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AN ABSTRACT OF THE THESIS OF BERNARD W. BANKSfor the degree DOCTOR OF PHILOSOPHY (Name) (Degree) in MATHEMATICS presented on (Major Department) (Date) Title: TIME DERIVATIVES OF OBSERVABLES AND APPLICATIONS Redacted for Privacy Abstract approved: Stuart Newberger LetA andH be self -adjoint operators on a Hilbert space. Conditions for the differentiability with respect totof -itH -itH <Ae cp e 9>are given, and under these conditionsit is shown that the derivative is<i[HA-AH]e-itHcp,e-itHyo>. These resultsare then used to prove Ehrenfest's theorem and to provide results on the behavior of the mean of position as a function of time. Finally, Stone's theorem on unitary groups is formulated and proved for real Hilbert spaces. Time Derivatives of Observables and Applications by Bernard W. Banks A THESIS submitted to Oregon State University in partial fulfillment of the requirements for the degree of Doctor of Philosophy June 1975 APPROVED: Redacted for Privacy Associate Professor of Mathematics in charge of major Redacted for Privacy Chai an of Department of Mathematics Redacted for Privacy Dean of Graduate School Date thesis is presented March 4, 1975 Typed by Clover Redfern for Bernard W. Banks ACKNOWLEDGMENTS I would like to take this opportunity to thank those people who have, in one way or another, contributed to these pages. My special thanks go to Dr. Stuart Newberger who, as my advisor, provided me with an inexhaustible supply of wise counsel. I am most greatful for the manner he brought to our many conversa- tions making them into a mutual exchange between two enthusiasta I must also thank my parents for their support during the earlier years of my education.Their contributions to these pages are not easily descerned, but they are there never the less. Finally, I want to thank Don Solomon, Greg Fredricks and Gary Greenfield for the special atmosphere of enthusiasm for mathematics that has existed between us during our years as graduate students.I learned much from them. TABLE OF CONTENTS Chapter Page INTRODUCTION 1 The Born Interpretation 1 Ehrenfe st cs Theo rem 2 Bound and Unbound States 3 Stone's Theorem for Real Hilbert Spaces 4 TIME DERIVATIVES OF UNBOUNDED OBSERVABLES 5 Potentials and States 32 The Polynomials Po /ax) 34 APPLICATIONS 37 Ehrenfest's Theorem 37 2 3 The Coulomb Potential on L(R) 38 The Case of Smooth Potentials 40 The Behavior of the Mean 43 The Absolutely Continuous States 47 The Wave Operators 48 BOUNDED OBSERVABLES 55 Multiplication Operators 59 Application 61 Integral Operators 62 A REAL STONE'S THEOREM 65 The Complexification of H 65 Operators on Hc 66 Complexification of a Real Operator 70 A Spectral Theorem for Real Operators 71 Further Properties of El and E2 78 The Uniqueness of E1 and E2 82 A Function Calculus for A 84 Stone's Theorem--Real Case 85 BIBLIOGRAPHY 92 TIME DERIVATIVES OF OBSERVABLES AND APPLICATIONS INTRODUCTION The Born Interpretation Let 9be a normalized state function defining the state of a quantum mechanical system.For a long time in the development of quantum mechanics it was not clear how to interpret 9 physically. 2 In the 1920's Max Born proposed that I 9 I should be regarded as the probability distribution of the position of the particle in state cp. Thus the probabilitTthat the particle is in regionR is I912dx. Suppose thatH = Ho + Vis the Hamiltonian of a particle in a potential fieldV, then in the Schrodinger picture the fundamental dynamical law of quantum mechanics is that if i(x, t)is the state t, then4(x, t)- iFhp(x, t). at time at That is,Ox, t)satisfies the Schrodinger wave equation.Well known considerations connected with Stone's theorem show that the solution inL2(R1)is itH Ox, t)-= e where may be thought of as the initial state at t = 0.Thus,eitHco is the state at any timet. For eacht,eitHis unitary, and so if 11911= 1 then II itHii 2 itHitH 1.The Born interpretation now says that some self -adjoint operators can be interpreted as observable quanti- ties.For such an operatorA the quantity <AeitHco, eitHcp>is 2 the expected value of A when the system is in the state e1t2. Indeed, ifAis multiplication byx., which has a natural inter- ) E Rn), pretation as the j-coordinate of position(x =(x1" xn itH itH e yo> x. eitHyo I 2dx, which is exactly the then<x.e j Rn expected value of the classicalposition function x x.when 2 leitH(pi is viewed as the probability distribution of position at timet. If we setA(-o)(t) =<AeitHco, eitHcp>then A(yo) is a real valued function oft.We may therefore ask a very general question: What is the behavior ofA(co)(t)? This paper provides some answers to this general question. Ehrenfest's Theorem dx Consider a particle of mass 1.In classical mechanics dt d2x is the momentum and is the force ifx(t) is the position of dt2 the particle. In the quantum mechanics the expected value of momentum in coordinate direction is given by<i eitHcp, eitHcp>.The thejth a x. question arises whether the derivative of the mean ofx,(?.(49)(t)), 3 3 is the expected value of momentum. Moreover, is the second deriva- tive of the mean ofx.,(x'((p)(t)), equal to the mean or expected a V < eitHitHccp,e > ?Paul value of the force which is given by Ox. 3 3 Ehrenfest [9, 455] asserted that the answer is yes!Note that in bibliographic citations the first number locates the reference in the bibliography and the second is the page number. Ehrenfest's justifi- cation of his assertion was not rigorous, nor could it be, for he gave no hypotheses. Here we shall give sufficient conditions under which Ehrenfest s Theorem is true. This will involve us in the question of the differentiability of means to which the bulk of Chapter I is devoted.It is in Chapter II that Ehrenfest's Theorem is taken up. Bound and Unbound States The theory of self -adjoint operators on a Hilbert space recognizes both eigenstates and absolutely continuous states. We can ask what the behavior ofx.(99)(t) is when cp is an eigenstate or an absolutely continuous state.For an eigenstate, isx.(P)(t), bounded? an absolutely continuous state, does x.- (cpFor)(t)converge to astconverges to co? We address ourselves to these questions in Chapter II. In Chapter III we shall give some results on the differentiability of the functionA((p)(t) where Ais assumed bounded. Under this assumption we no longer require Ato be self-adjoint. 4 Stone's Theorem for Real Hilbert Spaces LetU(t)be a strongly continuous group of unitary operators on a complex Hilbert space.Then Stone's theorem says that itH U(t) = e , for some unique self-adjoint operator H, and Ut(t) = iHeit. In Chapter IV we shall formulate and prove Stone's theorem for real Hilbert spaces. To some extent Chapter IV is independent of the preceding chapters. On the other hand,it is quite reasonable to ask about the differentiability of means on real Hilbert spaces.But then a Stone's theorem in the real case becomes indispensable, and so the material of Chapter IV is not as independent of the main theme of this paper as it might at first appear. 5 I.TIME DERIVATIVES OF UNBOUNDED OBSERVABLES LetA and H be self-adjoint operators on a Hilbert space A((p)(t) <Ae-itHco, . Consider the expression -itHco>defined fortE R.In this chapter we shall provide conditions onA, H and A--,((p)(t,) exists and is equal to 9such thatdA(9)(t)dt 1 -itH -itH <i[HA-AHje 9, e co>. In particular we shall study the cases n where"S-1= L (R ) and Ais a polynomial inx = (x1, ..., xn),or a , a a , Ais a polynomial in In these two cases ax ax ° ax we shall assume thatH = + Vwhere I-I is the self-adjoint H0 0 realization of the Laplacian =- L2and V is a suitable ax. potential. We shall begin by considering any self -ad joint operatorsA and Hon a Hilbert spacefi The first theorem gives a general criterion for the differentiability of A(co)(t).IfAis any operator we shall useD(A)to denote the domain ofA. We shall assume all operators are densely defined, but not not necessarily bounded unless it is so stated. Theorem 1.1.Let A be a symmetric operator andH a self- ad joint operator on a Hilbert space Suppose E -it 0(pH - e D(HA-AH),e-itHcp, D(A), andhAe II is bounded itH -itH for alltin a neighborhood ofto' andA((P)(t) = <Ae-c o, e ep>. 6 Then -itoH -itoH A1(9)(to) =<i[HA-AH]e 9, e 49> ° Proof.Without loss of generality setto = 0,for otherwise we can -itoH replace 9 bye p in the following argument. itH -itH (9)0) = lim<e Ae 9,9> - <Acp, (p> t itH-itH eitH-itHAe -e e A9 = lim < (P> t 0 -itHco-e-itHA9 -itH = lim< ,e 9> t 0 e-itHco-Aco-fe - itHA9 -A9] - itH = lim< , e > tø.0 e-itH49-A9 -itH e-itHAco-A9 -itH lim< ,e (p> - lim < ,e 9> t 0 t 0 Ae-itH49-A49 -itH tourn< , e co> + <iHAV 49> We next show that -itH Ae 9 - A9 - urn < ,eitH(p> = . t 0 We have 7 Ae-itHyo-Acp ,e-itH(p> <-iA1-1(p, co> I -itH =I < e cp,Ae (p> --itH-Ae-itH(p>+ AeitH(p> - < A(p>I sinceAis symmetric, I-1(pE D(A), and by adding and subtracting <-iHcp, Ae-itH(p>.
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