GATE 2023 PH – Question 15

Quantum Mechanics · orbital and spin angular momenta · 1 mark · Multiple choice

$H$ is the Hamiltonian and $\mathbf L,L_z$ are orbital angular momentum and its z component. The hydrogen 1s state is an eigenfunction of which set?

$H,L^2,L_z$
$H,\mathbf L,L^2,L_z$
$L^2,L_z$ only
$H,L_z$ only

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Correct answer: (B) $H,\mathbf L,L^2,L_z$

Explanation

The 1s state of hydrogen has the quantum numbers $n=1$, $l=0$, $m=0$.

It is a stationary state, so it is an eigenfunction of the Hamiltonian $H$.
With $l=0$ the angular part is the constant $Y_0^0$, which is unchanged by any rotation. Therefore $\mathbf L\psi=0$ for **every** component $L_x$, $L_y$, $L_z$, and also $L^2\psi=0$. So it is an eigenfunction of the vector $\mathbf L$ (all three components), of $L^2$ and of $L_z$.

The full set is $H$, $\mathbf L$, $L^2$ and $L_z$ (option B). Option A leaves out the components $L_x$ and $L_y$, which are also eigen-operators with eigenvalue zero for this state.