Category: Part 6: Atomic & Nuclear

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 For the hydrogen $2p$ state ($n=2$, $l=1$, $m=0$), find $\langle r\rangle$ and $\langle r^2\rangle$. 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 oscillation/wave problem: Find the energy levels of a quantum harmonic oscillator and the spacing between adjacent levels. Exact solution of the Schrödinger equation gives: $$E_n = \hbar\omega(n + \frac{1}{2}), \quad n = 0, 1, 2, $$ Key features: Zero-point energy $E_0 = \hbar\omega/2 > 0$ (quantum) Equally spaced levels: $\ Given…

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 oscillation/wave problem: A free particle has a Gaussian wave packet at $t=0$. Show that the packet spreads over time and find the spreading rate. Initial packet: $\psi(x,0) = Ae^{-x^2/(4\sigma_0^2)}e^{ip_0x/\hbar}$ has width $\sigma_0$. The momentum spread: $\Delta p = \hbar/(2\sigma_0)$ (Fourier transform of Gaussian). Aft Given Information Mass $m$ and spring constant…

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 Analyze the circuit: Define the probability current and show it satisfies $\partial\rho/\partial t + \nabla\cdot\mathbf{j} = 0$. $$\mathbf{j} = \frac{\hbar}{2mi}(\psi^*\nabla\psi – \psi\nabla\psi^*)$$ Multiplying the Schrödinger equation by $\psi^*$ and its conjugate by $\psi$, then subtracting: $$\frac{\partial|\psi|^2 Given Information Resistance values $R_1, R_2, \ldots$ as specified EMF $\mathcal{E}$ and internal resistance $r$ of battery…

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 oscillation/wave problem: Verify $\psi_0 = Ae^{-\alpha x^2/2}$ satisfies the harmonic oscillator Schrödinger equation and find $E_0$. Substituting into $(-\hbar^2/2m)d^2\psi/dx^2 + m\omega^2x^2\psi/2 = E\psi$: $$(-\hbar^2/2m)(\alpha^2x^2-\alpha) + m\omega^2x^2/2 = E$$ Matching $x^2$-terms: $\alpha = m\omega/\hbar$. Constant Given Information Mass $m$ and spring constant $k$ (or equivalent), or wave parameters Initial conditions (amplitude $A$,…