GATE 2019 CY – Question 63

Physical Chemistry · Rotational Motion: Angular momentum operators (orbital and spin), spherical harmonics and their properties. · 2 marks · Numerical answer

For spin-up $|\alpha\rangle$, $\langle\alpha|S_xS_y-S_yS_x|\alpha\rangle=i\hbar^2a$. Find a (one decimal place).

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Correct answer: 0.5 to 0.5

Explanation

The angular-momentum commutator is $[S_x,S_y]=i\hbar S_z$. A normalized spin-up state has Sz eigenvalue ℏ/2. Therefore its expectation is iℏ(ℏ/2)=iℏ²/2, so a=0.5.

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