# Quantum Chemistry

**Topics:**Quantum mechanics, Hilbert space, Classical mechanics

**Pages:**280 (85317 words)

**Published:**June 29, 2013

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Essential Quantum Mechanics

GARY E. BOWMAN

Department of Physics and Astronomy Northern Arizona University

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Great Clarendon Street, Oxford OX2 6DP Oxford University Press is a department of the University of Oxford. It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide in Oxford New York Auckland Cape Town Dar es Salaam Hong Kong Karachi Kuala Lumpur Madrid Melbourne Mexico City Nairobi New Delhi Shanghai Taipei Toronto With oﬃces in Argentina Austria Brazil Chile Czech Republic France Greece Guatemala Hungary Italy Japan Poland Portugal Singapore South Korea Switzerland Thailand Turkey Ukraine Vietnam Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries Published in the United States by Oxford University Press Inc., New York c Gary E. Bowman 2008 The moral rights of the author have been asserted Database right Oxford University Press (maker) First published 2008 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this book in any other binding or cover and you must impose the same condition on any acquirer British Library Cataloguing in Publication Data Data available Library of Congress Cataloging in Publication Data Data available Typeset by Newgen Imaging Systems (P) Ltd., Chennai, India Printed in Great Britain on acid-free paper by Biddles Ltd., King’s Lynn ISBN 978–0–19–922892–8 (Hbk.) ISBN 978–0–19–922893–5 (Pbk.) 1 3 5 7 9 10 8 6 4 2

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Contents

Preface 1 Introduction: Three Worlds 1.1 Worlds 1 and 2 1.2 World 3 1.3 Problems The 2.1 2.2 2.3 Quantum Postulates Postulate 1: The Quantum State Postulate 2: Observables, Operators, and Eigenstates Postulate 3: Quantum Superpositions 2.3.1 Discrete Eigenvalues 2.3.2 Continuous Eigenvalues Closing Comments Problems ix 1 1 3 4 5 6 8 10 11 12 15 16 19 19 19 22 24 26 28 28 30 32 33 36 36 36 39 41 41 42 43 43 45

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2.4 2.5 3

What Is a Quantum State? 3.1 Probabilities, Averages, and Uncertainties 3.1.1 Probabilities 3.1.2 Averages 3.1.3 Uncertainties 3.2 The Statistical Interpretation 3.3 Bohr, Einstein, and Hidden Variables 3.3.1 Background 3.3.2 Fundamental Issues 3.3.3 Einstein Revisited 3.4 Problems The Structure of Quantum States 4.1 Mathematical Preliminaries 4.1.1 Vector Spaces 4.1.2 Function Spaces 4.2 Dirac’s Bra-ket Notation 4.2.1 Bras and Kets 4.2.2 Labeling States 4.3 The Scalar Product 4.3.1 Quantum Scalar Products 4.3.2 Discussion

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vi Contents

4.4

4.5 5

Representations 4.4.1 Basics 4.4.2 Superpositions and Representations 4.4.3 Representational Freedom Problems

47 47 48 50 52 53 54 56 56 57 59 61 61 62 62 64 68 68 68 69 70 71 72 72 73 75 75 77 78 82 83 83 85 86 86 87 88 91 93 95 95 97 97

Operators 5.1 Introductory Comments 5.2 Hermitian Operators 5.2.1 Adjoint Operators 5.2.2 Hermitian Operators: Deﬁnition and Properties 5.2.3 Wavefunctions and Hermitian Operators 5.3 Projection and Identity Operators 5.3.1 Projection Operators 5.3.2 The Identity Operator 5.4 Unitary Operators 5.5 Problems Matrix Mechanics 6.1 Elementary Matrix Operations 6.1.1 Vectors and Scalar Products 6.1.2 Matrices and Matrix Multiplication 6.1.3 Vector Transformations 6.2 States as Vectors 6.3 Operators as Matrices 6.3.1 An Operator in Its Eigenbasis 6.3.2 Matrix Elements and Alternative Bases 6.3.3 Change of Basis 6.3.4 Adjoint, Hermitian, and Unitary Operators 6.4 Eigenvalue Equations 6.5 Problems Commutators and...

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