2005AugQualQUANT - Ben Sauerwine Practice for Qualifying...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
Ben Sauerwine Practice for Qualifying Exams Problem Source: CMU Qualification Exam Day 2 (August 2005) Consider a particle of mass m moving in a one-dimensional potential with infinitely high walls placed at a x ± = . The potential vanishes between the walls. (a) Write down the eigenenergies and the corresponding eigenstates of the system in the x basis. The eigenstates, even and odd, are () x a n i n n e C x 2 π ψ = The corresponding eigen-energies are given by: () () () x E x x V x m ψψ = + 2 2 2 2 h For: 2 2 2 2 = a n m E n h (b) Suppose that at time t = 0, the particle is in a state that can be well approximated by the wave function 2 2 δ x Ne x = , where a << . (i) Determine N and then determine the wave function at some later time . 0 t πδ θ 1 2 2 2 2 2 00 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 = = = = = = = ∫∫ + N dx x rdrd e dxdy e dy e dx e dx x dx e dx x r y x y x x The wave function at a later time is given by
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
() ( ) ( ) () () ()() t x x m t x x m t x x x m t x x m t x t x t x t i t x x m φψ δδ δ = + = = Ψ = Ψ Ψ = Ψ 1 4 4 2 16 4 2 4 2 , 2 , , , 2 2 2 2 2 2 4 2 2 2 2 2 2 2 2 2 2 h h h h h h Results of this form indicate that separation of variables is not possible in this case.
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 04/07/2012 for the course PHYSICS 767 taught by Professor Dr.jaouni during the Spring '12 term at Abu Dhabi University.

Page1 / 5

2005AugQualQUANT - Ben Sauerwine Practice for Qualifying...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online