348-problem15 - ordinates we say that it is separable and as you see its solution is trivial in those cases A separ-able Hamiltonian means that the

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
Chem. 540 Instructor: Nancy Makri PROBLEM 15 Consider a particle moving on the xy plane. Suppose that the motion of the particle on the plane can be described by a Hamiltonian 1 2 ˆ ˆ ˆ H H H = + where the operators 1 ˆ H and 2 ˆ H describe the motion of a particle in the x and in the y direction, re- spectively: 2 1 1 ˆ ˆ ˆ ( ) 2 x p H V x m = + , 2 2 2 ˆ ˆ ˆ ( ) 2 y p H V y m = + . Suppose we call ( ) n x φ the eigenfunctions of 1 ˆ H with eigenvalues n ε and ( ) k y ψ the eigenfunc- tions of 2 ˆ H with eigenvalues k ξ . Show that the eigenfunction of the total Hamiltonian ˆ H are products of the type ( ) ( ) n k x y . Find the corresponding eigenvalues. Whenever the Hamiltonian can be decomposed into a sum of operators involving different co-
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: ordinates, we say that it is separable, and as you see its solution is trivial in those cases. A separ -able Hamiltonian means that the motion of the particle in each coordinate is independent of the mo-tion in the other coordinates. This is not the case in general; any potential field that depends simul -taneously on x and on y would make the x motion coupled to the y motion; in that case, the eigen-states would no longer be expressible in product form....
View Full Document

This note was uploaded on 02/08/2012 for the course CHEM 540 taught by Professor Mccall during the Fall '08 term at University of Illinois, Urbana Champaign.

Ask a homework question - tutors are online