IGNOU MPH 4 2026 - Show that in the terms of the and basis vectors

Question

Show that in the terms of the $|\uparrow\rangle$ and $|\downarrow\rangle$ basis vectors defined by
$\hat{S}_z|\uparrow\rangle = \frac{\hbar}{2}|\uparrow\rangle$ ; $\hat{S}_z|\downarrow\rangle = -\frac{\hbar}{2}|\downarrow\rangle$ , we can write: $\hat{S}_x = \frac{\hbar}{2} \left[ |\uparrow\rangle\langle\downarrow| + |\downarrow\rangle\langle\uparrow| \right]$

19 Jan 2026

Answer :

Word Count : 197

The states (|\uparrow\rangle) and (|\downarrow\rangle) are defined as eigenstates of (\hat S_z) with eigenvalues (+\hbar/2) and (-\hbar/2) respectively. Any spin operator can be represented in this two–dimensional basis by its action on these states. The spin operators satisfy the angular momentum commutation relations, and the ladder operators are defined as [ \hat S_\pm = \hat S_x \pm

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