Quantum Mechanics II - Variational Method and Harmonic Potential, Assignments of Quantum Mechanics

Download Quantum Mechanics II - Variational Method and Harmonic Potential and more Assignments Quantum Mechanics in PDF only on Docsity! PHY662 - Quantum Mechanics II HWK #5, Due Tues., Feb. 17, at the start of class • Reading: Read through p. 435 of Shankar by Tuesday, Feb. 17. • This is a short (5 pt.) homework. 1. Variational method introduction. (a) Gaussian integral. Suppose you had a unnormalized wave function ψ(α, x) = e−αx 2 , with α > 0, with the single spatial coordinate x. What is the normalized wave function? (b) Shankar Problem 16.1.1, spelled out in detail : Use ψ(α, x) = e−αx 2 (properly normalized; with α > 0) as a trial ground-state wave func- tion for a single particle of mass m moving in one dimension (x- coordinate). The potential for the particle is the harmonic potential V = 12mω 2x2. Find the value of α, αmin, that minimizes 〈H〉. Com- pare the energy estimate from this variational solution ψ(αmin, x) with the exact solution for the ground-state energy for this potential. (c) Modifying the potential. Now consider adding a δ-function to the potential to represent a repulsive point at the origin. Let the total potential be V = 12mω 2x2 + uδ(x). Compute the resulting change in 〈H〉 = 〈ψ(α, x)|H|ψ(α, x)〉 (i.e., the change from the calculation in part (b)). Then, using your work from part (b), write the new full equation for α that gives a minimum for 〈H〉. Just write out the equation (4th-order polynomial in √ α) that you would need to solve in order to exactly determine this optimal value for α. You do not need to solve this equation. (d) Consistency check. Study this 4th-order equation in √ α qualitatively : how does the addition of a repulsive potential at the origin change the estimated ground state wave function? That is, how does αmin for u > 0 compare, qualtitatively, with αmin for the case u = 0? How does this change in α change the width of the probability distribution for the particle? 1