Spin-Orbit Interaction Radial Dependence
The spin-orbit interaction energy ($H_{SO}$) for an electron in a central field is proportional to the dot product of orbital and spin angular momenta ($\vec{l}\cdot\vec{s}$) and depends on the radial part $f(r)$ as given: $H_{SO} = f(r)\vec{l}\cdot\vec{s}$.
The radial function $f(r)$ is determined by the potential $V(r)$ created by the charge distribution. The standard form is:
$f(r) \propto \frac{1}{r}\frac{dV(r)}{dr}$Electric Field in Uniform Sphere
Consider an electron moving inside a uniformly charged sphere of radius $R$. The electric field $E(r)$ at a distance $r$ from the center ($r < R$) depends on the enclosed charge. Using Gauss's Law, the electric field is found to be proportional to the radius:
$E(r) \propto r$Potential Gradient Calculation
The electric field is related to the potential $V(r)$ by $E(r) = -\frac{dV(r)}{dr}$. Therefore, the gradient of the potential has the following radial dependence:
$\frac{dV(r)}{dr} = -E(r) \propto -r$Determining $f(r)$ Dependence
Now, substitute the radial dependence of the potential gradient into the expression for $f(r)$:
$f(r) \propto \frac{1}{r}\frac{dV(r)}{dr}$ $f(r) \propto \frac{1}{r}(-r)$ $f(r) \propto -1$Conclusion
The result $f(r) \propto -1$ shows that $f(r)$ is a constant value, independent of the radial distance $r$. This corresponds to Option A.
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).- $H$ is the Hamiltonian, $\vec{L}$ the orbital angular momentum and $L_z$ is the $z$-component of $\vec{L}$. The $1s$ state of the hydrogen atom in the non-relativistic formalism is an eigen function of which one of the following sets of operators?
- 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:
$Y_0^0 = \sqrt{\frac{1}{4\pi}}$, $Y_1^0 = \sqrt{\frac{3}{4\pi}} \cos\theta$, $Y_1^{\pm 1} = \mp \sqrt{\frac{3}{8\pi}} \sin\theta e^{\pm i\phi}$
