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GATE 2022 CY – Question 59

Physical Chemistry · Hydrogen and Hydrogen-like Atoms: Atomic orbitals; radial and angular distribution functions and their probabilities; Atomic units, Multi-electron atoms: Orbital approximation; electron spin; Pauli exclusion principle; Slater determinants. Variational method and secular determinants; first order non-degenerate perturbation techniques. · 2 marks · Numerical answer

The electron in $He^+$ has normalized wavefunction $\Psi(r,\theta,\phi)=\sqrt{3/8}R_{21}Y_1^0-i\sqrt{7/16}R_{10}Y_0^0+xR_{32}Y_2^0$. Here $R_{nl},Y_l^m$ are radial and angular eigenfunction components and x is unknown. The probability of measuring energy $-2/9$ atomic unit is ______ (two decimal places).

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Correct answer: 0.18 to 0.20

Explanation

Hydrogenic He+ has En=−Z²/(2n²)=−2/n²a.u., so−2/9 corresponds to n=3. The first two components have probabilities 3/8 and 7/16. Normalization leaves |x|²=1−3/8−7/16=3/16=0.1875. Rounded **0.19**; the phase−i does not alter probabilities.