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Question: A hydrogen atom is located in a quadrupole field, which gives it a perturbation $$H_1=A(x^2-y^2)$$ where $A$ is some constant. Calculate the quantity $\langle 211|H_1|21\bar{1}\rangle$, one of the two nonzero elements of the degenerate perturbation matrix.

Attempt: First, I took $R^2\equiv x^2+y^2$, thus allowing my to simplify the perturbation:

$$\frac{H_1}{R^2}=A\left (\frac{x^2}{R^2}\right ) -\left (\frac{y^2}{R^2} \right)$$

$$=A\sin^2(\theta)(\cos^2(\phi)-\sin^2(\phi))=\sin^2(\theta)\cos(2\phi)$$

$$=\sin^2(\theta)\frac{1}{2}(\exp(i2\phi)+\exp(-i2\phi))=N(Y_{22}+Y_{2\bar{2}}) $$

where $Y_{\ell\,m}$ are the spherical harmonics and $N$ is a normalization constant to be determined. Now, when I calculate the matrix element, I obtain

$$A\langle 211|(x^2-y^2)|21\bar{1}\rangle=AN\langle 211|Y_{22}+Y_{2\bar{2}}|21\bar{1}\rangle $$

$$=AN\int dr\,d\theta\,d\phi r^2 \sin(\theta) R_{21}^*Y_{11}^*(Y_{22}+Y_{2\bar{2}})R_{21}Y_{11} $$

Shouldn't this be zero from orthogonality conditions? It doesn't seem to make sense that it should be zero. What did I do incorrectly? Any help would be appreciated.

share|improve this question
    
what are x and y? operators? –  Timtam Feb 8 at 8:09
1  
In the second line of your last equation, $Y_{11}$ should be replaced by $Y_{1\bar{1}}$. –  higgsss Feb 8 at 9:51

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