Midterm 7 Solutions

Bn ha l f a a continuing this recursion relation we

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: Ha L F a ` `† n - 1 `† `† n = Ha L + BN , Ha L F a ` `† n ` `† n - 2 `† `† `† n `† n - 1 BN , Ha L F = Ha L + :Ha L + BN , Ha L F a > a Continuing this recursion relation, we find that ` `† n `† n BN , Ha L F = n Ha L ` Therefore, the eigenstates of N are of the form n\ = `† Ha Ln n! 0\ where ` N 0^ = 0 as ` N `† Ha Ln ` N `† Ha Ln n! 0\ = Bn `† Ha Ln n! 0\ + `† Ha Ln n! ` N 0\F =0 n! 0\ = n `† Ha Ln n! 0\ Printed by Mathematica for Students MidtermExam7Solutions.nb ü Part c The condition that we have equal probability of measuring E = yH0L\ = 1 2 1 2 — w or E = 3 2 — w implies (from (b)) B 0\ + ‰Â f 1\F where we determine the relative phase ‰Â f using mw— 2 X px \ = as ` X px \ = YyH0L px yH0L] ` ¬༼ px =  m—w 2 `† ` Ja - aN from (a) =  2 mw— 2 `† ` ` BX0 + ‰- f X1 F Ja - aNB 0\ + ‰Â f 1\F ¬༼ a 0] = 0 and inner product on 2\ vanishes =  2 mw— 2 BX0 + ‰- f X1 FB 1\ - ‰Â f 0\F ‰Â f - ‰-  f N 2 = mw— 2 J X px \ = mw— 2 sin f Thus, the constraint on the average momentum implies sin f = 1 ‰Â f =  Therefore yH0L\ = 1 2 B 0\ +  1\F ü Part d We have YHx, tL = Xx yHtL\ = Xx ‰ - ` ÂH t — y H0L\ where ‰ - ‰ - ` ÂH t — y H0L\ = 1 y H0L\ = 1 ` ÂH t — 2 2 ‰ - B‰ ` ÂH t - — B 0\ +  1\F...
View Full Document

This note was uploaded on 02/01/2014 for the course PHYC 491/496 taught by Professor Akimasamiyake during the Fall '13 term at New Mexico.

Ask a homework question - tutors are online