A hydrogen electron is in the state $n=3,l=2,m=-2$. If the variance of its orbital angular-momentum component is $(\Delta L_y)^2=\alpha\hbar^2$, find α.
In an $|l,m\rangle$ state, $\langle L_y\rangle=0$ and symmetry about z gives $\langle L_x^2\rangle=\langle L_y^2\rangle$. Since $L^2=L_x^2+L_y^2+L_z^2$, $\langle L_y^2\rangle=[l(l+1)-m^2]\hbar^2/2$. Substitution gives $(6-4)\hbar^2/2=\hbar^2$, so α = 1.
More Orbital and spin angular momenta questions
GATE 2020 PH Q61– A hydrogen atom is in an orbital angular-momentum state l,m=l. In the vector model, its…
GATE 2025 PH Q45– An electron of mass m and charge q is initially spin up, pmatrix1\\0pmatrix. A constant…
GATE 2026 PH Q37– Consider the Pauli matrices x=pmatrix0&1\\1&0pmatrix, y=pmatrix0&-i\&0pmatrix,…
GATE 2023 PH Q15– H is the Hamiltonian and L,L z are orbital angular momentum and its z component. The…
GATE 2023 PH Q25– An atom with non-zero magnetic moment has an angular momentum of magnitude √12 ħ. When a…
GATE 2023 PH Q33– In the vector model of angular momentum applied to atoms, what is the minimum angle in…