IGNOU MPH 4 2026 - i) Write down the angular momentum states and calculate the

Question

i) Write down the angular momentum states $|j, m_j\rangle$ and calculate the matrix elements of $\hat{J}_x, \hat{J}_y$ and $\hat{J}_z$ for $j = 1/2$ .
ii) For the angular momentum state $|3,1\rangle$ show that $J_+|3,1\rangle = \hbar\sqrt{10}|3,2\rangle$ and $J_-|3,1\rangle = 2\hbar\sqrt{3}|3,0\rangle$ .

19 Jan 2026

Answer :

Word Count : 486

Let's solve this step by step manually, using standard angular momentum algebra. --- ### i) Angular momentum states for ( j = 1/2 ) and matrix elements of (\hat{J}_x, \hat{J}_y, \hat{J}_z) For ( j = 1/2 ), the possible ( m_j ) values are: [ m_j = +\frac{1}{2}, -\frac{1}{2} ] So the states are: [ \left| \frac{1}{2}, \frac{1}{2} \right\rangle, \quad \left| \frac{1}{2}, -\frac{1}{2} \right\rangle ] Step 1: (\hat{J}_z) matrix By definition: [ \hat{J}_z \left| j, m_j \right\rangle = \hbar m_j \left| j, m_j \right\rangle ] So in the basis ( { \left| \frac{1}{2}, \frac{1}{2} \right\rangle, \left| \frac{1}{2}, -\frac{1}{2} \right\rangle } ): [ \hat{J}_z = \begin{bmatrix} \hbar/2 & 0 \ 0 & -\hbar/2 \end{bmatrix} ] --- Step 2: Ladder operators (\hat{J}*+) and (\hat{J}*-) [ \hat{J}_\pm \left| j, m_j \right\rangle = \hbar \sqrt{ j(j+1) - m_j(m_j \pm 1) } , \left| j, m_j \pm 1 \right\rangle ] * For ( \hat{J}_+ ): [ \hat{J}_+ \left| \frac{1}{2}, -\frac{1}{2} \right\rangle = \hbar \sqrt{\frac{1}{2} \left(\frac{1}{2}+1\right) - \left(-\frac{1}{2}\right)\left(-\frac{1}{2}+1\right)} , \left| \frac{1}{2}, \frac{1}{2} \right\rangle ] [ \frac{1}{2} \cdot \frac{3}{2} - (-\frac{1}{2} \cdot \frac{1}{2}) = \frac{3}{4} - (-\frac{1}{4})

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