385lec17 - PHYS 385 Lecture 17 Particle in a 3D box Lecture...

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PHYS 385 Lecture 17 - Particle in a 3D box 17 - 1 © 2003 by David Boal, Simon Fraser University. All rights reserved; further resale or copying is strictly prohibited. Lecture 17 - Particle in a 3D box What's Important: particle in a box Fermi energy Text : Gasiorowicz, Chap. 9. In this lecture, we address the situation in which localized interactions are unimportant, so that particle wavefunctions span an entire system, perhaps even as large as a star. We start by considering the motion of a particle between two reflective walls in one dimension, and then generalize the result to three dimensions. Application of our findings to neutron stars are made in the next lecture. Particle in a one-dimensional box Let's consider first the motion of a particle in one dimension between two perfectly reflective walls (from Lec. 9): As there is no potential energy gradient in the region between the walls, the one- dimensional (time independent) Schrödinger equation reads - h 2 2 m d 2 u ( x ) dx 2 = E u ( x ) (1) The solution to this equation has the usual plane-wave form, u ( x ) = (1/ N ) sin( n π x / L ) (2) which satisfies the boundary conditions that u ( x ) = u ( L ) = 0, where L is the length of the box. From o π sin 2 ( n ) d = π /2, the normalization constant N is 1/ N 2 = ( L / π ) o π sin 2 ( n ) d = ( L / π )•( π /2) = L /2. (3) The allowed values of the particle's momentum p obey p = h k = h 2 π / (2 L / n ) = n π h / L . (4)
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