The question asks to identify the set of operators for which the 1s state of the hydrogen atom is a simultaneous eigenfunction in the non-relativistic formalism.
Identifying Key Quantum Numbers
The 1s state corresponds to the following quantum numbers:
- Principal quantum number: $n=1$
- Azimuthal (orbital) quantum number: $l=0$
- Magnetic quantum number: $m_l=0$
Eigenfunctions and Commuting Operators
A quantum state can be a simultaneous eigenfunction of multiple operators only if those operators commute. The operators relevant here are the Hamiltonian ($H$), the square of the orbital angular momentum ($L^2$), the z-component of the orbital angular momentum ($L_z$), and the orbital angular momentum vector ($\vec{L}$).
- Hamiltonian ($H$): The 1s state is the ground state and thus an eigenfunction of $H$ with the lowest energy eigenvalue $E_1$.
- Angular Momentum Squared ($L^2$): The eigenvalue of $L^2$ is given by $l(l+1)\hbar^2$. For the 1s state, $l=0$, so the eigenvalue is $0(0+1)\hbar^2 = 0$. Therefore, the 1s state is an eigenfunction of $L^2$.
- Angular Momentum z-component ($L_z$): The eigenvalue of $L_z$ is given by $m_l\hbar$. For the 1s state, $m_l=0$, so the eigenvalue is $0\hbar = 0$. Therefore, the 1s state is an eigenfunction of $L_z$.
In the context of the hydrogen atom's central potential, $H$, $L^2$, and $L_z$ all commute with each other ($[H, L^2] = 0$, $[H, L_z] = 0$, $[L^2, L_z] = 0$). This allows for simultaneous eigenfunctions.
Operator $\vec{L}$ Consideration
The vector operator $\vec{L} = (L_x, L_y, L_z)$ does not commute with $L_z$ (e.g., $[L_x, L_z] = i\hbar L_y \neq 0$). Therefore, a state cannot be a simultaneous eigenfunction of $\vec{L}$ and $L_z$. While the 1s state has $l=0$, meaning $\vec{L}$ acting on it results in the zero vector ($0$), the operator $\vec{L}$ itself cannot be included in a set of operators that are simultaneously diagonal with $L_z$. The standard basis states are typically defined by eigenvalues of $H$, $L^2$, and $L_z$.
Conclusion
Based on the quantum numbers ($n=1, l=0, m_l=0$) and the commutation relations, the 1s state of the hydrogen atom is a simultaneous eigenfunction of $H$, $L^2$, and $L_z$. This corresponds to Option 1.
Important Questions from Angular Momentum Operators Eigenvalues Clebsch Gordan
An electron in the Coulomb field of a proton is in the following state of coherent superposition of orthonormal states $\psi_{nlm}$
$\Psi = \frac{1}{3}\psi_{100} + \frac{1}{\sqrt{3}}\psi_{210} - \frac{\sqrt{5}}{3}\psi_{320}$
Let $E_1, E_2$, and $E_3$ represent the first three energy levels of the system. A sequence of measurements is done on the same system at different times. Energy is measured first at time $t_1$ and the outcome is $E_2$. Then total angular momentum is measured at time $t_2 > t_1$ and finally energy is measured again at $t_3 > t_2$. The probability of finding the system in a state with energy $E_2$ after the final measurement is $P/9$. The value of $P$ is ______________ (in integer).- An atom with non-zero magnetic moment has an angular momentum of magnitude $\sqrt{12}\hbar$. When a beam of such atoms is passed through a Stern-Gerlach apparatus, how many beams does it split into?
- In the vector model of angular momentum applied to atoms, what is the minimum angle in degrees (in integer) made by the orbital angular momentum vector and the positive $z$ axis for a $2p$ electron?
A particle has wavefunction
$\psi(x,y,z) = N ze^{-\alpha(x^2+y^2+z^2)}$,
where $N$ is a normalization constant and $\alpha$ is a positive constant. In this state, which one of the following options represents the eigenvalues of $L^2$ and $L_z$ respectively?
Some values of $Y_l^m$ are:
$Y_0^0 = \sqrt{\frac{1}{4\pi}}$, $Y_1^0 = \sqrt{\frac{3}{4\pi}} \cos\theta$, $Y_1^{\pm 1} = \mp \sqrt{\frac{3}{8\pi}} \sin\theta e^{\pm i\phi}$- The spin $ \vec{S}$ and orbital angular momentum $ \vec{L}$ of an atom precess about $ \vec{J}$, the total angular momentum. $ \vec{J}$ precesses about an axis fixed by a magnetic field $ \vec{B}_1 = 2B_0 \hat{z}$, where $B_0$ is a constant. Now the magnetic field is changed to $ \vec{B}_2 = B_0( \hat{x} + \sqrt{2} \hat{y} + \hat{z})$. Given the orbital angular momentum quantum number $l = 2$ and spin quantum number $s = 1/2$, $ \theta$ is the angle between $ \vec{B}_1$ and $ \vec{J}$ for the largest possible values of total angular quantum number $j$ and its $z$-component $j_z$. The value of $ \theta$ (in degree, rounded off to the nearest integer) is ________
