IGNOU MPH 4 2026 - and are the orthonormal basis states of a two-d

Question

$|1\rangle$ and $|2\rangle$ are the orthonormal basis states of a two-dimensional Hilbert space. A Hermitian operator $\hat{A}$ in this basis is given by the spectral representation: $\hat{A} = 3|1\rangle\langle1| - 1|2\rangle\langle2|$ . For a normalized state given by

$$|\psi\rangle = \frac{1}{2}|1\rangle + \frac{\sqrt{3}}{2}|2\rangle, \text{ calculate } \langle\hat{A}\rangle \text{ and the root mean square deviation in } \hat{A} .$

19 Jan 2026

Answer :

Word Count : 314

The given Hermitian operator is [ \hat{A} = 3|1\rangle\langle 1| - 1|2\rangle\langle 2| ] and the normalized state is [ |\psi\rangle = \frac{1}{2}|1\rangle + \frac{\sqrt{3}}{2}|2\rangle. ] First, the expectation value of (\hat{A}) in state (|\psi\rangle) is [ \langle \hat{A} \rangle = \langle \psi | \hat{A} | \psi \rangle. ] Expanding using the basis states: [ \langle \psi | \hat{A} | \psi \rangle = \left(\frac{1}{2}\langle 1| + \frac{\sqrt{3}}{2}\langle 2|\right) \left(3|1\rangle\langle 1| - 1|2\rangle\langle

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