Category: Part 6: Atomic & Nuclear

Problem Statement Solve the momentum/collision problem: Find the magnitude of electron spin and its z-component. What does spin-1/2 mean physically? Electron spin $s = 1/2$: $$S = \sqrt{s(s+1)}\hbar = \sqrt{3/4}\hbar = \frac{\sqrt{3}}{2}\hbar = 9.13\times10^{-35} \text{ J·s}$$ z-components: $S_z = m_s\hbar = \pm\hbar/2$ The spin-1/2 particle has only two Given Information Masses $m_1$, $m_2$ and initial…

Problem Statement Analyze the circuit: Problem 6.90 — Schrödinger: Current Through Barrier See problem statement for all given quantities. This problem applies fundamental physics principles to the scenario described. The solution requires identifying the relevant conservation laws and equations of motion, then solving systematically wit Given Information Resistance values $R_1, R_2, \ldots$ as specified EMF…

Problem Statement Plot and interpret the radial probability densities $P(r) = r^2|R_{nl}|^2$ for the 1s, 2s, and 2p states. 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…

Problem Statement Solve the work-energy problem: For a particle in state $\psi = c_1\phi_1 + c_2\phi_2$ (superposition of energy eigenstates), find $\langle E\rangle$ and $\Delta E$. $\phi_1, \phi_2$ are energy eigenstates with $\hat{H}\phi_i = E_i\phi_i$, normalized: $|c_1|^2 + |c_2|^2 = 1$. $$\langle E\rangle = \langle\psi|\hat{H}|\psi\rangle = Given Information Mass $m$, velocity $v$, height $h$, or…

Problem Statement State the Bethe-Weizsäcker semi-empirical mass formula and explain each term physically. 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…

Problem Statement Solve the momentum/collision problem: Find the magnitude of the orbital angular momentum for $l = 0, 1, 2, 3$. $L = \sqrt{l(l+1)}\hbar$: $l$ Subshell $L/\hbar$ $L$ (J·s) 0 s 0 0 1 p $\sqrt{2}$ $1.49\times10^{-34}$ 2 d $\sqrt{6}$ $2.58\times10^{-34}$ 3 f $\sqrt{12}=2\sqrt{3}$ $3.65\times10^{-34}$ Note: Bohr model incorrectly gives $L = n Given Information…

Problem Statement Solve the nuclear physics problem: Find the binding energy per nucleon for $^{56}$Fe (the most stable nucleus). Atomic mass $M(^{56}\text{Fe}) = 55.9349$ u. $^{56}$Fe has $Z=26$ protons and $N=30$ neutrons. Using atomic masses (which include electron masses, so we use hydrogen mass $m_H = 1.00783$ u): $$\Delta m = 26m_H + 30m_n –…

Problem Statement Find the most probable distance for the $2s$ and $2p$ states 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…

Problem Statement Solve the momentum/collision problem: Show that $L_x$ and $L_y$ do not commute: $[L_x, L_y] = i\hbar L_z$. Angular momentum operators: $L_x = yp_z – zp_y$, $L_y = zp_x – xp_z$, $L_z = xp_y – yp_x$ $$[L_x, L_y] = [yp_z – zp_y, zp_x – xp_z]$$ Expanding using $[x_i, p_j] = i\hbar\delta_{ij}$ and $[x_i, x_j]…

Problem Statement A free particle is described by a plane wave $\psi = Ae^{i(kx-\omega t)}$. Find the probability density and current. 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…