E-Book Content
Solutions Manual Fundamentals of Quantum Mechanics: For Solid State Electronics and Optics C.L. Tang Cornell University Ithaca, N. Y. Cambridge University Press All rights reserved. No part of this book may be reproduced in any form or by any means without explicit permission in writing from the author and the publisher. Chapter 2 2-1. (a) ψ(x) [in units of A] 4 2 −4 −2 0 4 2 x 6 (b) +∞ 1 6 −∞ −4 1 1 = ∫ | Ψ ( x ) | 2 dx = | A | 2 [ ∫ ( 4 + x) 2 dx + ∫ ( 6 − x ) 2 dx ] = ∴ A= (c) 1 5 3 10 = 1 250 | A |2 3 . , by inspection. Next, find σ2 first: 1 6 −4 1 σ 2 = | A |2 [ ∫ ( x − 1) 2 ( 4 + x) 2 dx + ∫ ( x − 1) 2 ( 6 − x) 2 dx ] = therefore, < x="" 2=""> = σ2 + < x="">2 = 7 2 2 - 1 . 5 2 ; (d) The answer to this question is tricky due to the discontinuous change in the slope of the wave function at x = -4, 1, and 6. Taking this into account , < k.="" e.=""> = − 2-2. 2 2 h 3 3h (0 ⋅1−5 ⋅ 2+ 0 ⋅ 1) = 2m 250 50m Given 2 sin(3 πx/a),for 0 < x="">< a="" ψ="" (x="" )=" " a="" 0,="" for="" x="" ≤="" 0="" and="" x="" ≥="" a.=""> (a) < h="">=− 2 h 2 2m a ∂ a 2 ∫ sin( 3πx /a) ∂x sin( 3πx /a) dx = 2 0 2 2 9π h 2ma 2 (b) h2 ∂2 ˆ H sin( 3πx / a ) = − sin( 3πx / a) = E sin( 3πx / a) 2 2m ∂x ∴ E = 2 2 9π h 2ma 2 . (c) 9 π 2h −i t 2 2 Ψ ( x , t )= sin( 3πx / a)e 2ma a . (d)