Remarks on time-energy uncertainty relations

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1 Remarks on time-energy uncertainty relations arxiv:quant-ph/ v1 9 Jul 2002 Romeo Brunetti an Klaus Freenhagen II Inst. f. Theoretische Physik, Universität Hamburg, 149 Luruper Chaussee, D Hamburg, Germany. February 1, 2008 Deicate to Huzihiro Araki on the occasion of his seventieth birthay Abstract Using a recent construction of observables characterizing the time of occurence of an effect in quantum theory, we present a rigorous erivation of the stanar time-energy uncertainty relation. In aition, we prove an uncertainty relation for time meaurements alone. 1 Introuction Time-energy uncertainty relations playe an important rôle in the early iscussions on the physical interpretation of quantum theory [5]. But contrary to the position-momentum uncertainty relation, their erivation an even precise formulation suffer from the ifficulty of assigning an observable (in the sense of selfajoint operators) of quantum theory with the measurement of time [10][8]. Meanwhile, as avocate long ago by Luwig [7], it is wiely accepte that the concept of observables shoul be generalize, by allowing not only selfajoint operators (corresponing to projection value measures) but also positive operator value measures [9], an it was quickly realize that one can fin such measures which transform covariantly uner time 1

2 translations an fulfil therefore all formal requirements for a time observable [3][6][13][15]. In a recent paper [2] we gave the first (to our knowlege) general construction of such measures starting from an arbitary positive operator which may be interprete as the effect whose occurence time is escribe by the measure; this notion of time is closely relate to the concept of time of arrival but in contrast to this our construction always leas to positive operator value measures (cfr. the book in [3]). All that is one in a strict quantum language, no classical ieas or generalizations of quantum mechanics are involve (we stress that the construction is vali also in quantum fiel theory, a subject which we plan to eal with in the future). In the present paper we show that for these time observables the usual time-energy uncertainty relation hols, an that in aition, provie the Hamiltonian is positive, one fins an uncertainty relation for time measurements alone which takes the form T const H with a universal constant. Arguments for these relations have been given by several authors [1][3][11][13][16], mainly in special situations an often only on a heuristic level. It is the aim of the present note to show that these relations can be rigorously erive for an arbitrary time translation covariant positive operator value measure. 2 Covariant Naimark-Stinespring Dilation Let H be a Hilbert space an (U(t)) a strongly continuous unitary group on H escribing the time evolution. The occurence time of an effect is, following [2], escribe by a covariant positive operator value measure F, i.e. for any Borel subset B of the real line R we have a positive contraction F(B) on H, an the operators F(B) satisfy the conitions F( n B n ) = n F(B n ) if the sets B n are pairwise isjoint an where the sum on the r.h.s. converges strongly, F(R) = 1, F( ) = 0, U(t)F(B)U( t) = F(B + t). 2

3 We want to construct from these ata an extene Hilbert space K with a projection P onto the subspace H, a unitary group (V (t)) on K which reuces on H to the original time translation (U(t)) an a covariant projection value measure E on K such that PE(B)P = F(B)P. Let K 0 be the space of boune piecewise continuous functions on R with values in H. On this space we introuce the positive semiefinite scalar prouct Φ, Ψ = (Φ(t), F(t)Ψ(t)). H can be isometrically embee into K 0 by ientifying the elements of H with constant functions. The enlarge Hilbert space K is then efine as the completion of the quotient of K 0 by the null space of the scalar prouct. On K 0 we efine (E(B)Φ)(t) = { Φ(t), t B 0, t B (V (t)φ)(s) = U(t)Φ(s t), PΦ = F(t)Φ(t). E(B) an V (t) map the null space into itself, P annihilates it. Therefore they are well efine operators on K an form the esire covariant ilation. In particular, (E, V ) is a system of imprimitivity over R [14] an therefore unitarily equivalent to a multiple of the Schröinger representation. 3 The general Time-Energy Uncertainty Relation Using the covariant ilation escribe in the previous section we can use the stanar uncertainty relation in the Schröinger representation on L 2 (R), Φ (x) Φ ( 1 i x ) 1 2 3

4 which hols for all wave functions Φ in the intersection of the omains of x an 1. This follows from the valiity of the canonical commutation i x relations in the sense of quaratic forms, (xφ, 1 i x Ψ) (1 Φ, xψ) = i(φ, Ψ) i x which may be erive from the fact that 1 is the generator of translations i x U(a), an that, by Stone s Theorem [12], a U(a)Φ is strongly ifferentiable for Φ in the omain of 1. Namely, we have i x (xφ, 1 i x Ψ) (1 Φ, xψ) i x = 1 i a a=0 ((xφ, U(a)Ψ) (U( a)φ, xψ)) = 1 i a a=0 ((xφ, U(a)Ψ) (Φ, (x + a)u(a)ψ)) = 1 i a a=0( a)(φ, U(a)Ψ) = i(φ, Ψ). We can now state the general time-energy uncertainty relation: Let F be a time translation covariant positive operator value measure, an let H enote the Hamiltonian. Let Φ be a unit vector in the omain of the Hamiltonian for which the secon moment of the probability measure µ(t) = (Φ, F(t)Φ) is finite. Then we have the uncertainty relation Φ (T F ) Φ (H) 1 2 where Φ (T F ) is the square root of the variance of µ an Φ (H) = ( HΦ 2 (Φ, HΦ) 2 ) 1 2 is the usual energy uncertainty. Proof: We use the covariant ilation escribe in Section 2. For Φ H we have (Φ, E(B)Φ) = (Φ, F(B)Φ), hence Φ is in the omain of efinition of the selfajoint operator T E efine by the projection value measure E. Moreover, since the ilate time translations V (t) restrict on H to the original time translations, Φ is also in the 4

5 omain of the generator K of V. But K an T E satisfy the canonical commutation relation in the sense of quaratic forms, thus Φ fulfils the uncertainty relations with respect to T E an K. The esire time-energy uncertainty relation now simply follows from the equalities Φ (T E ) = Φ (T F ), Φ (K) = Φ (H). It is clear that the tricky point was to fin a useful representation of the Hilbert space K with which we coul reuce the computation to the stanar position-momentum uncertainty relation. However, we stress that this representation only plays an auxiliary rôle, no physical interpretation has to be associate with it. (An essentially equivalent erivation may alreay be foun in [15].) 4 Uncertainty of Time The replacement of projections by positive operators in the escription of time observables leas to an intrinsic uncertainty. We will assume in this section that the Hamiltonian is positive. Uner this conition, we will show that the minimal time uncertainty is inversely proportional to the expectation value of the energy. Let Φ be a unit vector for which the time uncertainty Φ (T F ) an the expectation value of the Hamiltonian are finite. We use the same covariant ilation as before. Because of the positivity of H, H must be containe in the spectral subspace of K corresponing to the positive real axis. We may realize K as the space of square integrable functions L 2 (R, L) where L is the Hilbert space which escribes the multiplicity of the Schröinger representation. K acts by multiplication an T E as generator of translations. Since Φ is in the omain of T E, it is absolutely continuous, an since H K + = L 2 (R +, L), Φ has to vanish at x = 0. Hence Φ is in the quaratic form omain q of the operator 2 on K x 2 + with Dirichlet bounary conition at x = 0 (symbolically 2 x 2 D ). Since the problem is invariant uner time shifts we may assume that the expectation value of T F vanishes, an to etermine the infimum (over Φ) of the quantity (Φ, 2 x 2 DΦ)(Φ, xφ) 2, 5

6 it woul be sufficient to take it over the set S = {Φ K +, Φ = 1, Φ q( 2 x 2 D ) q(x)}. We use the following relation which is vali for a, b > 0, ab 2 4 = inf λ>0 27λ 2(a + λb)3. The relation may be verifie by noting that the argument of the infimum assumes the value of the left sie for λ = 2a, hence it suffices to check the b inequality ab λ 2(a + λb)3, a, b, λ > 0. Setting c = λb, we obtain the equivalent inequality a( c 2 )2 ( a + c 3 )3. Taking now the logarithm on both sies we fin again an equivalent inequality which is a irect consequence of the concavity of the logarithm. We therefore obtain the following relation 2 inf (Φ, Φ S x DΦ)(Φ, xφ) 2 2 = inf inf 4 2 Φ S λ>0 27λ2(Φ, ( x 2 D + λx)φ) 3. We may perform on the right han sie first the infimum over Φ. We then can exploit the behaviour of the operator 2 x 2 D +x uner scale transformations. Namely, let (D(µ)Φ)(x) = µ 1 2 Φ(µx) be the unitary scale transformations on K +. Then we have D(λ 1 3 ) 1 ( 2 x 2 D + λx)d(λ 1 3 ) = λ 2 3 ( 2 x 2 D + x). Since the set S is scale invariant, the infimum over Φ is inepenent of λ. We thus obtain 2 inf (Φ, Φ S x DΦ)(Φ, xφ) 2 = c3, 6

7 where c is the infimum of the spectrum of 2 x 2 D + x. The spectrum of this operator is a pure point spectrum [12][4]. Its eigenfunctions are Φ n (x) = Ai(x λ n ), with eigenvalues λ n where Ai is the Airy function an λ n are its zeros. The smallest eigenvalue is λ 1 = So we finally arrive at the uncertainty relation Φ (T F ) H Φ with = Some comments are in orer now: 1. The new relation gives a rather large boun if compare to the original time-energy uncertainty, inee we have Φ (T F ) 2 H 2 Φ = Φ (T F ) 2 ( H 2 Φ + Φ (H) 2) , the exact largest lower boun of the left han sie being 9/4. Let us also notice that the boun is universal, i.e., oes not epen on the etails of the Hamiltonian H. 2. The state relation is covariant, i.e., energy shifts o not change it. In case the infimum of the Hamiltonian is not zero we may change H with H inf(σ(h)) 1, where σ(a) is the spectrum of the operator A an 1 is the unit operator on the Hilbert space. 3. We have an explicit formula for the state with minimum uncertainty, namely the state Φ 1 (x) = Ai(x λ 1 ). Its shape shows how the energy spectrum has to be istribute in orer to have minimal ispersion in time. (Recall that the variable x labels the energy of the system.) 4. In the light of the last remark one woners whether it woul be possible to prepare such a kin of state in a laboratory an check the relation explicitely. 5. The same relation hols for the raial momentum of the system in place of T F an by replacing the Hamiltonian by the raius. 7

8 References [1] Aharonov, Y., Oppenheim, J., Popescu, S., Reznik, B., an Unruh, W.G., Phys. Rev. A, 57 (1998) [2] Brunetti, R., Freenhagen, K., Time of occurence observable in quantum mechanics, Phy. Rev. A, in press. [3] Busch, P., The time energy uncertainty relation, in Time in Quantum Mechanics, e. Muga J. G., Sala Mayato R. an Egusquiza I. L., Lecture notes in Physics vol.72, Springer-Verlag, Berlin-Heielberg-New York (2002) an other references therein. [4] Flügge, S., Practical Quantum Mechanics, Springer-Verlag, Berlin, [5] Heisenberg, W., Z. Phys., 69 (1927) 56. [6] Kijowski, J., Rep. Math. Phys., 6, (1974) 362, an Phys. Rev. A, 59, (1999) 897. [7] Luwig, G., Founations of Quantum Mechanics, Springer-Verlag, New York, [8] Muga, J. G., Leavens, C. R., Phys. Rep. 338, (2000) 353. [9] Naimark, M. A., Izv. Aca. Nauk. SSSR Ser. Mat. 4, (1940) 277. [10] Pauli, W., General Principles of Quantum Mechanics, Spinger-Verlag, New-York, [11] Pfeifer, P., Fröhlich, J., Rev. Mo. Phys., 67 (1995) 759. [12] Ree, M., Simon, B., Functional Analysis, secon eition, Acaemic Press, New York, [13] Srinivas, M. D., Vijayalakshmi, R., Pramana, 16 (1981) 173. [14] Varaarajan, V. S., Geometry of Quantum Theory, vol.ii, D. van Nostran Cp., Princeton New-Jersey, [15] Werner, R., J. Math. Phys., 27 (1986)

9 [16] Wigner, E.P., On the time-energy uncertainty relation, in: Aspects of Quantum Theory, e. by A. Salam, E. P. Wigner, Cambrige University Press, Cambrige Mass