Category: Part 6: Atomic & Nuclear

Problem Statement Find ionization energy and first Lyman line wavelength for He$^+$ ($Z=2$) and Li$^{2+}$ ($Z=3$). 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…

Problem Statement Solve the quantum/modern physics problem: Solve the quantum/modern physics problem: A line at $\lambda = 500$ nm has natural linewidth $\Delta\lambda = 10^{-4}$ nm. Find the excited state lifetime. $$\Delta E = hc\Delta\lambda/\lambda^2 = (6.626\times10^{-34})(3\times10^8)(10^{-13})/(5\times10^{-7})^2 = 7.95\times10^{-26} \text{ J}$$ $$\tau Given Information Frequency $\nu$ or wavelength $\lambda$ of radiation Work function $\phi$ of…

Problem Statement Solve the oscillation/wave problem: Electrons accelerated through $V = 54$ V diffract from Ni crystal at $\theta = 50°$. Find the interplanar spacing. $$\lambda = h/\sqrt{2m_eeV} = 6.626\times10^{-34}/\sqrt{2\times9.109\times10^{-31}\times54\times1.6\times10^{-19}} = 1.67 \text{ Å}$$ Bragg condition with glancing geometry: $d = \lambd Given Information All quantities, constants, and constraints stated in the problem above Physical…

Problem Statement Solve the quantum/modern physics problem: Solve the quantum/modern physics problem: Find the electron recoil angle $\alpha$ as a function of photon scattering angle $\varphi$ in Compton scattering. From momentum conservation: $$\cot\alpha = \frac{\nu – \nu’\cos\varphi}{\nu’\sin\varphi} = (1 + h\nu/m_ec^2)\cot(\varphi/2)$$ For $\varphi = 90° Given Information Frequency $\nu$ or wavelength $\lambda$ of radiation Work…

Problem Statement Solve the optics problem: Solve the optics problem: Find the force exerted by $P = 1.0$ kW beam on a perfectly reflecting mirror. Each reflected photon transfers $2p = 2h\nu/c$ of momentum. Total force: $$F = 2P/c = 2\times1000/3\times10^8 = 6.67\times10^{-6} \text{ N} = 6.67 \mu\text{N}$$ Answer: $F = 2P/c \approx 6.7$ μN…

Problem Statement Show using the uncertainty principle that electrons cannot be confined within a nucleus of $R=10^{-14}$ m. 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…

Problem Statement Solve the oscillation/wave problem: Solve the quantum/modern physics problem: Show $v_{phase} = c^2/v > c$ and $v_{group} = v$ for relativistic particles. $$v_{phase} = \omega/k = E/p = \gamma mc^2/(\gamma mv) = c^2/v > c$$ $$v_{group} = d\omega/dk = dE/dp = pc^2/E = \gamma mvc^2/(\gamma mc^2) = v \checkmark$$ Note: $v_{phase}\times v Given…

Problem Statement Solve the work-energy problem: Solve the quantum/modern physics problem: What fraction of photon energy ($E_0=0.10$ MeV) is transferred to the electron at $\theta=180°$? $$\frac{T}{E_0} = \frac{2E_0/m_ec^2}{1+2E_0/m_ec^2} = \frac{2\times0.10/0.511}{1+2\times0.10/0.511} = \frac{0.391}{1.391} = 28.1\%$$ Answer: $\approx 28\%$ of th Given Information Mass $m$, velocity $v$, height $h$, or other given quantities Any forces doing work…

Problem Statement Solve the quantum/modern physics problem: Solve the quantum/modern physics problem: At $\lambda_1=0.30$ μm, $V_1=1.73$ V; at $\lambda_2=0.40$ μm, $V_2=0.60$ V. Find $h$ and $\phi$. $$h = \frac{e(V_1-V_2)}{c(1/\lambda_1-1/\lambda_2)} = \frac{1.6\times10^{-19}\times1.13}{3\times10^8\times8.33\times10^5} = 7.2\times10^{-34} \text{ J·s}$$ $$\ph Given Information Frequency $\nu$ or wavelength $\lambda$ of radiation Work function $\phi$ of metal (if photoelectric) Planck’s constant $h…

Problem Statement Show that the quantum emission frequency for $n\to n-1$ approaches the classical orbital frequency for large $n$. 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…