Category: Part 6: Atomic & Nuclear

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: 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 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 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…

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 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 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 $A$,…

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 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 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,…