Category: Part 6: Atomic & Nuclear

Problem Statement Ag atoms through field gradient $dB/dz=15$ T/m over $L=0.10$ m at $v=500$ m/s. Screen at $D=0.20$ m. Find beam separation. 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…

Problem Statement Solve the quantum/modern physics problem: List all quantum states of hydrogen at $n=3$ and find the total degeneracy. $l=0,1,2$; $m_l = -l, ,+l$; $m_s = \pm1/2$ $3s$: 2 states ($l=0$, $m_l=0$) $3p$: 6 states ($l=1$, $m_l=-1,0,+1$) $3d$: 10 states ($l=2$, $m_l=-2, ,+2$) Total: $2n^2 = 2\times9 = 18$ degenerate states (ignoring fine structure…

Problem Statement Solve the quantum/modern physics problem: Calculate $\Delta x\cdot\Delta p_x$ for the hydrogen 1s state and verify the uncertainty relation. By symmetry $\langle x\rangle = \langle p_x\rangle = 0$. Then: $$\langle x^2\rangle = \langle r^2\rangle/3 = a_0^2 \implies \Delta x = a_0$$ $$\langle p_x^2\rangle = \langle p^2\rangle/3 = \hbar^2/(3a_ Given Information Frequency $\nu$ or…

Problem Statement Solve the oscillation/wave problem: Find the most probable electron-nucleus distance in the hydrogen ground state. $$\psi_{1s} = (\pi a_0^3)^{-1/2}e^{-r/a_0}, \quad P(r) = 4\pi r^2|\psi|^2 = \frac{4r^2}{a_0^3}e^{-2r/a_0}$$ Maximum: $dP/dr = 0 \implies 2r – 2r^2/a_0 = 0 \implies r_{mp} = a_0 = 0.529$ Å ✓ Mean value: $\langle r\rangle = Given Information Mass $m$…

Problem Statement Solve the quantum/modern physics problem: Electron ($T=1.0$ eV) tunnels through a barrier $U_0=5.0$ eV, $a=1.0$ nm. Find the transmission coefficient. $$\kappa = \sqrt{2m_e(U_0-T)}/\hbar = \sqrt{2\times9.109\times10^{-31}\times4\times1.6\times10^{-19}}/1.055\times10^{-34} = 1.024\times10^{10} \text{ m}^{-1}$$ $$T \approx e^{-2\kappa a} = e^ Given Information Frequency $\nu$ or wavelength $\lambda$ of radiation Work function $\phi$ of metal (if photoelectric) Planck’s constant…

Problem Statement Solve the 1D infinite square well of width $L$. Find energy eigenvalues and normalized wavefunctions. 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…

Problem Statement Solve the quantum/modern physics problem: Solve the quantum/modern physics problem: Sodium photodiode: $P = 1.0$ mW at 589 nm, photocurrent $I = 1.0$ μA. Find quantum efficiency. $$N_{ph} = P/E_{ph} = 10^{-3}/(3.37\times10^{-19}) = 2.97\times10^{15} \text{ s}^{-1}$$ $$N_e = I/e = 10^{-6}/1.6\times10^{-19} = 6.25\times10^{12} \text{ s}^{-1}$ Given Information Frequency $\nu$ or wavelength $\lambda$ of…

Problem Statement Solve the nuclear physics problem: Estimate the $K_\alpha$ energy and wavelength for tungsten ($Z=74$) via Moseley’s law. $$E_{K_\alpha} = 10.2(Z-1)^2 = 10.2\times(73)^2 = 10.2\times5329 = 54356 \text{ eV} \approx 54.4 \text{ keV}$$ $$\lambda = 1240/54400 = 0.0228 \text{ nm}$$ (Actual: 59.3 keV — Moseley’s law is approximate) Given Information All quantities, constants, and…

Problem Statement Solve the work-energy problem: Solve the quantum/modern physics problem: Electron confined to $L = 1.0$ nm box: minimum kinetic energy by uncertainty principle. $$T_{min} \sim \hbar^2/(2m_eL^2) = (1.055\times10^{-34})^2/(2\times9.109\times10^{-31}\times(10^{-9})^2) = 6.1\times10^{-21} \text{ J} = 38 \text{ meV}$$ Exact quantum re Given Information Mass $m$, velocity $v$, height $h$, or other given quantities Any forces…

Problem Statement Find wavelengths of all photons emitted when H drops from $n=4$ to lower levels. 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…