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
- Consider two particles with angular momenta $j_1 = 2\hbar$ and $j_2 = \hbar/2$. If the expression
$$|j = 5/2, m = 3/2\rangle = \begin{cases} c_1|j_1 = 2, m_1 = 1\rangle|j_2 = 1/2, m_2 = 1/2\rangle + \\ c_2|j_1 = 2, m_1 = 2\rangle|j_2 = 1/2, m_2 = -1/2\rangle \end{cases}$$
gives an eigenstate of the total angular momentum of the two particles, using standard notation. Which of the following is true?
(Hint: $\hat{J}_{\pm}|j, m\rangle = \sqrt{j(j + 1) - m(m \pm 1)} |j, m \pm 1\rangle$) - A system of three non-identical spin $\frac{1}{2}$ particles has the Hamiltonian $H = \frac{A}{\hbar^2} (\vec{S}_1 + \vec{S}_2) \cdot \vec{S}_3$, where $\vec{S}_1, \vec{S}_2$ and $\vec{S}_3$ are the spin operators of particles labelled $1,2$ and $3$ respectively and $A$ is a constant with appropriate dimensions. The set of possible energy eigenvalues of the system is
- $H$ 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?
