Fall 07 Phys 143l HW 11 Solutions

Fall 07 Phys 143l HW 11 Solutions - Physics 143 Fall Term...

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

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Physics 143 Fall Term 2007 HOMEWORK 11 Due: November 28 Problem 1: (BFG Ch.10 Problem 1) Consider two particles in an infinite well. The quantum state of two particles in an infinite well is: Eq. 1022 or 1023 1 uS x , x u x u x u x u x NS 1 uA x , x u x u x u x u x NA Normalization: u u dx dx Therefore, 1 u x u x NS 1 NS |u | x |u | x 1 u x u x x u x u u x u x x u x u u x u x x u x u dx dx |u | x |u | x dx dx x u x u 2 |u | u u dx 1 NS The same for uA x , x , we obtain: 2 NA |u | u u dx 1 The wave function of single particle in an infinite well is: nx u sin L L Do the following integral: 2 mx nx sin u u dx sin dx L L L L L cos L cos L dx NS NA (If m n, plug in the above expressions, NS NA 2 2 |u | |u | , 1, m 0, m u u dx u u dx n n 21 21 0 0 2 2 21 21 1 1 4 0 P a g e | 13 Problem 2: (BFG Ch.10 Problem 5) Two identical free spinless bosons in a one-dimensional infinite well. For spinless boson, we shall use symmetric wave function: (from Problem 1) 1 u x u x u x u x uS x , x NS The wave function of single particle in this infinite well is: nx u sin 2a For the ground state, both bosons occupy the ground state of single particle: (m 1 u x u x u x u x uS x , x 4 Notice that m n here, so NS 4 but not 2 uS x , x n case) 1 x x sin sin 2a 2a a For the first excited state, one of boson occupies the first excited state of single particle: (m 1 u x u x u x u x uS x , x 2 Columbia University (NY) 4.3 University of TexasAustin n case uS x , x 1 2a sin x x sin 2a a sin x x sin 2a a Problem 3: (BFG Ch.10 Problem 7) Two identical free spin-up fermions in a one-dimensional infinite well. Two fermions with same spin, due to Pauli Principle, should occupy different energy state and have antisymmetric wave function: 1 uA x , x u x u x u x u x 2 For the ground state, one occupies the ground state and the other occupies the first excited state of single particle: 1 uA x , x u x u x u x u x 2 2a The wave function vanishes at the following point x , x (0, 0), (a, a), (a, 2a), (2a, a), (2a, 2a) uA x , x 1 sin x x sin 2a a sin x x sin 2a a (Actually, under the following condition, the wave function will also vanishes: x either x or x is equal to 0 or 2a, or, x P a g e | 23 Problem 4: (BFG Ch.10 Problem 8) Three identical free spinless bosons in a one-dimensional infinite well. For spinless boson, we shall use symmetric wave function and for the ground state, all the bosons occupy the ground state of single particle: uS x , x Nu x u x u x x x x N sin sin sin 2a 2a 2a For the first excited state, one of the bosons occupies the first excited state of single particle: uS x , x u x u x u x u x u x u x Nu x u x u x uA x , x N sin x x x sin sin a 2a 2a x x x sin sin sin 2a 2a a sin x x sin 2a a sin x 2a November 30, 2007 P a g e | 33 ...
View Full Document

This note was uploaded on 04/27/2008 for the course PHYSICS 143L taught by Professor Scholberg during the Fall '07 term at Duke.

Ask a homework question - tutors are online