Category: Part 6: Atomic & Nuclear

Problem Statement Use the trial wavefunction $\psi = Ae^{-\alpha r}$ and the variational principle to find the ground state energy 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…

Problem Statement The $2s$ and $2p$ states of hydrogen are degenerate in the Bohr model but split in reality. Explain the Lamb shift. 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 nuclear physics problem: Find the binding energy of the deuteron ($^2$H), whose mass is $m_d = 2.01355$ u. (Proton mass $m_p = 1.00728$ u, neutron mass $m_n = 1.00867$ u.) The binding energy equals the mass defect times $c^2$: $$\Delta m = m_p + m_n – m_d = 1.00728 + 1.00867…

Problem Statement For a finite potential well of depth $U_0$ and width $2a$, find the condition for bound state energies. 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…

Problem Statement Solve the quantum/modern physics problem: Explain the four quantum numbers $n$, $l$, $m_l$, $m_s$ for the hydrogen atom and their physical meaning. Quantum Number Name Values Physical Meaning $n$ Principal $1,2,3, $ Energy: $E_n = -13.6/n^2$ eV; average distance $\sim n^2a_0$ $l$ Orbital $0,1, ,n-1$ Orbital angular momentum: $L = \sqrt{l(l+ Given Information…

Problem Statement State the WKB approximation for bound states and derive the Bohr-Sommerfeld quantization condition. 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 nuclear physics problem: Explain alpha decay qualitatively using quantum tunneling. Why do heavier nuclei decay more slowly? An alpha particle ($^4$He nucleus) inside a heavy nucleus faces a Coulomb barrier due to the nuclear + electrostatic potential. Although classically it cannot escape, quantum tunneling allows a nonzer Given Information Nuclide symbol,…

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 Analyze the circuit: For a particle of energy $E > U_0$ incident on a potential step of height $U_0$, find the reflection and transmission coefficients. Left ($x Right ($x>0$): $\psi = Te^{ik_2x}$, $k_2 = \sqrt{2m(E-U_0)}/\hbar$ Matching at $x=0$: $1+R = T$ and $k_1(1-R) = k_2T$ $$R = \frac{k_1-k_2}{k_1+k_2}, \quad T_{a Given Information Resistance…

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…