Physics 486 Homework 8 - Spring 2007: Perturbation Theory in Quantum Mechanics, Assignments of Quantum Physics

Download Physics 486 Homework 8 - Spring 2007: Perturbation Theory in Quantum Mechanics and more Assignments Quantum Physics in PDF only on Docsity! Physics 486 Homework #8 Spring 2007 1). Charged oscillator in an electric field – Consider a particle having charge e and mass m, which is confined to a one-dimensional harmonic oscillator potential characterized by an angular frequency ω. A weak electric field is turned on, giving rise to a small perturbing potential H’ = eεx. (a). Find an exact expression for the energy eigenvalues for the new Hamiltonian H=Ho+H’ {Hint: you do not need to do a complicated calculation for this! Start with the new Hamiltonian for this problem, and perform a variable change y = x + eε/(mω2) to get something that looks like the original harmonic oscillator Hamiltonian.} (b). Calculate the first-order correction to the energy {Hint: use raising and lowering operators} (c). Calculate the second-order correction to the energy {Hint: use raising and lowering operators} (d). Calculate the first-order correction to the unperturbed wavefunction {Hint: use raising and lowering operators} 2). Excited state of the one-dimensional harmonic oscillator - In this problem, you will use variational methods to estimate the first excited state energy of a particle confined to this potential. (a). Sketch the trial wavefunction ( ) 2, xtr x Bxe αψ α −= , where B is a normalization constant and α is a variational parameter, and explain why this is an appropriate trial wavefunction for determining the first excited state energy. (b). Determine the normalization constant B. (c). Use variational methods and the trial wavefunction in (a) to obtain estimates of the first excited state energy and wavefunction for this potential. 3). Griffiths 6.1: Suppose we put a perturbing delta-function potential, H’ = αδ(x-a/2), in the center of an infinite square well potential of width a, where α is a constant. (a). Find the first-order corrections to the allowed energies. Explain why the energies are not perturbed for even values of n. (b). Find the first three nonzero terms in the expansion of the first-order correction to the ground state, Ψ1(1).