Category: Part 6: Atomic & Nuclear

Problem Statement List all allowed electric-dipole transitions from $n=3$ levels of hydrogen. 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 field equation governs…

Problem Statement Solve the oscillation/wave problem: Solve the quantum/modern physics problem: Estimate the ground state energy of a harmonic oscillator using the uncertainty principle. $E = (\Delta p)^2/(2m) + m\omega^2(\Delta x)^2/2$ with $\Delta p = \hbar/(2\Delta x)$. Minimizing over $\Delta x$: $$\frac{dE}{d(\Delta x)} = 0 \implies \Delta x^2 = \ Given Information Mass $m$ and…

Problem Statement Solve the work-energy problem: Solve the quantum/modern physics problem: Find the de Broglie wavelength of an electron with the same kinetic energy as a photon of $\lambda_0 = 0.50$ nm. $$E = hc/\lambda_0 = 1240/0.50 = 2480 \text{ eV}$$ $$\lambda_e = h/\sqrt{2m_eE} = \lambda_0\sqrt{E/(2m_ec^2)} \cdot 2/1 = 0.50\sqrt{2480/(2\times Given Information Mass $m$, velocity…

Problem Statement Solve the kinematics problem: Find maximum speed of photoelectrons from K ($\phi = 2.25$ eV) at $\lambda = 200$ nm. $$T_{max} = 1240/200 – 2.25 = 6.20 – 2.25 = 3.95 \text{ eV} = 6.32\times10^{-19} \text{ J}$$ $$v_{max} = \sqrt{2T_{max}/m_e} = \sqrt{2\times6.32\times10^{-19}/9.109\times10^{-31}} = 1.18\times10^6 \text{ m/s}$$ Given Information All quantities, constants, and constraints…

Problem Statement Find the minimum X-ray wavelength from a tube at $V = 40$ kV. 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 Solve the momentum/collision problem: Solve the quantum/modern physics problem: Derive $\cot\alpha = (1 + h\nu/m_ec^2)\cot(\theta/2)$ for the electron recoil angle. From momentum conservation ($\varepsilon = h\nu/m_ec^2$): $$\cot\alpha = \frac{p_0 – p’\cos\theta}{p’\sin\theta} = \frac{\nu – \nu’\cos\theta}{\nu’\sin\theta}$$ Using $\nu’ Given Information Masses $m_1$, $m_2$ and initial velocities $u_1$, $u_2$ as given Type of collision: elastic…

Problem Statement Solve the quantum/modern physics problem: Solve the quantum/modern physics problem: A 1.0 mW laser at $\lambda = 633$ nm emits how many photons per second? $$E_{ph} = hc/\lambda = 1240/633 = 1.96 \text{ eV} = 3.14\times10^{-19} \text{ J}$$ $$N = P/E_{ph} = 10^{-3}/3.14\times10^{-19} = 3.18\times10^{15} \text{ photons/s}$$ Frequency $\nu$ or Given Information Frequency…

Problem Statement Analyze the circuit: Analyze the circuit: Explain why doubling light intensity doubles photocurrent but does not change stopping voltage. Stopping voltage is set by $eV_s = h\nu – \phi$ — it depends on photon energy (frequency), not on the number of photons. Intensity determines the photon flux: double the intensity, do Given Information…

Problem Statement Solve the oscillation/wave problem: Solve the quantum/modern physics problem: At what photon energy does Compton scattering become significant? What is the physical meaning of $\lambda_C$? $\lambda_C = h/(m_ec) = 2.426$ pm. The fractional wavelength shift is $\Delta\lambda/\lambda \sim \lambda_C/\lambda = h\nu/(m_ec^2)$. Compton effec Given Information Mass $m$ and spring constant $k$ (or equivalent),…

Problem Statement Solve the momentum/collision problem: Solve the quantum/modern physics problem: Show that photon spin $\hbar$ leads to the electric dipole selection rule $\Delta l = \pm 1$. Photons are spin-1 bosons with $m_s = \pm1$ (helicity states for a massless particle; $m_s=0$ is forbidden). In an atomic transition, total angular momentum must be Given…