always works to use [itex]T^{\left(2\right)}_{\pm 2} = T_{\pm 1}^{\left(1\right)}T_{\pm 1}^{\left(1\right)}[/itex]. The full set of operators is
[tex]\begin{align*}
T^{(2)}_{\pm 2} &= T^{\left(1\right)}_{\pm 1} T^{\left(1\right)}_{\pm 1}\\
T^{(2)}_{\pm 1} &= \frac{1}{\sqrt{2}}\left( T^{\left(1\right)}_{\pm 1} T^{\left(1\right)}_0 + T^{\left(1\right)}_0 T^{\left(1\right)}_{\pm 1} \right)\\
T^{(2)}_{0} &= \frac{1}{\sqrt{6}}\left( T^{\left(1\right)}_{+1} T^{\left(1\right)}_{-1} + T^{\left(1\right)}_{-1} T^{\left(1\right)}_{+1} + 2T^{\left(1\right)}_0 T^{\left(1\right)}_0 \right)
\end{align*}[/tex]
 
  • #4
Yes, you can also read them out by analogy with the spherical harmonics.. good luck with the rest!
 
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  • #5
Goddar said:
Yes, you can also read them out by analogy with the spherical harmonics..
How's that?
 
  • #6
well, look at the spherical harmonics remembering that z = cosθ/r, x2+y2=sin2θ/r2, etc... if you discard the numerical factors, you'll see the pattern.
 
  • #7
oops, in the last post r factors should be multiplied on the right, not divided..
 

FAQ: Y^2 - x^2 in the [itex]\mid n\ell m \rangle[/itex] basis - tensor Op.

1. What is the [itex]\mid n\ell m \rangle[/itex] basis in the context of Y^2 - x^2?

The [itex]\mid n\ell m \rangle[/itex] basis is a set of quantum states that describe the energy, angular momentum, and magnetic moment of an electron in an atom. It is commonly used in the study of atomic and molecular systems.

2. How is Y^2 - x^2 related to the [itex]\mid n\ell m \rangle[/itex] basis?

Y^2 - x^2 is a tensor operator that represents the square of the orbital angular momentum in the [itex]\mid n\ell m \rangle[/itex] basis. This means that it describes the probability of finding an electron in a particular orbital state, based on its angular momentum.

3. What is a tensor operator?

A tensor operator is a mathematical object that represents a physical quantity in quantum mechanics. It is used to describe the properties of particles, such as their position, spin, and angular momentum, in a particular basis.

4. How is Y^2 - x^2 calculated in the [itex]\mid n\ell m \rangle[/itex] basis?

The calculation of Y^2 - x^2 in the [itex]\mid n\ell m \rangle[/itex] basis involves applying the appropriate operators to the quantum states in the basis. The result is a set of values that represent the square of the orbital angular momentum for each state.

5. What is the significance of Y^2 - x^2 in quantum mechanics?

Y^2 - x^2 is an important quantity in quantum mechanics as it helps to describe the behavior and properties of particles in an atom. It is used in calculations related to atomic and molecular systems, and has applications in fields such as spectroscopy and quantum chemistry.

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