Category: HC Verma Part 2: Modern Physics

Problem Statement Solve the quantum/modern physics problem: A hydrogen atom in the ground state absorbs a photon of wavelength 97.2 nm. Find the excited state it reaches and whether the photon is Lyman series. $E_{photon} = hc/\lambda$; match to $E_n – E_1$ Step 1: $E_{photon} = 1240/97.2 = 12.76$ eV Step 2: $E_n – E_1…

Problem Statement Solve the oscillation/wave problem: Find all wavelengths in the Balmer series that lie in the visible region ($400-700$ nm). $1/\lambda = R_H(1/4 – 1/n^2)$ for $n = 3, 4, 5, $ $n=3$: $1/\lambda = R_H(1/4-1/9) = R_H\times5/36$; $\lambda = 656$ nm (H$\alpha$, red) ✓ $n=4$: $1/\lambda = R_H(1/4-1/16) = R_H\times3/16$; $\lambda = 486$…

Problem Statement Solve the work-energy problem: Find the series limit (shortest wavelength) of the Balmer series and the series limit of the Lyman series. Explain their ratio. Series limit: $n\to\infty$; $1/\lambda = R_H/n_{low}^2$ Step 1: Lyman limit ($n_{low}=1$): $1/\lambda_L = R_H = 1.097\times10^7$; $\lambda_L = 91.2$ nm Step 2: Balmer limit Given Information Mass $m$,…

Problem Statement Calculate the ground state energy of positronium (electron + positron bound state). Positronium has reduced mass $\mu = m_e/2$. 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…

Problem Statement Solve the oscillation/wave problem: Find the wavelength of light emitted by He$^+$ ($Z=2$) in the transition $n=4 \to n=2$. $\frac{1}{\lambda} = R_H Z^2\left(\frac{1}{n_2^2} – \frac{1}{n_1^2}\right)$ Step 1: $$\frac{1}{\lambda} = 1.097\times10^7\times4\times\left(\frac{1}{4} – \frac{1}{16}\right) = 4.388\times10^7\times\frac{3}{16}$$ Given Information Mass $m$ and spring constant $k$ (or equivalent), or wave parameters Initial conditions (amplitude $A$, phase $\phi$)…

Problem Statement For large quantum numbers, show that the frequency of radiation emitted in the transition $n \to n-1$ equals the classical orbital frequency. (Correspondence principle) 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…

Problem Statement Solve the magnetic field/force problem: Find the magnetic moment of an electron in the first Bohr orbit of hydrogen. (This gives the Bohr magneton.) Magnetic moment: $\mu = iA = \frac{ev}{2\pi r}\cdot\pi r^2 = evr/2$ Also $\mu = eL/(2m_e) = e\hbar/(2m_e) = \mu_B$ (Bohr magneton) Step 1: $$\mu_B = \frac{e\hbar}{2m_e} = \frac{1.6\times10^{- Given…

Problem Statement Solve the momentum/collision problem: Find the angular momentum of the electron in the $n = 3$ orbit of hydrogen. Bohr quantization: $L_n = n\hbar$ Step 1: $$L_3 = 3\hbar = 3\times1.055\times10^{-34} = 3.165\times10^{-34}\text{ J·s}$$ $$\boxed{L_3 = 3.165\times10^{-34}\text{ J·s} = 3\hbar}$$ Given Information Masses $m_1$, $m_2$ and initial velocities $u_1$, $u_2$ as given Type…

Problem Statement If hydrogen is excited to the $n = 4$ level, how many distinct spectral lines can be emitted? 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…

Problem Statement Solve the work-energy problem: What is the minimum energy needed to excite hydrogen from ground state to the third excited state ($n = 4$)? $\Delta E = E_4 – E_1 = -13.6/16 – (-13.6) = 13.6(1 – 1/16)$ eV Step 1: $E_1 = -13.6$ eV; $E_4 = -13.6/16 = -0.85$ eV Step 2:…