On Conservation Laws in Quantum Mechanics

On conservation laws in quantum mechanics Yakir Aharonova;b, Sandu Popescuc, and Daniel Rohrlichd aSchool of Physics and Astronomy, Tel Aviv University, Tel Aviv 69978 , Israel. bDepartment of Physics, Chapman University, Orange CA, USA. cH. H. Wills Physics Laboratory, University of Bristol, Tyndall Avenue, Bristol BS8 1TL and dPhysics Department, Ben-Gurion University of the Negev, Beersheba 8410501, Israel. We raise fundamental questions about the very meaning of conservation laws in quantum mechanics and we argue that the standard way of defining conservation laws, while perfectly valid as far as it goes, misses essential features of nature and has to be revisited and extended. I. INTRODUCTION II. SUPEROSCILLATIONS Essential to this paper is a mathematical structure we Conservation laws, such as those for energy, momen- call \superoscillation". Common wisdom assumes that tum, and angular momentum, are among the most fun- no function can oscillate faster than its fastest Fourier damental laws of nature. As such they have been inten- component. Yet as we show here, there is a large class sively studied and extensively applied. First discovered of functions for which this assumption fails. Indeed we in classical Newtonian mechanics, they are at the core of have found functions that oscillate, on a given interval, all subsequent physical theories, non-relativistic and rel- arbitrarily faster than the fastest Fourier component. An ativistic, classical and quantum. Here we present a para- example of such a function is the following: doxical situation in which such quantities are seemingly not conserved. Our results raise fundamental questions 1 + α 1 − α N about the very meaning of conservation laws in quantum f(x) = eix=N + e−ix=N ; (1) mechanics and we argue that the standard way of defin- 2 2 ing conservation laws, while perfectly valid as far as it goes, misses essential features of nature and has to be where α is a positive real number, jxj ≤ πN, and N is a revisited and extended. large integer. An extensive discussion of the properties of this function, first introduced in [4, 5], appears in [6{8]. To display its basic properties, we first write it, via the That paradoxical processes must arise in quantum me- binomial formula, as chanics in connection with conservation laws is to be ex- pected. Indeed, on the one hand, physics is local: causes and observable effects must be locally related, in the N X sense that no observations in a given space-time region f(x) = c(n; N; α)ei(2n=N−1)x ; (2) can yield any information about events that take place n=0 outside its past light cone [1]. On the other hand, mea- surable dynamical quantities are identified with eigenval- where the c(n; N; α) are constants; ues of operators and their corresponding eigenfunctions 1 N are not, in general, localized. Energy, for example, is a c(n; N; α) = (1 + α)n(1 − α)N−n: (3) property of an entire wave function. However, the law 2N n of conservation of energy is often applied to processes in From (2) one can see that f(x) is a sum over wave num- which a system with an extended wave function interacts bers k = 2n=N − 1, ranging from -1 to 1. with a local probe. How can the local probe \see" an ex- n p tended wave function? What determines the change in Now consider this function in the region jxj . N. arXiv:1609.05041v1 [quant-ph] 16 Sep 2016 energy of the local probe? These questions lead us to un- Here we can approximate the exponentials by their first cover quantum processes that seem, paradoxically, not to order Taylor expansion and obtain conserve energy. 1 + α x 1 − α x N The present paper (which is based on a series of unpub- f(x) ≈ (1 + i ) + (1 − i ) 2 N 2 N lished results, first described in [4] and [5]), presents the paradox and discusses various ways to think of conserva- iαxN = 1 + ≈ eiαx: (4) tion laws but does not offer a resolution of the paradox. N A subsequent paper will present our resolution. The rea- son for publishing the paradox and resolution separately is that the paradoxical effect stands alone - its existence Hence, in the restricted region, f(x) behaves as an os- is independent of attempts to explain it - while readers cillation of wave number α. But, crucially, α need not may disagree with our proposed resolution. be smaller than 1. By taking α 1, we ensure that in 2 p the region of validity of the approximation, jxj . N, the function f(x) oscillates with wave number α 1 although all its Fourier components have wave number smaller than 1. In other words, a superposition of long wavelengths, the longest being 2πNp and the shortest being 2π, can, in the region jxj . N, oscillate with the much shorter wavelength 2π/α. Furthermore, the region of these \superoscilations" can be made arbitrarily large and include arbitrarily many wavelengths by taking N sufficiently large. Note that there is no contradiction with the Fourier theorem since the region where this function is (almost) identical to an oscillation of a frequency not contained by its Fourier decomposition does not extend over the entire region where the function is defined. a) Although it is not essential for our present paper, it 1/α is interesting to note that outside the superoscillatory region f(x) increases exponentially. This is a generic property of functions with superoscillatory regions. III. THE EXPERIMENT b) 1 The type of effect we describe here is common for all conserved quantities that depend on the shape of the wave function all over the space including energy, momentum and angular momentum. Here we will focus on energy, FIG. 1: (a) A photon in the box in the specially prepared for which the proof is more intuitive. low-energy quantum state which, in the central region, oscillates with a spatial frequency greater than that of any Fourier Consider a box of length 2πNa (where a is some unit components (see (b)). As the opener is passing by, it opens the box and extracts the photon, if the photon is there. The length) that contains a single photon in the state (x) shape of the wave function is not accurate but simply illus- i trative; the true wave function is exponentially larger away (x) = f(x=a) − f ∗(x=a) (5) from the center and has a more complicated shape. N with f(x) defined in Eq. (4) and α 1. Here N is a normalisation factor. (x) has properties similar to f but it obeys the boundary conditions (−πNa) = (πNa) = 0 at the walls of the box. (See Fig. 1.) E = α >> Emax = 1: (8) From now on however, for simplicity, we take a = 1 In other words, in the box we have a low-energy photon, and we work in the usual units ~ = c = 1. Given the relation between wavelength, frequency and which in the center of the box looks like a high-energy energy for the photon, the decomposition (2) shows that photon. the photon is in a superposition of different energy eigen- states with wave numbers Suppose now that a mechanism that we will callp the \opener" opens the box in the center for a time T . N kn = (2n=N − 1) (6) and inserts a mirror, such that if the photon hits the mirror, it comes out of the box (as in Fig. 1). This happens all smaller than or equal to 1 (in absolute value), corre- if the photon is situated at a distance not larger than sponding to energy eigenvalues T from the mirror; otherwise it cannot get there while the box is open. The probability for this to happen is at most R T j (x)j2dx. En = jknj (7) −T with the maximal energy Emax = 1.p On the other hand, Now suppose we find the photon out of the box. What we also know that in the region jxj . N around the cen- is its energy? ter of the box, the wave function of the photon resembles that of a monochromatic photon with wave number α Naively we would think that the emerging photon must hence of energy have one of the energies En = j2n=N − 1j ≤ 1 that it had 3 originally in the box - after all, reflection from a mirror doesn't change the spectrum of light. On second thought, however, we realize that this cannot be so. Indeed, the box was openp only for a time T . In the entire region jxj ≤ T ≤ N around the opening, the wave function of the photon was essentially indistinguishable from a plane wave of high energy E. The information revealing that the photon was not a true monochromatic photon of this high energy is contained in the shape of the wave function in regions situated farther from the opening than T ; relativistic causality dictates that this information could not arrive at the opening during the time it was open. Hence the photon that emerges from the box must be in FIG. 3: The initial and final distribution of the energy of a state which is identical to that in which a genuine pho- the opener for the cases when the photon emerges from the ton of energy E would emerge from the box. But for this box. One would expect the final energy of the opener to second case it is trivial to see what happens: the mir- be lower than the initial, to compensate for the increase of ror just reflects the photon out of the box but it doesn't the energy of the photon, but this does not happen; instead the final energy distribution is the same as the initial (up to change it frequency and energy.

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