Quantum Mechanics Homework 5: Solving Schrodinger Equation for Various Potentials, Assignments of Quantum Mechanics

Download Quantum Mechanics Homework 5: Solving Schrodinger Equation for Various Potentials and more Assignments Quantum Mechanics in PDF only on Docsity! Quantum Mechanics A (PHY 5645): Homework 5 DUE: Friday Nov 14 P1(20 points): Solve the Schrodinger equation for a 3D, spherically symmetric harmonic oscillator with a potential energy V (r) = 1 2 mω2r2. P2(20 points): Given an attractive central potential of the form V (r) = −V0e−r/a solve the Schrodinger equation for the s states. Obtain the equation for the eigenvalues. Solve this equation numerically and find the smallest value of λ = √ 8ma2V0/h̄ for which the potential traps a bound state. Determine the answer to within three significant digits. (Hint: Substitute ξ = e−r/(2a) and note that ( z2 ∂2 ∂z2 + z ∂ ∂z + z2 − ν2 ) f(z) = 0 is the Bessel equation.) P3(20pts): Find the bound state energies and wavefunctions for a Schrodinger particle moving in the x− y plane interacting with the potential V (x, y) = − Ze 2 √ x2 + y2 . You can ignore the wavefunction normalization, but rewrite the bound state wavefunctions in terms of the confluent hypergeometric function 1F1(a, c, z) = 1 + a c z 1! + a(a + 1) c(c + 1) z2 2! + . . . What is the degeneracy associated with each bound state energy level? (Hint: Exploit the cylindrical symmetry. Use series method to solve the radial part of the Schrodinger equation.) P4(20pts): Find the wavefunctions for the bound states for a particle moving in 3D potential V (x, y, z) = 0; for r < a (1) V (x, y, z) = ∞; for r ≥ a (2) Find the equation satisfied by the eigenenergies. Numerically, find the energies, in units of h̄ 2 2ma2 , of the lowest 3 energies in each of the angular momentum channels: l = 0, l = 1, and l = 2. (For this part, ignore the 2l + 1-fold degeneracy.) P5(20pts): The electron of a hydrogen atom is in its ground state. Determine 〈r〉, 〈r2〉 and the most probable value r0 for this state. (r = √ r2)