On the Ehrenfest theorem of quantum mechanics

Transcription

1 On te Erenfest teorem of quantum mecanics Gero Friesecke an Mario Koppen arxiv: v [mat-p] 0 Jul 2009 Marc 28, 2009 Abstract We give a matematically rigorous erivation of Erenfest s equations for te evolution of position an momentum expectation values, uner general an natural assumptions wic inclue atomic an molecular Hamiltonians wit Coulomb interactions. INTRODUCTION A basic result in te pysics literature asserts tat te mean position an momentum of a quantum system in R wit Hamiltonian evolve classically, H = j= 2 2m j x 2 j +Vx t X j ψt = P j ψt, m j 2 t P j ψt = V ψt 3 Erenfest s equations, [Er27]. Here ψt L 2 R enotes te state of te quantum system at time t, assume to evolve via te time-epenent Scröinger equation i ψt = Hψt, 4 t A ψt stans for te mean or expecte value ψt, Aψt of a linear operator or observable A on L 2 R, f, g = f g is te usual L 2 inner prouct, X an R P are te position an momentum operators see 8, 9 below, an te potential V : R R an te force V : R R act by multiplication. Te evolution of mean values as in 2, 3, besies being of interest in its own rigt, plays an important role in te stuy of quantum-classical coupling in molecular ynamics see e.g. [BNS96, GKZC04]. Also, a priori estimates relate to 2, 3 are utilize in te proofs of important results in scattering teory [SS87, Gra90, Der93, DG97]. Te euristic justification, wic can be foun in any quantum mecanics textbook, goes as follows. Wen H an A are ermitean, formally ifferentiating te mean of A an substituting into te Scröinger equation yiels t A ψt = i [H,A] ψt, 5 Zentrum Matematik, Tecnisce Universität Müncen, Boltzmannstrasse 3, Garcing, Germany, gf@ma.tum.e, mario@ma.tum.e

2 were[h,a]enoteste commutatorha AH. WenH isofte form anais a component of te position or momentum operator, formal evaluation of te commutator gives 2, 3. For interesting iscussions in te pysics literature on te sortcomings of te formal argument see [ADVGD0, ADV00a, ADV00b, Hil73]. Also, it is straigtforwar to make Erenfest type equations rigorous uner sufficiently stringent assumptions, suc as: H is self-ajoint an A is boune an leaves te omain of H invariant; or: H is a Scröinger operator wit smoot potential, A is a component of te position or momentum operator, an ψt belongs to Scwartz space for all t, te Scwartz norms being uniformly boune wen t belongs to a boune interval. To our knowlege, a rigorous version of 2, 3 uner satisfactory assumptions, wic inclue in particular te basic atomic an molecular Hamiltonians wit Coulomb interactions, is so far missing from te literature. It is owever reaily obtaine by combining known results in particular, [Hun66, RS78] wit stanar functional analytic arguments, as we point out in tis note. Making rigorous sense of te terms appearing in te abstract Erenfest equation 5 wen H an A are unboune self-ajoint operators requires, in particular: Well-efineness of te expecte value ψt, Aψt for all t. Wen te latter is interprete as an inner prouct, tis means invariance of DA uner te propagator e ith of eq Differentiability of ψt, Aψt wit respect to t. 3 Well-efineness of te expecte value of te commutator HA AH for all t, in some suitable sense. As turns out, is essentially sufficient for 2, 3, an te valiity of eq. 5: Teorem.. Abstract Erenfest teorem Let H an A be two ensely efine linear operators on a Hilbert space H suc tat: H H : DH H is selfajoint, A : DA H is ermitean H2 e ith leaves DA DH invariant for all t R H3 For any ψ 0 DA DH, sup t I Ae ith ψ 0 < for I R boune. Ten for ψ 0 DA DH, te expecte value A ψt, ψt := e ith ψ 0, is continuously ifferentiable wit respect to t an satisfies eq. 5, te rigt an sie being efine as te following quaratic form: [H,A] ψ := Hψ,Aψ Aψ,Hψ ψ DA DH. 6 Remarks Assumption H2 is neee to make te expecte value A ψt well efine see, an H3 will be useful in establising its ifferentiability see 2 an te proof below. Note tat H2 an H3 ol automatically wen A is relatively boune wit respect to H. 2 Definition 6 allows us to make sense of te rigt an sie of 5 witout te nee to consier te often complicate omains of te composite operators AH an HA, let alone proving teir invariance uner e ith. Note tat tese omains nee not be ense wen A an H are general self-ajoint operators, an o not even contain C 0 R in basic examples from pysics see below. Of course, wen ψ belongs DAH DHA were DAH := {ψ DH Hψ DA}, 6 reuces to te classical efinition ψ,ha AHψ. Example Let H be te yrogen atom Hamiltonian an A te momentum operator, H = 2 +Vx, Vx = / x, x R3. On C 0 R3 we ave i[h,a] = Vx = x/ x 3. But x/ x 3 oes not map C 0 R 3 into L 2 R 3, because te singularity at zero beaves like x 2 an ence V φ 2 L 2 = R Vxφx 2 x = 2

3 for all φ C 0 R3 wit φ It is a eeper fact is tat if A is in aition assume to be self-ajoint ten H an H2 imply H3. A proof of tis fact, as well as a counterexample sowing tat ermiteanity oes not suffice for tis implication, will appear elsewere [FS]. 4 Te main point in te proof is a weak convergence argument wic exploits te boun in H3; see Section 2. Next, we escribe our ensuing results on equations 2, 3. Let H be a Hamiltonian of te form 2 H = T+Vx, T = 2m j= j x 2, m,..,m > 0, H = L 2 R, DH = H 2 R, j 7 for conitions on V suc tat H is well efine an self-ajoint on H 2 R see below an let A a component of te position or momentum operator, X j ψx = x j ψx, DX j = {ψ L 2 R x j 2 ψx 2 x < }, 8 R P j = i, DP j = {ψ L 2 R p j ψp L 2 R }, 9 were ψ enotes te Fourier transform of ψ, efine on smoot rapily ecaying functions by ψp = R e ip x ψxx. Clearly, X j an P j are ermitean. In orer to guarantee well-efineness an self-ajointness of H on DH, by te Kato- Rellic teorem it suffices to assume tat V : R R is a real-value locally integrable function suc tat ψ Vψ is relatively boune w.r.t.t wit relative boun α <. 0 By tis one means tat for all ψ H 2 R te function Vψ belongs to L 2 R an Vψ α Tψ +C α ψ, for some constants α < an C α. Prototypical are te electronic Hamiltonian of a general molecule, = 3N, x = y,..,y N, y j R 3, H el = 2 y + N vy i + i= i<j N y i y j, vy i = α=m Z α y i R α, an te full electron-nuclei Hamiltonian of suc a molecule, = 3N + M, x = y,..,y N,R,..,R M, y i R 3, R α R 3, H = H el M α= 2M α Rα + α<β M Z α Z β R α R β. Here te M α > 0 an Z α > 0 are te masses an carges of te nuclei, an atomic units ave been use so tat = an te electrons ave mass an carge. Corollary.2. Time evolution of expecte position an momentum Let H be given by 7, wit V : R R being a real-value locally integrable function satisfying te Kato-Rellic conition 0. Ten: i X j ψt is continuously ifferentiable wit respect to t for any ψ 0 DX j DH, an satisfies te equation t X j ψt = i Hψt,X j ψt X j ψt,hψt. 3

4 ii P j ψt is continuously ifferentiable wit respect to t for any ψ 0 DH, an satisfies te equation t P j ψt = i Hψt,P j ψt P j ψt,hψt. Proof It suffices to ceck tat te ypoteses of Teorem. are satisfie. Clearly, H ols for bot X j an P j. In te case of X j, H2 an H3 are satisfie by te results of [Hun66, RS78]. As regars P j, H2 an H3 are satisfie since DH DP j = DH, P j is relatively boune wit respect to H i.e. P j ψ C Hψ + ψ for all ψ DH, as is easily euce from te Kato- Rellic conition, an ψt an Hψt are time-invariant. Note, owever, tat so far we ave not fully erive te classical Erenfest equations for position an momentum, since it remains to be verifie tat te rigt an sies in i an ii agree wit te classical rigt an sies in 2, 3. In particular, in orer to recover te classical expression V ψt, i.e. te quaratic form of te multiplication operator corresponing to te j t component of te force, aitional assumptions on te potential are neee. Teorem.3. Let H be given by 7, were V : R R is real-value, belongs to te Sobolev space W, loc R of locally integrable functions wit locally integrable weak erivatives, an V an V satisfy te Kato-Rellic conition 0. Ten: i X j ψt is continuously ifferentiable wit respect to t for any ψ 0 DX j DH, an satisfies 2. ii P j ψt is continuously ifferentiable wit respect to t for any ψ 0 DH, an satisfies 3. An important tecnical point regaring te assumptions of Teorem.3 is tat te Kato-Rellic conition is not require for V, but only its square root. Tis is relate to te fact tat V is only neee as a quaratic form, see Sec. 3. Hence te Hamiltonians, satisfy te assumptions of te teorem. Tis is because te square root of te graient of a typical term in V looks te same as te term itself, yi y i R α /2 = y i R α. Note tat for tese Hamiltonians, V itself fails te Kato-Rellic conition it oes not even map C 0 to L 2, see te example in Remark 2 above. 2 Proof of abstract Erenfest teorem Proof. We begin by recalling te properties of te strongly continuous one-parameter unitary group e ith generate by a self-ajoint operator H see e.g. [RS80]. For all t, e ith leaves DH invariant an commutes on DH wit H; moreover for any ψ 0 DH, t ψt = e ith ψ 0 is a continuously ifferentiable map from R to H an satisfies i tψt = Hψt for all t. In orer to sow te existence of ψt+,aψt+ ψt,aψt ψt,aψt =lim, t 0 we use te ecomposition ψt+, Aψt+ ψt, Aψt = Aψt+, ψt+ ψt + ψt+ ψt an sow te existence of te limits of te two terms on te RHS separately. Since ψt+ ψt ihψt strongly in H for 0, te secon term converges 4, Aψt

5 to i Hψt, Aψt. Moreover, te first term goes to i Aψt, Hψt provie we can sow tat Aψt+ Aψt weakly in H for 0. To tis en, fix t R an coose a sequence { j } R satisfying j 0. Ten by H3, te set {Aψt+ j } is boune in H. By weak compactness of te unit ball, tere exists a subsequence again labelle by j suc tat Aψt+ j f for some f H. We claim tat f = Aψt. To see tis, coose any φ DA an calculate using te weak convergence of Aψt+ j, te ermiteanity of A, te continuity of ψt in t an te fact tat ψt DA f, φ = lim j 0 Aψt+ j, φ = lim j 0 ψt+ j,aφ = ψt,aφ = Aψt,φ. Since DA is ense in H, te claim follows, an since te argument is vali for all sequences j 0, it follows tat Aψt+ Aψt. Tis completes te proof of ifferentiability of A ψt an of eq. 5. It remains to sow tat te erivative 2Im Hψt, Aψt is continuous in t. Inee, for 0 we ave Hψt+ Hψt stronglyin H an, as just sown, Aψt+ Aψt weakly in H, completing te proof. 3 Derivation of te classical Erenfest equations Proof of Teorem.3 i Tanks to Corollary.2, we only nee to evaluate te abstract expecte value 6 of te commutator an sow tat i Hψ,X j ψ X j ψ,hψ = ψ, P j ψ for all ψ H 2 R DX j. 2 m j For functions ψ C 0 R tis follows from an elementary calculation. Te general case will follow from an approximation argument, but ue to te presence of te unboune operator X j, a little care is neee. First, consier functions ψ H 2 R wit compact support. Clearly it suffices to approximate ψ by a sequence ψ ε of C 0 functions in suc a way tat te four terms appearing insie te inner proucts, Hψ ε, X j ψ ε, P j ψ ε an ψ ε, converge in L 2 to te corresponing terms for ψ. Coose a ball B R 0 containing te support of ψ. Consier te following stanar approximation obtaine by mollification: ψ ε x = χ ε ψx = R χ ε x yψyy, were χ ε x = ε χx/ε, ε 0,, χ C 0 R, χ = 0 outsie B 0, R χ =. Ten see e.g. [Eva98] ψ ε C 0, ψ ε ψ in H 2. Since H, P j an I are continuous operators from H 2 to L 2 in case of H tis follows from 0, Hψ ε Hψ, P j ψ ε P j ψ an ψ ε ψ in L 2. Finally, suppψ ε, suppψ B R+ 0, an X j is boune on te subspace of L 2 functions wit support in B R+ 0, so X j ψ ε X j ψ in L 2. Tis establises 6 for compactly supporte H 2 functions. Finally let ψ be a generalfunction in DH DX j = H 2 DX j. Let χ C 0 wit χ0 =. Ten ψ R x := χx/rψx is a compactly supporte H 2 function, so 2 ols for ψ R by te previous step, an it is straigtforwar to ceck tat ψ R ψ in H 2. Moreover X j ψ R converges to X j ψ in L 2 since X j ψ R = X j ψ R an X j ψ L 2. Consequently Hψ R, X j ψ R, P j ψ R an ψ R converge in L 2 to te corresponing terms for ψ, establising 2 in te general case. Proof of Teorem.3 ii As in te proofof i, it only remains for us to evaluate te abstract expecte value 6 of te commutator an to sow tat i Hψ,P j ψ P j ψ,hψ = ψ, V ψ }{{}}{{}, 3 =:Qψ =:Q ψ 5

6 for all ψ H 2 R. We start by consiering ψ C0 R. In tis case, an elementary calculation sows tat Qψ = Vx ψx 2 x. 4 R Due to te assumption V W, loc R an te fact tat ψx 2 C0 R, we can integrate by parts to obtain 3 for all ψ C0 R. To establis 3 for all ψ H 2, by te ensity of C0 in H 2 it suffices to sow tat te quaraticforms Q an Q are continuouson H 2. As regarsq, tis follows from te fact tat H an P j are continuous operators from H 2 to L 2. As regars Q, tis follows by writing Q ψ = fψ, gψ, f := V, g := sgn V V an noting tat ψ fψ, ψ gψ are continuous operators from H 2 to L 2 by te relative bouneness of f wit respect to T an te fact tat T : H 2 L 2 is continuous. Te proof of Teorem.3 is complete. References [ADV00a] Vial Alonso an Salvatore De Vincenzo. Erenfest-type teorems for a one-imensional Dirac particle. Pys. Scripta, 64: , [ADV00b] Vial Alonso an Salvatore De Vincenzo. On te Erenfest teorem in a one-imensional box. Nuovo Cimento B, 54:55, [ADVGD0] Vial Alonso, Salvatore De Vincenzo, an Luis González-Díaz. Erenfest s teorem an Bom s quantum potential in a one-imensional box. Pys. Lett. A, 287-2:23 30, 200. [BNS96] [Der93] [DG97] [Er27] [Eva98] [FS] F. Bornemann, P. Netteseim, an C. Scütte. Quantum-classical molecular ynamics as an approximation to full quantum ynamics. J. Cem. Pys., 05: , 996. J. Derezinski. Asymptotic completeness of long-range n-boy quantum systems. Ann. Mat., 38: , 993. J. Derezinski an C. Gerar. Scattering teory of classical an quantum N-particle systems. Springer-Verlag, 997. P. Erenfest. Bemerkung über ie angenäerte Gültigkeit er klassiscen Mecanik inneralb er Quantenmecanik. Zeitscrift für Pysik, 45:455, 927. L. C. Evans. Partial Differential Equations. American Matematical Society, 998. G. Friesecke an B. Scmit. In preparation. [GKZC04] M. Griebel, S. Knapek, G. Zumbusc, an A. Caglar. Numerisce Simulation in er Molekülynamik. Springer-Verlag, [Gra90] [Hil73] G.M. Graf. Asymptotic completeness for n-boy sort-range quantum systems: A new proof. Comm. Mat. Pys., 32:73 0, 990. R.N. Hill. A paraox involving te quantum mecanical Erenfest teorem. American Journal of Pysics, 45: ,

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