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**Quantum State Calculation for Hydrogen Atom**

Calculate \(\langle x^2 \rangle\) for the quantum state \(\{n = 2, \, l = 1, \, m = 1\}\) for a hydrogen atom. Notice that this average can be expressed more simply as \(\langle 2,1,1|x^2|2,1,1 \rangle\), where the ket \(|n, l, m \rangle\) defines the quantum state of a hydrogen atom. However, as discussed in class, the spin quantum number is not included.

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