Category: Part 6: Atomic & Nuclear

Problem Statement Solve the kinematics problem: Solve the kinematics problem: Using the Bohr model, find the radius and velocity of the electron in the first orbit of hydrogen. Balancing Coulomb force with centripetal, plus quantization $m_evr = \hbar$: $$a_0 = \frac{4\pi\varepsilon_0\hbar^2}{m_ee^2} = 0.529 \text{ Å}$$ $$v_1 = \frac{e^2}{4\pi\va Given Information Initial velocity $u$ (or $v_0$)…

Problem Statement A state has lifetime $\tau = 10$ ns. Find the minimum frequency uncertainty and relative linewidth at $\lambda = 0.50$ μm. 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…

Problem Statement Solve the quantum/modern physics problem: Solve the quantum/modern physics problem: Estimate the minimum kinetic energy of a neutron confined in a nucleus of radius $l = 1.3\times10^{-14}$ m. $$\Delta p \sim \hbar/l; \quad T_{min} = \hbar^2/(2m_nl^2)$$ $$= (1.055\times10^{-34})^2/(2\times1.675\times10^{-27}\times(1.3\times10^{-14})^2)$$ $$= Given Information Frequency $\nu$ or wavelength $\lambda$ of radiation Work function $\phi$ of metal…

Problem Statement Solve the quantum/modern physics problem: Solve the quantum/modern physics problem: Estimate the minimum kinetic energy of an electron confined to $\Delta x = 0.20$ nm. $$\Delta p \sim \hbar/\Delta x = 1.055\times10^{-34}/2\times10^{-10} = 5.28\times10^{-25} \text{ kg·m/s}$$ $$T_{min} = (\Delta p)^2/(2m_e) = (5.28\times10^{-25})^2/(2\times9 Given Information Frequency $\nu$ or wavelength $\lambda$ of radiation Work function…

Problem Statement Solve the oscillation/wave problem: Solve the quantum/modern physics problem: At what kinetic energy does the electron’s de Broglie wavelength equal its Compton wavelength? $\lambda_{dB} = \lambda_C$ requires $p = m_ec$. Then: $$E = \sqrt{(m_ec^2)^2 + (m_ec^2)^2} = \sqrt{2}m_ec^2$$ $$T = E – m_ec^2 = (\sqrt{2}-1)m_ec^2 = 0.4142\times0 Given Information Mass $m$ and spring…

Problem Statement Solve the oscillation/wave problem: Thermal neutrons with $T = 0.025$ eV fall on a crystal with plane spacing $d = 0.20$ nm. Find the first-order Bragg reflection angle. $$\lambda = h/\sqrt{2m_nT} = 6.626\times10^{-34}/\sqrt{2\times1.675\times10^{-27}\times4\times10^{-21}} = 1.81 \text{ Å} = 0.181 \text{ nm}$$ Bragg’s law ($n=1$): $2d Given Information All quantities, constants, and constraints stated…

Problem Statement Solve the oscillation/wave problem: Solve the quantum/modern physics problem: Find the de Broglie wavelength of an electron with kinetic energy equal to its rest mass energy ($T = m_ec^2$). Total energy: $E = 2m_ec^2$. Relativistic momentum from $E^2 = (pc)^2 + (m_ec^2)^2$: $$pc = \sqrt{4(m_ec^2)^2 – (m_ec^2)^2} = \sqrt{3}m_ec^2$$ $$p Given Information Mass…

Problem Statement Solve the oscillation/wave problem: Solve the quantum/modern physics problem: Find the de Broglie wavelength of: (a) electron with $T = 0.025$ eV; (b) thermal neutron at 293 K; (c) proton through $V = 1.0$ kV. $\lambda = h/\sqrt{2mT}$ (a) $T = 0.025\times1.6\times10^{-19} = 4\times10^{-21}$ J: $$\lambda_e = \frac{6.626\times10^{-34}}{ Given Information Mass $m$ and…

Problem Statement Solve the momentum/collision problem: Solve the quantum/modern physics problem: A photon of $h\nu = 0.50$ MeV is Compton-scattered at $\varphi = 60°$. Find the electron’s kinetic energy and recoil angle. $$\Delta\lambda = \lambda_C(1-\cos60°) = 2.426\times0.5 = 1.213 \text{ pm}$$ $$\lambda = hc/E = 1240/500000 \text{ nm} = 2.48 \text{ p Given Information Masses…

Problem Statement Solve the momentum/collision problem: Solve the quantum/modern physics problem: A photon with $E_0 = 0.15$ MeV is scattered by a stationary electron. Find the maximum kinetic energy of the recoil electron. Maximum energy transfer occurs at backscattering ($\theta = 180°$): $$T_{max} = \frac{2E_0^2/m_ec^2}{1 + 2E_0/m_ec^2} = \frac{2(0.15 Given Information Masses $m_1$, $m_2$ and…