1 Introduction

1 Introduction J. Gonzalo Muga1, Rafael Sala Mayato2, and Iñigo L. Egusquiza3 1 Departamento de Qu´ımica-F´ısica, Universidad delPa´ıs Vasco, Apdo 644. Bilbao, Spain 2 Departamento de F´ısica FundamentalII, Universidad de La Laguna, Tenerife, Spain 3 Department of TheoreticalPhysics and History of Science, University of the Basque Country, PO Box 644, 48080 Bilbao, Spain This chapter is a historical sketch of “time in quantum mechanics”. We have adopted a “physicist’s approach” to history, summarizing in rough chronological order some of the contributions underlying the topics and concepts treated in the book. The account is complemented with comments on later or recent repercussions of the original works, giving certain prominence to some of our own contributions. The time span of this “story” is almost 100 years, so necessarily many of the comments and references will be minimal, and even entire fields are absent, in particular those not related to the topics addressed in the subsequent chapters. 1.1 Role of Time in the Early Days of Quantum Theory In 1913 Bohr suggested that the interaction of radiation and atoms occurred by means of instantaneous transitions, “quantum jumps”, among the allowed atomic orbits [1]1. The jumps were accompanied by the absorption or emission of radiation, whose frequency corresponded to the energy difference of the stationary orbits, ν = ∆E/h. However, no mechanism for the timing of these transitions was provided. Soon Rutherford pointed out to Bohr that this lacuna was a “grave difficulty” of his theory, and Slater noted a contradiction between the assumed instantaneous character of the jumps and the observed narrow widths of the spectral lines. There was some interest in discovering an explanatory mechanism, but none was found. This led Einstein to propose a statistical theory for the transition probabilities in 1917. Heisenberg and the Göttingen group tried to solve the problems of the old quantum theory by creating a “truly discontinuous” theory, matrix mechanics, in which unobservables, in particular the orbits, would be eliminated, and in which visualizable models based on a space-time continuum would play no role at all. However, the timing of events did not quite fit in this scheme, as evidenced by a letter from Pauli [3] to Bohr in 1925 “In the newtheory, all physically observable quantities still don’t really occur. Absent, namely, are the time instants of transition pro- 1 The main source for this section is a recent book by Mara Beller [2] J.G. Muga, R. Sala Mayato, and I.L. Egusquiza (Eds.): LNP m72, pp. 1–29, 2002. c Springer-Verlag Berlin Heidelberg 2002 2 J. Gonzalo Muga, Rafael Sala Mayato, and Iñigo L. Egusquiza cesses, which are certainly in principle observable (as for example, are the instants of the emission of photoelectrons). It is nowmy firm conviction that a really satisfying physical theory must not only in- volve no unobservable quantities, but must also connect all observable quantities with each other. Also, I remain convinced that the concept of ‘probability’ should not occur in the fundamental laws of a satisfying physical theory.” He also speculated that perhaps time could be defined through the concept of energy, and wondered about the meaning of a time duration. Heisenberg answered a few days later [4]:2 “Your problem of ‘duration’ plays a fundamental role, and I’ve thought over several matters for domestic use. First, I believe that one can distinguish between a ‘coarse’ and a ‘fine’ duration. When, as in the newtheory, a point in space has no longer a fixed place, or when this place is still only defined formally and symbolically, then the same is true also of the time-point of an event. But there is al- ways given a rough duration, as also a rough place in space: with our geometric picture we shall still be able to achieve a rough picture of the phenomena. I think it is possible that this rough description is perhaps the only one we may ask for in the formalism.” Heisenberg’s words announce some aspects of his uncertainty paper. How- ever, things were not quite mature for that step as yet. The original agenda of matrix theorists was quite radical; they doubted that the “position of the electron in time” could be given any meaning. This opinion goes even beyond the impossibility of an accurate definition of conjugate variables, a statement that was formulated later. However, reacting to the success of the visualizable and continuous wave mechanics of Schrödinger, matrix theorists retreated eventually from the extreme original program. Even though Schrödinger had shown the formal equivalence between matrix and wave mechanics for bound systems, an interpretational war began, and the two approaches were for a while competing, each claiming a superior or more fundamental status. Ma- trix theorists reemphasized the importance of the discontinuities, and thus of the quantum jumps, since they were not evident in the continuum wave theory. 1.1.1 Born’s Collision Papers Born’s aim in his two 1926 collision papers was to harmonize the jumps and the wave picture [6,7]. For an electron-atom collision he interpreted the 2 Just one week later Heisenberg, in a letter to Einstein, wondered if the times of transition should be regarded as observable or not [5]. 1 Introduction 3 squared modulus of the stationary wave function coefficients at infinite dis- tances as probabilities for the electron “to be thrown” into a given direction for a given atomic state. The theory would therefore not provide the actual state of the atom after the collision, but rather the probability of a certain event, identified by Born as a quantum jump. In this description, however, the wave function considered was stationary, so the question of the timing of the event was not really addressed. Two early reactions to Born’s proposal were due to Pauli and Jordan: Pauli extended the probabilistic interpretation to the position of the electron, and opened the way for the return of Heisenberg to visualizable space- time concepts. According to Jordan, the probability interpretation of Born changed the status of transition probabilities from primary to secondary concepts, calculable now from the more elementary probabilities of the stationary states through their time variation [8]. He believed at that time that these were probabilities to be in the states, not “to be found”. However, the distinction is important with regard to a consistent “transition rate interpretation” of time derivatives. In general it is not possible to write the exact quantum mechanical equation for the derivative of a “population” Pa as a rate equation dP a = (W P − W P ) (1.1) dt ab b ba a b in terms of transition rates W , since the interference (non diagonal) terms of the density matrix also matter. In other words, one cannot visualize the ensemble as being divided into different subensembles until a measurement is performed, and therefore the probabilities under consideration are not probabilities to be in the state, as Jordan thought. Note, however, that in open systems or systems repeatedly measured, an equation such as (1.1) and a genuine transition rate interpretation may be justified. More on this in 1.6.2 below. 1.1.2 Pauli’s First Encyclopedia Article In 1926, Pauli still acknowledged in his first Encyclopedia article having doubts about the role of time, along the lines of his letter to Bohr [9]. He em- phasized that Einstein’s probabilistic treatment of absorption and emission was mute about the times of transition. Did this fact indicate a fundamental acausality, or was it due to the incompleteness of the theory? “This is very much debated, yet still an unsolved issue,” he concluded. He also insisted on the problem of the duration of the jump, suggesting that perhaps the pre- cision limit of the time of transition was of the same order of magnitude as the period of the light emitted, but admitted that he could not offer a more precise analysis. The duration question is still being studied nowadays, and discussed in Chap. 4 by L. S. Schulman. 4 J. Gonzalo Muga, Rafael Sala Mayato, and Iñigo L. Egusquiza 1.1.3 Uncertainty Relations Heisenberg’s 1927 paper is in part a response to Pauli’s questions [10]. It was largely based on the discussion of several thought experiments where the disturbance of the measuring device was treated in terms of classical particle space-time pictures and quantum jumping, possibly to counterattack Schrödinger’s wave picture. Heisenberg insisted on the instantaneous quantum jump concept, “... the time of transitions or ‘quantum jumps’ must be as concrete and determinable as, say, energies in stationary states.” However, in one of the idealized experiments discussed, he argued that the imprecision within which the instant of transition is specifiable, is given by ∆t in ∆E∆t ∼ h, where ∆E is the change of energy in the quantum jump. Without denying the fundamental importance of Heisenberg’s uncertainty paper, his introduction of the “principle of indeterminacy” is rather unsatis- factory and imprecise for most modern commentators. After more than seven decades, many of the issues involved have been clarified for position and mo- mentum (see e.g. a recent detailed analysis by M. Appleby where distinction is made between retrodictive errors, predictive errors, and disturbances, which lead to six uncertainty-like relations [11]), but similar clarity has not yet been achieved for time and energy. In a “note added in proof” Heisenberg already acknowledges the deficien- cies, pointed out by Bohr, of some of the arguments given to that date.

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