PHYS 5260: Quantum Mechanics II - Homework Set 1 - Prof. Leo Radzihovsky, Assignments of Quantum Mechanics

Download PHYS 5260: Quantum Mechanics II - Homework Set 1 - Prof. Leo Radzihovsky and more Assignments Quantum Mechanics in PDF only on Docsity! Leo Radzihovsky, Spring 2008 PHYS 5260: Quantum Mechanics - II Homework Set 1 Issued January 14, 2008 Due January 28, 2008 Reading Assignment: Shankar, Ch.16 1. Harmonic oscillator Use a variational theory with a Gaussian trial wavefunction ψ(x) = Ne−αx 2 (N proper normalization) determine the ground state wavefunction, i.e., the optimum α0, and the corresponding ground state energy E0, by minimizing E(α) = 〈H〉 (where H is the harmonic oscillator Hamiltonian) with respect to the variational parameter α. Compare your answers here with the exact results for α0 and E0. 2. (Shankar 16.1.3) For the attractive delta function potential V = −aV0δ(x) use the variational principle with a Gaussian trial wavefunction to calculate the upper bound on E0 and compare it to the exact answer −ma2V 20 /2h̄2 (from last semester, a problem that you should review). 3. Use the variational method with a variational function ψ(x) = Nxe−ax (N a proper normalization) for a 1D particle of mass m in a potential V (x) to determine the opti- mum value a0 of parameter a and the corresponding ground-state energy E0 = Egs(a0). Take V (x) = ∞, for x < 0 and V (x) = x, for 0 < x, 4. Helium atom within Hartree-variational approximation Fill in all the steps for the variational estimate (with Z as the variational parameter) of the ground state energy of a Helium atom sketched out in Shankar, Ch.16, that leads to E0 in Eq.16.1.16. 5. Estimate the tunneling-out probability (decay rate) at energy E for the potential V (x) = x, for 0 < x < d and V (x) = 0, for d < x using WKB approximation. 6. Use WKB approximation to compute a decay rate out of a 3D attractive short-ranged potential well for a particle in a state of orbital angular momentum ` and energy E. How does the result depend on E and `?