GATE CY 2022 Question Paper (06-Feb-2022) (Shift 2)

For three operators Â, B̂, and Ĉ, [Â, [B̂, Ĉ]] + [B̂, [Ĉ, Â]] =

The difference between the number of Gauche-butane interactions present in P and Q is

The calculated magnetic moment (in BM, rounded off to two decimal places) of a Ce³⁺ complex is

The state of the electron in a He⁺ ion is described by the following normalized wavefunction,

$\psi (\mathrm{r},\theta .\varphi )=\sqrt{\frac{3}{8}}{\mathrm{R}}_{21}(\mathrm{r}){\mathrm{Y}}_{10}(\theta ,\varphi )-\mathrm{i}\sqrt{\frac{7}{16}}{\mathrm{R}}_{10}{\mathrm{Y}}_{00}(\theta ,\varphi )+\mathrm{x}{\mathrm{R}}_{32}(\mathrm{r}){\mathrm{Y}}_{20}(\theta ,\varphi )$

Here, Rₙₗ and Y${}_{lm}$ represent the radial and angular components of the eigenfunctions of He⁺ ion, respectively, and 𝑥 is an unknown constant. If the energy of the ion is measured in the above state, the probability (rounded off to two decimal places) of obtaining the energy of $-\frac{2}{9}$ atomic unit is

A certain wavefunction for the hydrogen-like atom is given by

$\psi (\mathrm{r},\theta ,\varphi )=\frac{2^{\frac{1}{2}}}{81{\pi }^{\frac{1}{2}}}{\left(\frac{\mathrm{z}}{{\mathrm{a}}_0}\right)}^{5/2}\left(6-\frac{\mathrm{zr}}{{\mathrm{a}}_0}\right)\mathrm{r}{\mathrm{e}}^{-zr/3{\mathrm{a}}_0}\cos \theta$

The number of node(s) in this wavefunction is