The Philosophy of Quantum Mechanics (II) 1. The Two-Slit Experiment The central problem which the physical theory of quantum mechanics is designed to

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1 The Philosophy of Quantum Mechanics (II) 1. The Two-Slit Experiment The central problem which the physical theory of quantum mechanics is designed to explain is the extraordinary fact that t h e elementary particles (be they electrons, photons, or whatever) exhibit in their behaviour two prima facie irreconcilable phenomena, the properties of discreteness and of interference. This situation is sometimes referred to as wave-particle duality. This duality in no way presents a philosophical problem on its own. Rather, the philosophical problems arise from a n y attempt to interpret the theory which can successfully explain that duality and the physical problems arise from any attempt to interpret the theory which can successfully explain that duality and the physical facts associated with it. The problem is not whether we think of electrons o r photons as waves or as particles or as wavicles. We should think of them as quantum mechanics tells us to. The problem is one of forming a coherent and acceptable account of the way the world is and our place in it, given what quantum mechanics says about the microphysical universe.the basic phenomena of discreteness and interference are that all elementary particles exhibit the characteristics of classical particles, while at the same time aspects of their behaviour exhibit certain paradigmatic aspects of wave motion, namely constructive and destructive interference. This duality is best illustrated by the well-known two-slit experiment. Electrons, photons or atoms (indeed any quantum mechanical particles) can be used in this experiment, which consists of firing a stream of such particles (say, photons) from a point source toward a screen (say, a photographic plate) which acts as a detector recording their arrival. Between the source of the photons and the detector-screen a barrier is placed in which there are two very fine slits. Photons can arrive at the screen only by passing through the slits. When we perform the experiment the particle aspect of the photon s behaviour is manifested b y

2 the fact that the record of their arrival at the screen consists of very many tiny little spots on the photographic plate. Each tiny spot is equally bright and of a similar size. It is just as if the screen had been sprayed with minute bullets. The tiny spots, each representing the arrival of a photon, that is, the detection of a photon impact by the screen, exhibits the discreteness or graininess that one would expect of a stream of particles.however, if we look at the distribution of the tiny spots, a very surprising fact is revealed. the spots are not uniformly distributed opposite the two slits. They are scattered all over the screen, not in a haphazard manner, but in a series of bands, corresponding alternately to the arrival of very few and of many photons. Since the arrival of a photon is registered by a pin-prick of light, we therefore observe a series of dark and bright bands on the screen. But such a sequence of dark and light bands is just what one would expect from the destructive and constructive interference of coherent waves emanating from the two slits. If we close off one of the slits and run the experiment again, then again we observe a scattering of the discrete spots of light representing the photon arrivals over the whole screen, but with a definite maximum opposite the slit remaining open. If we measure the number of photons arriving at various positions on the screen per unit time, then we find that that rate (call it P1) is maximum opposite the open slit and tails off towards the extremities of the screen. Similarly if we close that slit and open the other, then the new distribution (P2) has a maximum opposite the open slit and tails off towards the extremities of screen.both these single slit experiments exhibit the spreading out of the discrete individual photon arrivals over t h e screen. Even though they passed through the single narrow slit, they do not all arrive directly opposite it, many arriving at points distant from t h e centre. If we now open both slits, we might expect that the distribution of the spots representing the arrival of the photons at the screen (call it P12) would simply be the sum of P1 and P2, the contribution of each slit. But

3 this is not what is observed. As noted, dark and bright interference bands now occur. P12 is not the sum of the two individual slit distributions, P1 plus P2, even though to arrive at the screen the photons must pass through one or other slit. In fact P12 is exactly the type of distribution obtained if the point source of particles had been replaced by a wave source and the detector measured the energy of the waves arriving at the screen after passing through the slits and constructively and destructively interfering as the wave fronts disturbed by their passage through the slits moved toward the screen.it appears that the photons are registered as discrete units when they arrive at the screen, but the distribution of their arrivals manifests wave-like interference. the precise nature of P12 at any point on the screen is in fact a function of the distance from that point to the point on the screen opposite the mid-point between the two slits, a n d represents the number of photons arriving at that distance from that centre point. By dividing the value of P12 at any position by the total number of photons employed in the experiment we have the relative probability of a photon arriving at that position. So the wave interference phenomenon manifest by the distribution P12 shows that it is the relative probability which manifests the wave-like interference. Thinking in terms of waves, then, it is not physical or matter waves which are interfering, but what we might call probability waves.it might be thought that the interference bands manifest by the distribution of hits on the screen is brought about by the photons interfering with each other, perhaps via collision or some unknown force operating after they have passed through the slits. That this is not so is shown by the fact that the experiment may be performed allowing only one photon at a time to pass through the apparatus, thus eliminating any interactions among the photons. If w e wait long enough, however, recording successive points of arrival on the screen, the interference pattern will gradually build up. This shows that

4 we have no hope of explaining the distribution of arrivals on the screen b y interaction between the photons. 2. Superposition These phenomena of discreteness and interference are paradigmatic of the class of phenomena that quantum mechanics successfully explains. At the heart of the explanation of such results is the superposition principle. To understand it, we need to look at how quantum mechanics describes the state of a physical system and how that state relates to observationally determinable qualities. According to quantum mechanics, the state of a system (say, an electron or an atom) is represented by a function of position and time, a function describing a wave motion. But it is a wave motion not in physical space, but in mathematical space, in particular, a vector space defined over the complex numbers (i.e. numbers of the form a + ib, where i is the square root of -1, and a and b are real numbers). What is important here is not the exact forms of t h e representation of a quantum mechanical state but the fact that in general those states are thought of as vectors in a vector space. (In order to picture a model of such a system, we may take ordinary three-dimensional space with Cartesian co-ordinate axes (Ox, Oy, Oz), and think of vectors a s rays emanating from the origin of the co-ordinates. This simple model will be sufficient to make the points we need.)according to quantum mechanics, we can think of the state of a physical system as specified by a complex-valued function, which can be represented as a vector in a certain vector space. Clearly, there is already a basic divergence from classical physics, for in that case states were represented as points in state-space, which itself was just the set of all such points, each point being represented as an n-tuple of real numbers. But in quantum mechanics w e need more structure to characterise the state of a system, for here w e think of the states as vectors in a vector space, that is, as complex-valued functions, each function being the mathematical representation of a

5 wave.there is a very well-known property of wave motion, namely, that if we add (or superimpose) one wave motion on another, we obtain a n e w wave motion. Quantum mechanical states are like waves in this respect: if we have two states of a system we can combine them (superimpose them) to obtain a new state (indeed, indefinitely many new states). There is no classical analogue in particle mechanics to this principle of adding states. Yet it is foundational in quantum mechanics.to see what revolutionary consequences it has, we need only note the following: suppose we have a system (say, a particle) such that when it is in state ϕ1, we know that it has a definite position (say, x1), and that when it is in state ϕ2, that it has a different position (say, x2). By the superposition principle there exists another state of the system ϕ1 + ϕ2. However, for the new state ϕ1 + ϕ2, quantum mechanics cannot assign a definite value for the position corresponding to this new superposed state. The very best it can do is assign to each possible value of the position a definite probability that if a measurement is carried out to ascertain the position of the particle, it will be found at that position. Thus, according to quantum mechanics, the particle has in the superposed state no definite position at all. Moreover, what we have said concerning position applied to any dynamical variable or observable of the system.this principle of superposition is the key to understanding the two-slit experiment. We have seen that, according to quantum mechanics, the correct description of the state of the particles when both states are open, is that they are in a superposed state of the form ϕ1 + ϕ2, where ϕ1 is the state-description of a particle which definitely passes through slit 1 and ϕ2 is the state-description of a particle which definitely passes through slit 2. When both slits are open t h e correct state-description is the superposition of these two statedescriptions. Once the state is given, quantum mechanics provides the rules for calculating the probability that a particle will arrive at a given

6 region on the screen. When this calculation is performed, we obtain the observed interference pattern from the probability distribution exactly. The important point is that that distribution corresponds to the superposed state. It is not the distribution that would be obtained if we computed the distribution on the assumption that the particle passed, we know not which, through either slit 1 or slit 2. That assumption would yield the joint distribution P1 + P2, but as we remarked above, because of the interference pattern, that is not what is observed. The correct distribution P12 is predicted by quantum mechanics when the state of the particles is the superposition ϕ1 + ϕ2. 3. Quantum Physics We have found three crucial differences between classical a n d quantum physics. First, that the state of a quantum mechanical system is represented by a complex-valued wave function, and not simply a point in state-space. Indeed quantum mechanics does not have a phase-space representation. Secondly, that like all wave-functions, they can be added or superposed. Thirdly, that quantum mechanics associates with states not exact values of observables but probability values, namely, the probability that when a measurement is made, the observable will be found to have a value within a certain interval. This last condition entails that if w e perform the same experiment a number of times, and exactly the same initial conditions obtain in each case, we will in general obtain different specific outcomes.this is completely at variance with the situation in classical physics, according to which, if we perform the same experiment, with exactly the same initial conditions, we will obtain exactly the same outcome. The reason for this is that the dynamical evolution in time of a classical system is described by a unique trajectory (a one-dimensional curve) in state-space. It follows that if we have exactly the same initial conditions, and the system obeys the same equation of motion as it will if the same experimental set-up is used, then the particles must follow

7 exactly the same trajectory, that is, pass through exactly the same sequence of states. Since each observable is a function of the state variables it follows in the classical case that identical states will give identical values for the observables.in quantum mechanics, how a system evolves through time cannot be the same as the classical evolution. Indeed, according to that theory there is not one type of evolution of states but two. If we consider an isolated system then its evolution will be like the classical case. It will follow a unique trajectory, the evolution of the states being governed by a deterministic wave-equation (the Schrödinger equation). If, however, the system interacts with another system, say, a measuring device, then a quite different evolution occurs. The system spontaneously jumps (a discontinuous change) into a state in which a given observable has a definite precise value. We can illustrate this point easily using the two-slit experiment. A photon, after interacting with the open slits 1 and 2, is in the superposed state which we have represented schematically by ϕ1 + ϕ2. Once in this state it evolves according to the deterministic Schrödinger wave equation until it interacts with the screen. As we have seen, while it is in the superposed state no definite position can be assigned to it. However, at the instant of interaction with t h e screen, the quantum state changes discontinuously (the wave packet ϕ1 + ϕ2 is reduced ) and in the new state the observable corresponding to position has acquired a definite, precise value, namely, the position of the tiny pin-prick of light on the screen registering the photon s arrival.what quantum mechanics allows us to do is compute, for each region of t h e screen. what the probability is that when reduction occurs the position of the event registering the photon s arrival will be in that region. Hence, in quantum mechanics, there are two forms of dynamical evolution, a deterministic form (when no interaction occurs) and a discontinuous jump (when there is interaction for measurement). This reduction of the wave packet is the core of the so-called measurement problem in quantum

8 mechanics, measurement being a special case of interaction between systems. Within classical physics there is no analogue of the discontinuous second type of evolution.