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Time in quantum mechanics Jan Hilgevoord University of Utrecht David Atkinson University of Groningen CLARENDON PRESS 2011 . OXFORD 0.1 The problem of time in quantum mechanics Many physicists believe that time constitutes a serious problem in quantum mechanics. The difficulty was epitomized in Wolfgang Pauli’s Handbook article of 1933 (Pauli 1933: 140): We conclude therefore that the introduction of an operator t must be renounced as a matter of principle, and that time t must necessarily be considered as an ordinary number (‘c-number’) in wave mechanics. (Translation: JH and DA). Pauli’s article signalled the conclusion of a period of rapid development of the new quantum theory that had been inaugurated by Heisenberg’s revolutionary paper of 1925 (Heisenberg 1925). An essential feature of quantum theory is that the dynamical variables of classical mechanics are represented by self-adjoint operators on a Hilbert space. The basic example is that of the position and momentum variables q and p of a particle, which are replaced by self-adjoint operators satisfying the commutation relation p q − qp = −ih̄ , (0.1) h , and h is Planck’s constant. If one assumes q and p to have where h̄ = 2π continuous eigenvalues running from −∞ to +∞ on the real axis, the well-known unique solution of (0.1) is q = q, p = −ih̄ d/dq, on a suitable space of functions (up to unitary equivalence, and modulo irreducibility). In a second famous article (Heisenberg 1927) Heisenberg sought to clarify the physical meaning of relation (0.1) by considering experiments in which the position and momentum of a particle could be measured. He concluded that these quantities could not both be measured with arbitrary precision in the same experiment. This was expressed by the uncertainty relation δp δq ∼ h̄ (0.2) where δp and δq are the precisions, or uncertainties, with which the values of p and q are known. Actually, by introducing appropriate definitions of the quantities δp and δq, relation (0.2) can be shown to be a direct consequence of relation (0.1). For example, in terms of the standard deviation as a measure of uncertainty, the inequality ∆p∆q ≥ 21 h̄ (0.3) can be derived. In the same article two more commutation relations are presented: Et − tE = −ih̄ or Jw − wJ = −ih̄ . (0.4) The quantities J and w are the conjugate variables that typically appear in the description of periodic systems. Classically J and w, like p and q, form a pair of conjugate variables, and so the second of the relations (0.4) is analogous to (0.1). The meaning of the first equation (0.4) is however much less clear. In some of the passages following equations (0.4), Heisenberg identifies E with J and t with w, calling t a ‘phase’, that is, t is considered to be an internal variable 2 of the system. This interpretation reflects the ‘or’ between the equations (0.4). But in other passages, and notably in the examples leading up to the uncertainty relation δE δt ∼ h̄, t is treated as a classical time parameter, clearly contradicting (0.4). Why did Heisenberg present the ‘same’ formula (0.4) in two different guises? In classical mechanics the time parameter is sometimes turned into an internal dynamical variable conjugate to (minus) the Hamiltonian of the system. Heisenberg may have had this in mind in connection with the first equation (0.4), although the minus sign in that equation would not then be correct. The notation also suggests a connection to Eq.(0.1). In relativity theory the momentum p and energy E of a particle are the components of a four-vector, and it is quite common to consider the position q of the particle and the time parameter t to form a four-vector also. The first equation (0.4) might then be seen as a natural complement of Eq.(0.1), as dictated by relativity theory. We shall come back to this matter in the next section. At about the same time it had become clear that Eqs.(0.4) can be mathematically problematic. In many cases (but not all) the range of the eigenvalues of J is the positive real axis, while the eigenvalues of w are angles in the interval [0, 2π]. It can be shown that self-adjoint operators whose eigenvalues satisfy these conditions cannot also satisfy the second relation (0.4). Also, it is not possible for the first equation (0.4) to be correct if E is bounded from below, and if the eigenvalue spectrum of t is the whole real axis (the latter condition being a necessary requirement if t is to be interpreted as the time parameter). Since in many cases the energy operator is known to be a well-behaved self-adjoint operator that is indeed bounded from below, it seems to follow that an acceptable time operator does not exist in quantum mechanics. Whence Pauli’s verdict mentioned at the beginning; and as a consequence the relations (0.4) have passed into oblivion. For a more complete account of this early period (see Hilgevoord 2005). The asymmetry between space and time that seems to be implied by Pauli’s statement, apparently contradicting the principles of relativity, has bothered physicists for a long time. Many proposals for circumventing the difficulty have been put forward, in particular a generalization of the axiom that observables must correspond to self-adjoint operators on Hilbert space. One can weaken the axiom to the postulate that “each observable is associated to a Positive Operator Valued Measure (POVM)” (Egusquiza, Gonzalo Muga and Baute 2002: 283; see also Busch 1989). POVMs are interesting in their own right, having many practical applications, but we shall not discuss them here, since we believe their use as a way of nullifying Pauli’s objection to be fundamentally misdirected. Again much attention has been given to finding an analogue of the uncertainty relation (0.2) in the case of energy and time. If time is not an operator, a relation of this type cannot exist. Nevertheless there do exist ‘uncertainty’ relations between energy and time of a different kind, for example the relation between the energy spread and the lifetime of a quantum state. The existence of such relations, in which t is an ordinary number, might suggest a certain similarity with momen- The problem dissolved 3 tum and position, but by the same token there appears to be a fundamental difference between position and time in quantum mechanics. Notwithstanding all these considerations, we shall show in the next section that quantum mechanics does not involve a special problem for time, and that there is no fundamental asymmetry between space and time in quantum mechanics over and above the asymmetry that already exists in classical physics. In Sect. 3 we study time operators in detail, and in Sect. 4 various uncertainty relations involving time are discussed. 0.2 The problem dissolved To see that time poses no problem for quantum mechanics, one must distinguish between two ways that it can appear in physics. First, time may figure as a general parameter of the theory, and secondly it may appear as a dynamical internal variable of some particular physical system described by the theory. In its first guise, time t is on a par with the spatial coordinates x, y, z. This is explicit in relativity theory, where t is added as a fourth coordinate to the space coordinates to form a relativistic four-vector (x, y, z, ct). Like the space coordinates, the time coordinate is independent of the physical systems the theory describes. In quantum mechanics the space and time coordinates remain c-numbers; neither x, y, z, nor t become operators. The space-time coordinates appear as parameters in the definition of well-known space-time symmetries, for example rotation invariance, spatial and temporal translation invariance, and Lorentz-invariance. In its second guise, time is a dynamical variable belonging to a particular physical system, and more than one such time variable may be present in the system. Like the other dynamical variables, dynamical time variables become operators in quantum mechanics. Examples of time operators are discussed in the following section, where we will see that time operators, although they are less common than position operators, are not fundamentally problematic in quantum theory. The work of Heisenberg that we discussed in the previous section clearly betrays a confusion between the two ways in which time occurs in physics. On the one hand, t in the first of Eqs.(0.4) is considered to be the analogue of w, an internal dynamical variable of the system; but on the other hand t is looked upon as the unique external time parameter. Also, Pauli seems to understand by t the general time parameter, the c-number character of which we have seen to be in fact unproblematic. The apparent problem of time arises when this time parameter is put on a par with dynamical position variables rather than with the coordinates of space. The confusion has proved to be quite persistent in the quantum mechanics literature, and in the remainder of this section we will try to see how this has come about. Ironically, the origin of the problem is to be found in space rather than in time, and its roots lie in classical mechanics. Much of fundamental physics deals with ‘point’ particles. A point particle is a material object that can have 4 a position, a momentum, a mass, an energy, a charge, etc. At any moment the particle is located at a point of space. Evidently a point particle and a point of space are very different things. Nevertheless they are not always clearly distinguished. Quite often the coordinates of space and the position variables of a point particle are denoted by the same symbols x, y, z (e.g. when one writes ψ(x, y, z, t) for the wave function of a particle). To avoid this confusion we shall denote the dynamical position variables of a particle by q = (qx , qy , qz ), reserving the symbols x, y, z for the coordinates of a point of space. The same confusion has led to the erroneous view that the position q of a particle and the time coordinate t form a relativistic four-vector. Though this may be true numerically, since the numbers qx , qy , qz can coincide with the numbers x, y, z, the variable q and the coordinate t are conceptually quite distinct. This becomes evident when there are several particles, which must share one and the same t. Conflating position variables of particles and space coordinates is responsible for the view that space is quantized whereas time is not, creating the false impression of an asymmetry between the treatment of space and time in quantum mechanics. We have seen that such an asymmetry does not exist, since neither the space coordinates x, y, z, nor the time coordinate t are quantized. If t is not the relativistic partner of q, what is the true partner of the latter? The answer is simply that such a partner does not exist; the position variable of a point particle is a non-covariant concept. It is an interesting fact that, whereas in classical physics the non-covariance of q may easily remain hidden because the pair q, t behaves as a four-vector, in quantum mechanics the non-covariance of q is very clear. Newton and Wigner were the first to show that an operator that represents the position of a particle must necessarily be non-covariant (Newton and Wigner 1949; Schweber 1962: 60-62). It is sometimes said that a worldline xµ (τ ) does provide a covariant description of a point particle, where xµ = (x, y, z, ct) and τ is the proper time along the curve. However, just as xµ is only a point of spacetime, conceptually unrelated to a particle, xµ (τ ) is just a curve in spacetime. And, just as the position q of a point particle at time t may coincide with the spatial components of the point xµ , so the orbit q(t) can coincide with the curve xµ (τ ) (though only if the tangent vector to the curve is timelike at every point). But the dynamical position variable q of the particle remains without a relativistic partner. Likewise, the dynamical time operators, to be discussed in the next section, are non-covariant quantities. We conclude that time gives rise to no special problem in quantum mechanics. For a fuller discussion see Hilgevoord (2005). 0.3 Time operators An ideal clock is a device or system which produces a reading that mimics coordinate time in much the same way that the position of a point particle mimics coordinate space. Let us consider this analogy in detail for a system of n free particles. The canonical commutation relations are [q x,i , px,j ] = i I δij [q x,i , q x,j ] = 0 = [px,i , px,j ] , (0.5) Time operators 5 for i, j = 1, 2, . . . , n, where q x,i , px,i are the linear operators representing the x-components of the positions and momenta, respectively, of the n particles. Similar relations apply of course for the y and z components. Units are such that h̄ = 1, and I is a unit operator on a suitable subspace of Hilbert space. In this section and the following one we shall consistently use the convention that operators are designated by bold letters, ordinary or c-numbers by normal fonts. A clock reading is represented by an eigenvalue of a clock variable operator, which may be either any real number between −∞ and ∞ (we then speak of the linear clock), or it may be limited to the interval [0, 2π] (this is the cyclic or periodic clock). There are interesting differences between the mathematics of the linear and of the cyclic clocks, and we will concentrate first exclusively on the linear clock, since the mathematics parallels that of the point particles, returning to the cyclic clock in Subsect. 3.2. 0.3.1 The linear clock Suppose that the clock variables associated with n clocks are represented by the linear operators τi , and let the canonically conjugate variables be called η i . The canonical commutation relations are [τi , η j ] = i I δij [τi , τj ] = 0 = [η i , η j ] . (0.6) Evidently these commutation relations are analogous to Eq.(0.5). ~, The generator of a translation in space is the total momentum operator P whereas the generator of a translation in time is the Hamiltonian H. Consider first an infinitesimal translation, dx, in the x-direction. This is generated by Px , so any canonical coordinate, Ω, changes under the transformation to Ω + dΩ, where d Ω = [Ω, P x ]dx/i . The y and z components of ~q i , and all the Cartesian components of ~pi , should remain unchanged under this transformation. For the system of n point particles, only the position operators q x,i should be changed under the translation generated by P x . Evidently the x-component of ~ = P n X ~pi i=1 generates precisely the required transformation. Thus ∂ q x,i = [ q x,i , Px ]/i = I , ∂x and so the operators q x,i are linear functions of coordinate space, indeed q x,i = q 0x,i + I x , (0.7) 6 where the q 0x,i are (operator) constants of integration (i.e. they are independent of x). The eigenvalues of the shifted operators Qx,i = q x,i − q 0x,i are all equal to x, and in this sense the particle positions mimic coordinate space. Indeed, it is from this fact that the pernicious error arises of identifying the shifted operators with coordinate space, x. These operators are equal to I x, not x, and the parallel distinction in the case of the clock is of cardinal importance in resolving much confusion about time in quantum mechanics. An infinitesimal time-translation, dt, is generated by H, changing any canonical coordinate Ω to Ω + d Ω, where dΩ = [Ω, H]dt/i . For a system of n ideal clocks, the clock variables τi should all be augmented under this transformation by dt, τ i → τ i + Idt , (0.8) the variables ηi being unchanged. In this way the ideal clocks are close analogues of the point masses: the position operators of the point masses are boosted in space by the total momentum operator, while the ideal clock variables are boosted in time by the Hamiltonian. The analogue of Eq.(0.7) is H = n X ηi . (0.9) i=1 This ensures that the clock variables transform as in Eq.(0.8), and that the ηi remain unchanged. The clock variables in fact satisfy dτi = [τi , H]/i = I , dt and so they are linear functions of time, τi = τi0 + I t . It is crucial to distinguish here coordinate time t from the clock readings that are the eigenvalues of the clock variables τi . In an eigenbasis |τ i = |τ1 , . . . , τn i of these operators, τi |τ i = (τi0 + t)|τ i . These n eigenvalues serve as indicators of coordinate time; but they are no more conceptually identical to it than the positions of n point particles are identical to coordinate space. The clock variables can be reset by defining T i = τ i − τ 0i , so that Ti |τ i = t|τ i. The eigenvalues of the reset clock variables, T i , are all equal to coordinate time t, much as the eigenvalues of the shifted position operators Time operators 7 representing the particle positions were all equal to x. The temptation to identify the reset clock variables with coordinate time is great, but it must be resisted: these variables are equal to It, not to t. It should be noted that the Hamiltonian (0.9) is not bounded from below, indeed its eigenvalues extend over the entire real line (cf. Subsect. 3.3). We use a Fourier integral to express the eigenvector |τ i as 1 |τ i = n (2π) 2 Z ∞ n X dη1 . . . dηn exp − i ηi τi |ηi , ∞ Z ... −∞ −∞ (0.10) i=1 where |ηi = |η1 , . . . , ηn i is an eigenvector of the variables η i . The inverse is |ηi = 1 n (2π) 2 Z ∞ Z ∞ ... −∞ n X dτ1 . . . dτn exp i ηi τi |τ i . −∞ (0.11) i=1 In the analogue of configuration space, a state vector |ψi is assigned a wave function ψ(τ ). This takes the form 1 ψ(τ ) ≡ hτ |ψi = n (2π) 2 Z ∞ Z ∞ ... −∞ n X ηi τi hη|ψi , dη1 . . . dηn exp i −∞ i=1 and, in this space, τi is represented by the number τi , while η i is represented by the differential operator −i∂/∂τi : hτ |η i |ψi = −i ∂ 1 ψ(τ ) = n ∂τi (2π) 2 Z ∞ Z ∞ ... −∞ n X ηi τi ηi hη|ψi . dη1 . . . dηn exp i −∞ i=1 Note that the Hamiltonian itself has the following representation as a differential operator on the space spanned by the eigenvectors of the clock variables: H ∼ −i n X ∂ . ∂τi i=1 This is perfectly well defined, whereas the occasionally suggested equivalence H = −id/dt is nonsense. It follows from Eqs.(0.9)-(0.10) that the unitary time evolution operator, U (t) = exp(−itH), induces the transformation U (t)|τ1 , . . . , τn i = |τ1 + t, . . . , τn + ti , (0.12) i.e. an eigenstate of the time operators simply evolves into an eigenstate of the same operators at a later time. 8 An intuitively appealing realization of the clock algebra (0.6),(0.9) is afforded by the following canonical transformation: q̂i = g τi2 2 − ηi mi g p̂i = mi g τi . These new canonical coordinates satisfy the standard commutation relations, and the Hamiltonian (0.9) takes on the form n 2 X p̂i − mi g q̂ i , (0.13) H = 2mi i=1 which we recognise as that of n freely falling point masses mi in a uniform gravitational field, g being the acceleration due to gravity. In this realization, the clock times are provided by the canonical momenta p̂i . This Hamiltonian is not bounded from below, of course, being simply (0.9) expressed in different variables. It is true that ideal clocks are ‘unphysical’ in a sense, but then so are point particles. Both are consistent with the formalism of quantum mechanics. Together they illustrate the similarity between the quantum mechanical treatment of indicators of position in space and indicators of position in time. 0.3.2 The cyclic clock An alternative realization of the commutation relations (0.6) is afforded by the cyclic clock, to which we alluded above. The eigenvalues of these clock variables, τi , are readings, τi , that are limited to [0, 2π]. Such clock variables resemble angle rather than position variables, and their conjugate variables, η i , resemble angular momenta rather than linear momenta. For notational simplicity we restrict our attention to just one cyclic clock, but any number could be treated. The eigenvectors can be expanded in a Fourier series: ∞ X 1 |τ i = √ e−imτ |ηm i , (0.14) 2π m=−∞ where |ηm i designates a discrete set of vectors, and |τ i refers to a continuous set of vectors in the same space, the two sets being related by the discrete Fourier transformation (0.14). The inverse of Eq.(0.14) is the finite Fourier integral Z 2π 1 |ηm i = √ dτ eimτ |τ i . (0.15) 2π 0 The eigenvector |ηm i of η evidently belongs to the eigenvalue ηm ≡ m, where m = 0, ±1, ±2, . . .. In the analogue of configuration space, a state vector |ψi is assigned a wave function ψ(τ ): ∞ X 1 eimτ hηm |ψi . (0.16) ψ(τ ) ≡ hτ |ψi = √ 2π m=−∞ Time operators 9 In this space, τ is represented by the number τ , and η by the differential operator d −i dτ : ∞ X d 1 hτ |η|ψi = −i ψ(τ ) = √ m eimτ hηm |ψi . dτ 2π m=−∞ The Hamiltonian of the cyclic clock is η, and the unitary time evolution operator is accordingly U (t) = exp(−itH) = exp(−itη). From Eq.(0.14), and recalling that ηm = m, we find U (t)|τ i = |τ + ti (mod 2π) . An operator on a Hilbert space of vectors is fully defined only if one gives its domain, which is a subset of vectors in the space that are mapped by the operator on to vectors in the space. It turns out that a careful definition of the relevant domains is of crucial importance in resolving certain conceptual difficulties with respect to the cyclic clock (and for the correct treatment of angular momentum), and we will now give the necessary attention to these matters. In particular, it will prove important to specify the domain of the unit operator appearing on the right of Eq.(0.6). It may be helpful first to explain some basic properties of unbounded linear operators on Hilbert space. We shall then briefly consider their relevance to the linear clock variables of the previous subsection, and then in more detail to those of the cyclic clock, where their application is indispensable. An operator Ω is defined by specifying a subset of the space called its domain, D(Ω), and a linear mapping of a vector in that domain to another vector in the space. The domain will not be the whole space if Ω is an unbounded operator; but if, for any |φi in the whole space, one can nevertheless find a |ψi in D(Ω) that is arbitrarily close to |φi in the sense of the norm, then the domain is said to be dense in the space. If, for every |ψ1 i and |ψ2 i in the domain, which is presumed dense, it is the case that hψ1 |Ωψ2 i = hΩψ1 |ψ2 i , then Ω is said to be Hermitian (or symmetric). This property is however not sufficient to guarantee the self-adjointness of Ω. For a fixed ψ2 in D(Ω), consider the subset of vectors in the Hilbert space comprising all ψ1 such that there is a φ in the space for which hψ1 |Ωψ2 i = hφ|ψ2 i . The set of all these ψ1 is called the domain of the operator adjoint to Ω . If Ω is Hermitian, and this adjoint domain is equal to D(Ω), then Ω is said to be self-adjoint. It is a basic assumption of quantum mechanics that any physical observable is represented by a self-adjoint operator on Hilbert space, for then a complete orthogonal set of eigenvectors exists (not admittedly always in the Hilbert space itself, but always in the extended space of the associated rigged Hilbert space, see Atkinson and Johnson 2002: 3). It should be noted that selfadjointness is a stronger constraint than mere Hermiticity, a fact that is ignored in elementary introductions to quantum mechanics. 10 For the linear clock, the algebra of the operators τ and η is isomorphic to that of the position and momentum operators of point particles, as we saw, and the following statements can be demonstrated by adapting standard proofs, for example those given in Yosida (1970: 198). In the space spanned by the eigend vectors of τ , the operators τ and η are represented by τ and −i dτ respectively. 2 Consider these as unbounded operators on the Hilbert space L (−∞, ∞), which is defined to be the R ∞ set of all square-integrable functions, i.e. all complex functions ψ(τ ) for which −∞ dτ |ψ(τ )|2 < ∞ , equipped with the inner product Z ∞ hφ|ψi = dτ φ∗ (τ )ψ(τ ) . −∞ From Yosida’s results we read off 1. τ is self-adjoint on the (dense) domain defined by the requirements that both ψ(τ ) and τ ψ(τ ) lie in L2 (−∞, ∞). 2. η is self-adjoint on the (dense) domain defined by the requirements that both ψ(τ ) and ψ 0 (τ ) lie in L2 (−∞, ∞), and moreover that ψ(τ ) be absolutely continuous, i.e. that ψ(τ ) be expressible in the form Z τ ψ(c) + dx g(σ) c where g(σ) is a locally integrable function. Returning to the cyclic clock, one defines the Hilbert space L2 (0, 2π) as the R 2π space of all complex functions ψ(τ ) for which 0 dτ |ψ(τ )|2 < ∞ , equipped with an inner product like that in L2 (−∞, ∞), except that the integration domain is limited to (0, 2π). Whereas τ is self-adjoint on L2 (0, 2π), η is not self-adjoint on the whole of the space. The most useful self-adjoint extension is defined by the requirements that both ψ(τ ) and ψ 0 (τ ) lie in L2 (0, 2π), that ψ(τ ) be absolutely continuous, and that ψ(0) = ψ(2π). This periodicity is in accord with the Fourier series representation Eq.(0.16). Consider however the commutation relation, [τ , η] = iI . (0.17) On what subspace of L2 (0, 2π) is I the unit operator? The difficulty is that, whereas ψ(τ ) must satisfy ψ(0) = ψ(2π) to qualify for inclusion in a domain on which η is self-adjoint, the function φ(τ ) = τ ψ(τ ) satisfies this condition only if ψ(0) = ψ(2π) = 0. Hence the domain on which the commutator (0.17) is valid is specified by the requirements that both ψ(τ ) and ψ 0 (τ ) lie in L2 (0, 2π), that ψ(τ ) be absolutely continuous, and that ψ(0) = ψ(2π) = 0. In Eq.(0.17), I is to be understood as the unit operator on this subspace alone. While there is no objection to this limitation, the restricted subspace being also dense in L2 (0, 2π), there are consequences for some of the uncertainty relations, as we will see in Sect. 4. Time operators 11 It may be noted that the above discussion of the cyclic clock variable and its conjugate is equally applicable to the angle, θ, and the orbital angular momentum, L, for the motion of a point particle in a central force field. The algebra of the operators τ and η for the cyclic clock is isomorphic to that of θ and L. In the physics literature it is sometimes asserted that if two operators satisfy the canonical commutation relation, then both must have continuous eigenvalues on the whole real axis. In fact, Pauli’s negative conclusion about the existence of a time operator mentioned at the beginning of this article was based on this belief. For a proof, Pauli referred to the first edition of Dirac’s famous book on quantum mechanics (Dirac 1930: 56). Our adaptation of his proof goes as follows: by repeated application of relation (0.17), one obtains the equality exp(icτ ) η exp(−icτ ) = η − cI , (0.18) where c is a real number. Let |ηi be some eigenvector of η belonging to the eigenvalue η. So η exp(−icτ )|ηi = exp(−icτ )(η − c)|ηi = (η − c) exp(−icτ )|ηi , (0.19) thus η − c is an eigenvalue of η, and since c may have any real value, so may the eigenvalues of η. Note that the eigenvector exp(−icτ )|ηi is guaranteed not to be the nullvector, since its norm is the same as that of |ηi, and that is surely nonvanishing.1 On interchanging the roles of η and τ we find a similar result for the eigenvalues of τ . Now we have seen that, in the case of the cyclic clock, η has discrete eigenvalues. In view of Dirac’s result, how can this be? To understand that we need to consider more carefully the conditions under which the above derivation is valid. In fact, in order to use the commutator (0.17) repeatedly, so as to obtain Eq.(0.18), we must find a domain on which the products τ η and ητ are welldefined, and on which both operators are self-adjoint. For the linear clock, this common domain is dense in the whole of the Hilbert space L2 (−∞, ∞), so there is no difficulty; but for the cyclic clock the matter is quite different. As we have noted, for the cyclic clock η is self-adjoint on a subspace of L2 (0, 2π) for which ψ(0) = ψ(2π), but the problem is that Eq.(0.17) does not hold on the whole of this space. In fact it can be shown that, if |ψi and |φi are any two vectors in the domain of η, hψ|[τ , η] − iI|φi = −2πiψ ∗ (2π)φ(2π) , (0.20) which means that Eq.(0.17) is simply not true on this space. It is only true on the subspace of the domain specified by ψ(0) = ψ(2π) = 0, but the difficulty is 1 In the second edition (1935: 94) of the book of Dirac just cited, the author adds that complex values of c are not allowed for physical reasons, since the putative eigenvector would blow up exponentially at infinity. The mathematical version of this objection is that such a complex value leads to a vector that is not contained in the extended Hilbert space. 12 that η is not self-adjoint on this restricted domain, and Eq.(0.18) does not hold, so the argument of Dirac does not go through. There is no space on which τ and η are self-adjoint, and on which Eq.(0.19) is valid. For further details, (see Kraus 1965 and Uffink 1990). 0.3.3 Discussion We have seen that the formalism of quantum mechanics allows for the existence of ideal clocks, and that the energy of such systems is unbounded. While the energy eigenvalues extend from −∞ to ∞, it is particularly the lack of a lower bound that is sometimes thought to be a serious defect, since it is feared that such a system could act as an infinite source of energy. However, as long as the system is isolated or coupled to a system that cannot take up infinite amounts of energy, nothing untoward will happen. In a sense, an ideal clock is better behaved than a point particle, for an eigenstate of such a particle’s position spreads with infinite velocity. By contrast, the eigenstates of time operators do not spread at all, but rather transform into eigenstates belonging to a different eigenvalue, cf. Eq.(0.12). We conclude that the existence of ideal clocks is perfectly consistent with the formalism of quantum mechanics. 0.4 Uncertainty Relations The commutation relation (0.1) and the inequality (0.3) are generally considered to embody the essential content of elementary quantum mechanics. They express two fundamental aspects of the theory: non-commutativity and irreducible uncertainty. In Sect. 3 we have seen that energy and time operators also exist satisfying the relation [τ, η] = iI on suitably defined domains. Concerning (0.3) we remark that although this inequality is traditionally considered to be the mathematical expression of the quantum mechanical uncertainty principle, it is actually quite unsatisfactory in this respect. The standard deviation is not the most obvious and certainly not the most adequate measure of uncertainty in quantum mechanics. For many perfectly normal quantum states the standard deviation diverges, and even when the wave-function approximates a δ-function the standard deviation may remain arbitrarily large. A consequence of this fact is that inequality (0.3) permits probability distributions of p and q to be simultaneously arbitrarily narrow, contrary to what might be expected from an uncertainty relation (Uffink and Hilgevoord 1985, Hilgevoord 2002). A more adequate measure of the spread of a probability distribution is the length Wα of the smallest interval on which a sizeable fraction α of the distribution is situated. An inequality of type (0.2) also holds for this measure: Wα (p)Wα (q) ≥ ca h̄ , if α ≥ 12 , (0.21) where cα is of order 1 (Landau and Pollak 1961, Uffink 1990). This relation expresses the intuitive content of the uncertainty principle in a much more satisfactory manner than does Eq.(0.3). Uncertainty Relations 13 As to the time and energy variables, since the linear clock is mathematically isomorphic to the case of p and q, we simply have ∆η∆τ ≥ 12 h̄ (0.22) and Wα (η)Wα (τ ) ≥ ca h̄ , if α ≥ 12 . (0.23) In the case of the cyclic clock the eigenvalues of η are discrete and proper eigenstates exist. For these states ∆η = 0, and this contradicts Eq.(0.22)! The problem is in fact well known, being analogous to the much discussed and mathematically identical case of angle and angular momentum. It turns out, however, that inequalities using more appropriate measures of uncertainty of the type (0.23) exist in this case too, as might be expected, since they are simply consequences of the Fourier transformation that relates the conjugate variables of the cyclic clock (Eqs. (0.14)-(0.15)). We refer to Uffink (1990) for a full discussion. In Sect. 1 we mentioned that the apparent lack of a counterpart to inequality (0.3) for a time operator stimulated people to look for uncertainty relations containing time as a parameter. Many instances of such relations in quantum mechanics are known, for example the familiar relation between the lifetime and energy-spread of a decaying state. An extensive discussion of such energy-time uncertainty relations is given in Busch (1989, 2000). Here we will briefly discuss a powerful and general class of relations which brings out the full symmetry between the four coordinates t, x, y, z. For a detailed discussion see Hilgevoord (1998). Basic to these relations is a notion of uncertainty that differs from the one in Eq.(0.3) and Eq.(0.21). It is an uncertainty concerning the state in which a quantum system finds itself (Hilgevoord and Uffink 1991). If a system is in quantum state |Ψi, the probability of finding it in a different state |Φi is generally non-zero; in fact this probability is given by the number |hΦ|Ψi|2 . The closer |hΦ|Ψi|2 is to 1 the harder it will be to distinguish between the two states by measurements. Accordingly, it is useful to define a number ρ as follows: |hΦ|Ψi| = 1 − ρ, with 0 ≤ ρ ≤ 1, and call it the reliability with which the states |Ψi and |Φi can be distinguished. If the states coincide this reliability is 0, whereas it has its maximal value 1 when the states are orthogonal. Let us apply these ideas to states that are translated with respect to each other in time and in space, respectively. For simplicity we consider only one space coordinate x, and we once more put h̄ = 1. The unitary operators U t (τ ) and U x (ξ) of translations in time and space are given by U t (τ ) = exp(−iHτ ) and U x (ξ) = exp(iPx ξ), (0.24) where H is the operator of the total energy and Px the x-component of the operator of the total momentum of the system, as in Sect. 3. Let the system be initially in state |Ψi. Then we define τρ to be the smallest time for which the following equality is valid: 14 |hΨ|U t (τρ )|Ψi| = 1 − ρ . (0.25) Similarly, ξρ is the smallest distance for which |hΨ|U x (ξρ )|Ψi| = 1 − ρ . (0.26) That is, the state |Ψi must be translated over at least an interval τρ in time to become distinguishable from the original state with reliability ρ, or it must be translated over at least a distance ξρ in space to become distinguishable with reliability ρ. We shall call τρ and ξρ the temporal and spatial translation widths of the state |Ψi. Both quantities have well-known physical meanings. If (1 − ρ)2 = 12 , τρ is the so-called half-life of the state; and relation (0.26) is closely related to Rayleigh’s criterion for the distinguishability of two spatially translated states, 1/ξρ being a generalization of the notion of the resolving power of an optical instrument. These translation widths are subject to very general uncertainty relations, which connect them to the width of the energy and momentum spectrum of the state |Ψi, respectively. Let |Ei and |Px i denote complete sets of eigenstates of Hand Px , where a possible degeneracy is ignored for simplicity. Using Eq.(0.24), we have Z Z hΨ|U t (τρ )|Ψi = |hE|Ψi|2 e−iτ E dE and hΨ|U x (ξρ )|Ψi = |hPx |Ψi|2 eiξPx dPx , (0.27) where the integrals may include summation over discrete eigenvalues. Define the overall widths Wα (E) and Wα (Px ) ofR the energy and momentum distributions as the smallest intervals such that Wα (E) |hE|Ψi|2 dE = α and similarly R |hPx |Ψi|2 dPx = α. It can then be shown (Uffink and Hilgevoord 1985, Wα (Px ) Uffink 1993) that τρ Wα (E) ≥ C(α, ρ)h̄ and ξρ Wα (Px ) ≥ C(α, ρ)h̄ (0.28) for ρ ≥ 2(1 − α). For sensible values of the parameters, say α = 0.9 or 0.8, and 0.5 ≤ ρ ≤ 1, the constant C(α, ρ) is of order 1. Inequalities (0.28) hold for all states |Ψi, and the only assumptions needed for their validity are the existence of the translation operators Eq.(0.24) and the completeness of the energy and momentum eigenstates. Furthermore, since in relativity theory t and ~x on the one hand, and E and P~ on the other, are united to form a 4-vector, the inequalities (0.28) lead to a relativistically covariant set of uncertainty relations. Let us expand a little on the physical meaning of inequalities (0.28). The first of these is a useful general expression of the relation between the lifetime and the energy spread of a state. Usually such an inequality is obtained in the approximation in which the decay is exponential, but here it is completely general. Though these inequalities deal with coordinate time and coordinate space, they also have relevance to time operators and position operators. This may be seen Uncertainty Relations 15 as follows. Taking position q as an example, let us see how the second inequality (0.28) relates to inequality (0.21). If the main part of the probability distribution |hq|Ψi|2 of q in state |Ψi is concentrated on an interval of length a, the width Wα (q) is of order a. In this case the translation width ξρ will be of the same order a. If, on the other hand, the distribution consists of a number of narrow peaks of width b, as in an interference pattern, the translation width ξρ will be of order b, whereas Wα (q) remains of the order of the overall width a of the distribution. Thus, inequalities (0.28) are stronger than (0.21). Next consider the case where a number of position operators q i are present in the system. Suppose the spread Wα (P ) of the total momentum spectrum is small. From Eq.(0.28) it follows then that the translation width ξρ must be large. This implies that the spread in all position variables of the system must be large. Conversely, if the spread in only one position variable is small, then ξρ is small and Eq.(0.28) implies that the spread in the total momentum must be large. Thus it is the total momentum that determines whether or not the position variables of a system can be sharply determined. Similar conclusions follow for the relation between the spread Wα (E) of the total energy and the spread of all time operators. It is the total energy that decrees whether or not the time variables of a system can be sharply determined. REFERENCES Atkinson, D., and Johnson, P.W. (2002), Quantum Field Theory. A Self-Contained Course (Rinton Press, Princeton). Busch, P. 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