All exercise S.Qs of Chapter#6: Applications to Quantum Mechanics of Quantum Mechanics-II. We have arranged all S.Qs for BS/MSc Physics students.
Q.1 why spin-orbit interaction arises?
Answer: Spin-orbit interaction is the iteration between spin and the orbit magnetic moment of atoms. Spin-orbit coupling in Hydrogen arises from the interaction between the electron’s spin magnetic moment and the proton’s orbital magnetic field $\mu_e=\frac{-eS}{m_\circ}$ when proton is being seen by electron’s perspective.
Q.2 Define and explain the role of fine structure constant (α) in energy splitting.
Answer: If we compare fine structure with Bohr energies, $\left(i.e\ E_n^0=\frac{-13.6}{n^2}\ eV\right)$ “fine structure is a tiny perturbation” smaller by a factor of $\alpha^2$, where
$$\mathbf{\alpha=\frac{e^2}{4\pi\varepsilon\hbar c}=\frac{1}{137.036}}$$
is known as fine structure constant which tells about the amount of splitting.
Q.3 Explain the spin-orbit coupling and consequences originating from this effect?
Answer: Spin-orbit interaction is the iteration between spin and the orbit magnetic moment of atoms. Spin-orbit coupling in Hydrogen arises from the interaction between the electron’s spin magnetic moment $\mu_e=\frac{-eS}{m_\circ}$ and the proton’s orbital magnetic field $\vec{B}$ when proton is being seen by electron’s perspective. Imagine an electron in orbit around the nucleus; From electron’s point of view, the proton is circling around the electron. This orbiting $+ve$ charge of proton sets up a magnetic field $\vec{B}$ in the electron frame, which exerts a torque on the spinning electron, tending to align its magnetic moment $\mu$ along the direction of field $\vec{B}$.
The energy associated with its torque is given by Hamiltonian
$$\mathbf{|H=-{\vec{\mu}}_e.\vec{B}}$$
To find H we must find the magnetic field of proton $\left(\vec{B}\right)$ and dipole moment of electron $({\vec{\mu}}_e)$.
Q.4 Differentiate between normal and anomalous Zeeman effect.
Answer: Historically, one distinguishes between the normal and an anomalous Zeeman effect. In normal Zeeman effect (spin of electron = 0) , each energy level splits into an odd number $\left(2\ell+1\right)$ of equally spaced levels, disagree with the experimental observations. In anomalous Zeeman effect (spin of electron is considered), each energy level splits into an even number $(2j\ +\ 1)$ of unequally spaced energy levels, agree with the experimental findings.
Q.5 Define Stark effect.
Answer: Stark effect is the splitting of a spectral line into several components in the presence of an electric field.
Q.6 Give the physical interpretation of the equation $\mathbf{{\hat{H}}_{SO}=\frac{e^2}{2m^2c^2r^3}\vec{S}.\vec{L}}$
Answer: Here
Hso=$\mathbf{\left(\frac{e^2}{8\pi\varepsilon_\circ}\right)\left(\frac{1}{m_\circ^2c^2r^3}\right)\left(\vec{S}.\vec{L}\right)}$
In quantum mechanics, in the presence of spin orbit coupling, the Hamiltonian does not commute or diagonalized with $\vec{L}$ and $\vec{S},$ so the spin and orbital angular momenta are not separately conserved. However ${\hat{H}}_{SO}$ commutes with $L^2,\ \ S^2$ and the total angular momentum $\vec{J}=\vec{L}+\vec{S}$ and hence these quantities are conserved. The eigen states of $\ \vec{L}$ and $\vec{S}$ are not good states to be used in perturbation theory, but the eigen states of |nljm〉 and $L^2,\ S^2,\ J^2$ are good to be used in the perturbation theory.
Q.7 What is meant by fine structure and hyperfine structure of hydrogen atom?
Answer: The splitting of state into two or more components, each representing a slightly different
wavelength is called fine structure. Fine structure is produced when an atom emits light in making the transition from one energy state to another.
More significant is the fine structure of hydrogen atom which is due to two distinct mechanisms.
- Spin-Orbit Coupling.
- Relativistic Correction.
The Hydrogen’s hyperfine structure is due to the magnetic interaction between the spin dipole moments of electron and spin of the proton.
Q. Draw the energy levels for S and P orbitals by using the $\Delta E=B\mu_B\left(m_l+2m_s\right)$
Answer:
Q.9 Draw the energy levels for $\mathbf{2}\mathbf{P}_{\mathbf{3}/\mathbf{2}\ }$ by using the $\mathbf{E_z^1=B\mu_Bg_jm_j i}$ if $g_j=4/3$.
Answer:
Q.10 Define the possible applications of Zeeman effect.
Answer: The Zeeman effect is very important in magnetic resonance imaging (MRI). And to
measure magnetic field strength of the sun and other stars.
