Category: HC Verma Part 2: Modern Physics

Problem Statement Find the wavelength emitted by Li$^{2+}$ ($Z=3$) in the transition $n=4 \to n=3$. Given Information All quantities, constants, and constraints stated in the problem above Physical constants used as needed (see Concepts section) Physical Concepts & Formulas This problem draws on fundamental physical principles. The key is to identify which conservation law or…

Problem Statement In a star’s spectrum, the Balmer $H_\alpha$ line (656 nm) is observed at 670 nm due to the Doppler effect. Find the recession speed of the star. Given Information All quantities, constants, and constraints stated in the problem above Physical constants used as needed (see Concepts section) Physical Concepts & Formulas This problem…

Problem Statement Solve the work-energy problem: The first excitation potential of a hydrogen-like atom is found to be 40.8 V. Identify the atom (find $Z$). First excitation potential = $(E_2 – E_1)/e = 13.6Z^2(1 – 1/4)$ V $= 10.2Z^2$ V Step 1: $10.2Z^2 = 40.8 \Rightarrow Z^2 = 4 \Rightarrow Z = 2$ Step 2:…

Problem Statement Solve the kinematics problem: In what orbit is the electron speed in hydrogen $v = 10^6$ m/s? $v_n = v_1/n$ where $v_1 = 2.18\times10^6$ m/s Step 1: $n = v_1/v = 2.18\times10^6/10^6 = 2.18$ Step 2: Since $n$ must be an integer, $n = 2$ gives $v_2 = 2.18\times10^6/2 = 1.09\times10^6$ m/s (closest).…

Problem Statement Solve the quantum/modern physics problem: A hydrogen atom is in the $n = 5$ state. List all possible photons that can be emitted as it cascades to the ground state. All transitions from levels $\leq 5$ down to $1$; $E_{photon} = 13.6(1/n_f^2 – 1/n_i^2)$ wait: $E_{photon} = 13.6(1/n_f^2 – 1/n_i^2)$ For emission $n_i…

Problem Statement Solve the oscillation/wave problem: Find the wavelength difference between the $H_\alpha$ line ($n=3\to2$, 656 nm) and the $H_\beta$ line ($n=4\to2$, 486 nm) in the Balmer series. Direct subtraction of known wavelengths Step 1: $\lambda_\alpha = 656.3$ nm (verified from Rydberg formula) Step 2: $\lambda_\beta = 486.1$ nm Step 3: $\Del Given Information Mass…

Problem Statement Solve the magnetic field/force problem: In a magnetic field $B = 1$ T, find the energy splitting between adjacent magnetic sublevels ($\Delta m_l = 1$) of a hydrogen atom. $\Delta E = \mu_B B$; Bohr magneton $\mu_B = 9.27\times10^{-24}$ J/T Step 1: $$\Delta E = \mu_B B = 9.27\times10^{-24}\times1 = 9.27\times10^{-24}\text{ J}$$ Step…

Problem Statement Solve the momentum/collision problem: A hydrogen atom in the $n = 2$ state collides with another hydrogen atom in the ground state. What minimum kinetic energy must the projectile have to excite the target from $n=1$ to $n=2$ (assuming perfectly inelastic collision at threshold)? Threshold in CM frame; for identical masses, threshold KE…

Problem Statement Solve the magnetic field/force problem: Find the magnetic field produced by the orbiting electron at the nucleus in the first Bohr orbit of hydrogen. Current in loop: $i = ef = ev/(2\pi r)$ $B = \mu_0 i/(2r)$ at centre of circular loop Step 1: $v_1 = 2.18\times10^6$ m/s; $r_1 = 0.529\times10^{-10}$ m Step…

Problem Statement A muon ($m_\mu = 207m_e$) is captured by a proton to form a muonic hydrogen atom. Find the ground state radius and energy. Given Information All quantities, constants, and constraints stated in the problem above Physical constants used as needed (see Concepts section) Physical Concepts & Formulas This problem draws on fundamental physical…