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The science we now call physics has its historic origins in what was formerly known as natural philosophy, a qualitative yet highly analytical system of thought which explained natural phenomena in terms of concepts we would regard as metaphysical. The seventeenth century, highlighted by the work of Galileo and Newton, inaugurated an era in which the theoretical basis of physics became predominantly mathematical, and physics (or mechanics as it was then called) asserted itself as a discipline that was independent of philosophy, operating without reference to the conceptual constraints of any system of metaphysics. This transition resulted in part from the perceived failure of natural philosophy to account for Galileo's observations, as well as from the fantastic predictive power of Newtonian mechanics, which reduced kinematic phenomena to a few elegant formulae. A new science based primarily on empirical observation and mathematical derivation yielded practical results that were more fruitful than qualitative ratiocination. This success appeared to vindicate the emancipation of physics from philosophy.
Nonetheless, the separation of physics from philosophy was more of a cultural choice than a scientific necessity. In fact, throughout the modern era, physicists have always made haphazard metaphysical or philosophical assumptions that implicitly underlie their theories, but these are rarely stated explicitly and almost never combined into a coherent system. This philosophical incoherence has not impeded the practical development of physics, because everyone agrees on the basic mathematical principles and empirical results which can be replicated by any competent physicist. This era of philosophical naivete may no longer be feasible, however, as modern physics appears to be approaching the limits of what can be physically observed, and touches upon fundamental questions of metaphysics. Such is eminently the case with quantum mechanics.
Most physicists interpret quantum mechanics according to the orthodoxy established by Niels Bohr and others at Copenhagen, with its principles of "superposition", "wave-particle duality", and the inability to observe a quantum system without altering it. The Copenhagen interpretation is a muddled, incoherent, quasi-subjectivist morass of ontological propositions, as is to be expected when a philosophy is constructed ex nihilo without reference to any established philosophical system. The physicist's disdain for philosophy as unverifiable speculation enables him to reject uncomfortable questions about wave-particle duality or the objective existence of unobserved events as meaningless, while in the next breath interpreting the mathematical formalisms of quantum mechanics to provide dubious answers to these supposedly meaningless questions. Emancipation from metaphysics allows physicists to assert logically incoherent theories as though they were profound, oscillating between subjectivist and realist interpretations ad hoc, and sometimes even claiming to have repudiated the principle of non-contradiction! Convinced of the explanatory completeness of their physico-mathematical theory, some theorists find any contradiction of the conventional interpretations of quantum mechanics with abstract philosophical logic only proves the limitations of the latter. The inability or unwillingness of most physicists to see an alternative to the Copenhagen interpretation results from an understandable lack of philosophical sophistication, especially in the metaphysics of being, where many appear to hold a simple binary notion of existence and a mechanistic notion of causality, both of which break down under quantum mechanics.
A sophisticated philosopher knows that there are multiple modes of being, such as potentiality and actuality, which may resolve quantum paradoxes without logical absurdity. Heisenberg, in fact, attempted to explain quantum mechanical states as neo-Aristotelian potentia, but his voice was drowned out by the Copenhagen chorus. The philosopher Karl Popper adopted a similar theory of "propensity", based on his discussions with Einstein, but the cultural disconnect between philosophy and physics has prevented his work from taking much effect. Physicists generally listen only to other physicists about physics, so the alternate ontological interpretations that have enjoyed some limited success in recent years are from D. Bohm, who argued the wavefunction was physically real, and a fashionable theory that considers the time-reversed "advanced wavefunction" as ontologically co-equal with the standard "retarded wavefunction". These alternate explanations have many merits, but they also have vision limited by the philosophical insularity of the physicists who formulated them. To reopen the dialogue between philosophy and physics, we must lay bare the absurdities of the Copenhagen theory and show how philosophically cogent thinking may provide the necessary remedy.
The conventional "Copenhagen interpretation" of quantum mechanics encompasses a broad range of opinions. Some theorists believe quantum paradoxes merely express the practical limitations of one's measuring apparatus, while others view them as descriptions of fundamental physical reality. This diversity of opinion makes Copenhagen ideology ill-defined, resulting in conceptual confusion in the teaching of quantum mechanics. Physics students interpret the theory as they or their professors see fit, but their real confidence is in the mathematical theory. The mathematical formalism of quantum mechanics, unlike its philosophical interpretations, has been experimentally verified as an excellent device for predicting probability distributions of events. We shall see that many of the conventional interpretations of quantum mechanics are at odds with this mathematical theory and with the logic upon which it is based.
Quantum mechanics, first and foremost, is a mathematical theory of physics. The mathematics is not especially arcane or counterintuitive; it is the linear algebra of operators with complex matrix elements. This mathematical theory is predicated on familiar concepts of mathematical logic, including the principle of non-contradiction. It would therefore be absurd to propose that quantum mechanical theory somehow transcends normal Aristotelian logic, since on the contrary the mathematical formalism's validity presupposes the validity of such logic.
In the mathematical formalism of quantum mechanics, physical states may be described as vectors in a complex Hilbert space. These state vectors are denoted using the shorthand "ket" notation, so the state ψ is written as: |ψ>. It may also be represented more explicitly as a column matrix/vector. The Hermitian conjugate, a row matrix/vector, is denoted by a "bra", or <ψ|. The "kets" may be thought of as existing in a column vector space, and the "bras" in a complementary row vector space. The norm of ψ is simply the product: <ψ|ψ>1/2. We introduced the "bras" in order to define the norm. This is the extent of the purpose and meaning of the so-called "dual space" of bras and kets; it has nothing to do with wave-particle duality, Schrödinger's cat paradox, or any of the other purported "dualisms" supposedly proven by quantum mechanics. The bra is simply the Hermitian mirror image of the ket. It makes no difference whether the bras or the kets are chosen to denote the wavefunction ψ, since the only physically observable quantities are the probabilities, which are found by squaring the magnitude of the components of ψ. This operation is invariant under Hermitian conjugation, so the same result is achieved whether we choose to use the bras or the kets as representing physical states. [A similar situation exists in special relativity with our choice of contravariant or covariant vectors.] By convention, physicists choose the kets (or column vectors) to denote the state of a quantum system.
By referring to the kets or column vectors as the "state" or "wavefunction", we make some interesting assumptions. Treating |ψ> as a "state", we attribute physical reality to a generally unobservable condition, as should be obvious from the fact that the coefficients of ψ may have imaginary values. The second name, "wavefunction", reflects the fact that ψ obeys Schrödinger's equation, a simple second-order differential equation that has the same mathematical form as the classical wave equation. This is where we derive the notion of particles having a "wave nature".
Properly understood, the "wave-particle duality" is a duality between the wavefunction and the particle it describes. We should note that the quantum wavefunction is not a physical wave! In the classical wave equation, ψ denotes a physically observable quantity, namely the amplitude of the wave. This may have spatial dimension, as in a mechanical wave, or dimensions of force per unit charge, and so on. In quantum mechanics, ψ is a dimensionless, complex-valued mathematical object whose sole relation to physical reality is its squared magnitude, which equals the probability of occupying a particular state. Similarity in mathematical form does not imply similarity in physical interpretation.
A similar realization occurred when the Michelson-Morley experiment forced physicists to abandon their belief in "ether", which arose in the first place from the conviction that light, being mathematically wavelike, must have a medium, as mechanical waves do. The analogy of course fails since the "wave" amplitude of a photon does not extend in real space, but is a function of the strength of its electric and magnetic fields. Since the electromagnetic wave is not a perturbation of real space, there is no reason to expect its conjectured "medium" to exist there either.
The distinction between a quantum wavefunction and a physical wave can be obscured when we consider the case where ψ is the spatial wavefunction, which measures the probability of being at position x, having continuous real values. In several celebrated experiments, such as Young's double-slit apparatus, the spatial distribution of scattered particle intensities resembles that of classical wave interference. Here, some physicists claim, is experimental evidence that the particle can become wavelike. Others, a bit more accurately, will say that the wavefunctions interfere with one another, while the particle remains a discrete object. I will deal with this topic at greater length after more fundamentals have been laid down, but for now consider that Schrödinger's equation, the mathematical basis of the supposed wave nature of particles, holds for all time for all systems, spatial or non-spatial. If Schrödinger's equation does indeed describe a wave nature of physical objects, then we must accept that these objects are wavelike at all times, not just in certain experimental setups. Of course this can not be the case, since we in fact observe particles as discrete objects, even in Young's experiment where they are individually detected at specific points. The result of Young's experiment and the like can be explained solely in terms of Schrödinger's equation, so there is no need to introduce a "wave nature" of particles, since Schrödinger's equation is fully consistent with particles existing in discrete form. The spatial wavefunction ψ remains a dimensionless, complex-valued mathematical object. When we determine the wavefunction as a function of position ψ(x), we are not describing a spatial wave, but projecting the ket vector onto real space, or calculating the product: |x> <x|ψ>. The spatial wavefunction ψ itself is not an object in real space and cannot be measured in meters.
Let us examine the composition of the state vector |ψ>. |ψ> coexists in an abstract Hilbert space with many other state vectors. We may arbitrarily construct a set of linearly independent vectors to act as coordinate axes, or a basis, of this Hilbert space being considered. For simplicity, we take them to be mutually orthogonal (as such a set may always be constructed by the Gram-Schmidt procedure). All vectors in the Hilbert space, including |ψ>, may be expressed as a linear combination of the basis vectors |φi>, where i is merely an index enumerating the vectors.
The product |φ1><φ1|ψ> is the projection of ψ onto |φ1>. Defining the norms of ψ and the orthogonal basis vectors φi to be unity is the same as saying that the sum of all |φi><φi|ψ> (for all vectors |φi> in the Hilbert space) equals the identity operator. This is nothing more than the closure relation from linear algebra, and it may be generalized for uncountably infinite dimensional Hilbert spaces (such as that of the spatial wavefunction) by introducing Cauchy sequences, a result which can be found in advanced linear algebra texts. Our basis vectors form an orthonormal complete set, and any vector in the Hilbert space may be expressed as a linear combination of them. [We should note that this "completeness" of the orthonormal basis is not the same as the mathematician's notion of the "completeness" of Hilbert space, which means that every Cauchy sequence in the space converges to a point in the space according to the metric defined by the norm.]
Until this point, we have retained the customary description of |ψ> as a physical state. It remains to be seen what physical sense, if any, can be made of this statement. In actual practice, a physicist does not begin not with ψ, but with the φi's out of which ψ is composed as a linear sum. The φi's are defined to be the eigenvectors of the physically observable dynamical quantities we wish to understand. These "observables" can be represented as Hermitian square matrices with the dimension of the Hilbert space. Since the matrices are Hermitian, they will have real eigenvalues, as is necessary since we can only measure real values for physical quantities.
The "completeness" of quantum mechanics is merely a mathematical statement that the eigenvectors of the observables completely span the Hilbert space. This is an algebraic truism, which does not in any way show that quantum mechanics "knows all", nor that there is no new physics to be discovered beyond what we can calculate. In informal physical terms, the completeness theorem says that the eigenstates of the observables span all possible states. "All possible states" should not be understood in the broadest sense, but only as applying to those possible states that are distinguishable by the observables (or dynamical quantities) considered. This does not in any way preclude the subsequent discovery of new quantum observables (as has happened many times since quantum mechanics was rashly declared to be a "complete" theory by the Copenhagen consensus). The dimensional extent of the Hilbert space is defined by the observables being considered; hence the completeness theorem provides no deeper revelation than the theorems of linear algebra.
The projections (or properly speaking, the inner products) <φi|ψ>, when squared, yield the probabilities of finding each eigenstate |φi>. The eigenstates are the only states that can be physically observed. The coefficients of the eigenvectors give the probabilities of being found in each eigenstate, and the measured values of the dynamical variables will always be eigenvalues of the observable.
We suspend, for the moment, the messy question of what constitutes making an observation, and instead consider the states themselves from a phenomenological perspective. As a physicist, one can only verify the eigenstates as real physical states. What then of intermediate states, the so-called "superposition of states" that supposedly occurs when no one is looking? Typically, physicists deny that there is any meaning in asking what happens between observations, yet in the next breath, they attempt to explain what happens between observations by the "principle of superposition" or some other sophistry. The idea that a particle can simultaneously exist in two eigenstates is utter nonsense. In fact, the mathematical formalism explicitly recognizes the mutual exclusivity of eigenstates by their orthogonality.
We understate the case when asserting that the eigenstates are the only states that may be physically observed, since they are the only states with any physical meaning at all. "Superpositions" of states are merely mathematical constructs that determine the probability of a system being found in each eigenstate. Treating the superposition as an actual physical state is an idea borrowed from classical wave mechanics, where any wave motion may be expressed as the mathematical sum, or superposition, of the normal modes of oscillation. Since, as discussed earlier, the mathematics of the quantum wavefunction is similar in form to that of classical waves, it is tempting to make the error that the physical interpretations must also be analogous. We can see immediately how the analogy fails, since in quantum mechanics the only possible states are the eigenstates, while the normal modes of a classical wave are not the only physically possible oscillatory states. Moreover, the wavefunction does not measure a physical quantity, but a probability amplitude, namely the probability of being found in a particular eigenstate. For the purposes of calculation, a physicist would not pay any penalty for pretending that the superposition is some physical state that exists "between observations" or "when nature's eyes blink", since the mathematical formalism is indifferent to physical interpretation. Nonetheless, this interpretation yields severe philosophical difficulties that contradict the principles of noncontradiction and empiricism which are the foundations of physical science.
Quantum mechanics upholds the principle of noncontradiction by keeping the eigenstates mutually orthogonal in probability space. Unlike the situation in classical wave mechanics, which deals with real Euclidean space, orthogonality in quantum Hilbert space directly implies mutual exclusivity: <φ|φ'> equals zero for all distinct eigenstates |φ>, |φ'>. So Schrödinger's cat is always either alive or dead, never both. We need not wonder how nature "knows" to present us with an eigenstate whenever we choose to look, nor do we need to propose that the act of observation forces the cat to become either alive or dead, concocting contrived theories of how the apparatus interferes with the experiment, when absolutely none of the preceding discussion, nor any of the formalism of quantum mechanics, takes any account of the structure of the apparatus. The quantum formalism is able to make extremely accurate predictions independently of the type of apparatus used. "Observation" in quantum mechanics does not in any way suppose the existence of a specific physical apparatus for procuring data.
This leads us to the next absurdity proposed by some Copenhagen theorists: the idea that merely knowing the system changes it physically, even though the existence of sentient life is not in any explicit way supposed by the mathematical formalism of quantum mechanics. Resolving this supposed paradox is simple enough if we keep in mind that quantum mechanics is a probabilistic theory. Consider, for example, the so-called "collapse of the wavefunction", of which Schrödinger's cat is a specific case. The mere act of observation causes the wavefunction to collapse into a spike, or delta function, about the eigenstate where it is observed. This, however, does not constitute a physical change in the system, since the wavefunction measures probability, not a physical dynamical quantity. The collapse of the wavefunction is nothing more fantastic than a simple corollary of conditional probability. Given that a particle is found in state |φ>, its probability of being found in |φ> is one, and the probability of being found elsewhere is zero. This phenomenon is utterly trivial and common, though misunderstood, in classical probability, and there is no reason to introduce a new interpretation here. Probability is probability, after all, is it not?
Many think not. Numerous attempts have been made to distinguish "classical probability" from "quantum probability", with most insinuating that the former is merely a function of ignorance while the latter reflects a deeper physical reality. Both of these interpretations are somewhat misguided, though perhaps forgivably, as probability can be a slippery concept. The misinterpretation of classical probability is more fatal, since it makes possible the claims of quantum exceptionalism.
In classical statistical mechanics, for example, we are usually taught that there is no real randomness involved, since the behavior of each molecule is assumed to be fully determined by Newton's equations of motion. Statistical mechanical phenomena, such as a drop of ink diffusing evenly throughout a liquid, but never reverting to a drop, are considered a result of our ignorance of initial conditions. Taken literally, this is obvious nonsense. As Karl Popper wittily observed, even if there was no one to supply the necessary nescience, the result of the experiment would be the same. It happens that the overwhelming majority of possible initial conditions will result in uniform diffusion. The result is caused not by ignorance, but preparation of the sample in one of the non-special cases.
Similarly, when tossing a coin, one might say that the equal number of heads and tails after many tosses is a result of ignorance of the initial conditions, but this is not quite right. It is by preparing the system in a suitable variety of initial conditions that we get the right distribution of results. While, microanalyzing, there is no "genuinely" random event, the experiment as a whole yields the results of a random macroevent. What we really have is an exercise in conditional probability. Given a distribution of initial conditions, a certain distribution of final conditions should result.
Further, we observe that statistics has very real, physically measurable implications which can be as deterministic as any classical mechanics problem. Pressure, friction, and temperature, after all, are purely statistical phenomena. Not too many people find statistical mechanics to be philosophically disturbing, since we can be comforted that there is no real randomness at the individual molecular level. However, an ensemble of classical particles could be treated as an object with a "wavefunction" that evolves in time as a function of conditional probability. Such a wavefunction would have real physical implications, and could be "collapsed" by making suitable measurements.
The quantum wavefunction differs from this classical wavefunction perhaps only in that it measures an intrinsic "propensity" (to use Popper's term) of the system to be in a given state. Wavefunctions thus act as Aristotelian potentia, though it is not clear that this is very different from the role of the classical wavefunction. In quantum mechanics, we must assume an initial state, or distribution of initial states, from which we derive a probability distribution of final states, as with the classical wavefunction. The only distinction is that in quantum mechanics we no longer assume determinism on the microscopic level. Whether the randomness we observe among fundamental particles is "genuine" or merely a result of statistical phenomena from an even smaller scale is an open question, though the openness of this question has been obscured by unfounded claims of the "completeness" of quantum mechanical theory. It is vain to argue that the fundamental particles are "structureless," for such a characterization only has meaning with respect to the known fundamental forces. At one time the nucleus was considered structureless; after the neutron was discovered, a "strong" force mediated by gluons had to be introduced, under which even the nucleons are not structureless. Regardless of whether quantum randomness is "genuine," it remains true that the relationship between the quantum wavefunction and the system it describes is fully analogous to that between our proposed classical wavefunction and a macroscopic system.
Quantum mechanical randomness seems different only because it describes the most fundamental level of matter we know, causing some to infer that this randomness is indeed "genuine". Tortured arguments about how the density matrix must be conceptually distinct from quantum superposition can be simply refuted by the student who uses either interpretation in applying the mathematical formalism with equal efficacy. As there is no mathematically compelling reason to distinguish between the types of probability, we turn to philosophical logic which indicates that the existence of "genuine" randomness in no way breaks the analogy between classical and quantum wavefunctions and their respective systems.
Affirming that the analogy is maintained is not equivalent to saying that the quantum and classical wavefunctions are identical. As noted above, the quantum wavefunction measures an actual propensity to be in a particular final state, given an initial state (or distribution of states). It is also important to consider that the wavefunction is a function of the system or experiment, not necessarily a particular particle. The idea of a ghostly wavefunction accompanying individual particles may be helpful for some problems, but leads to chaos and confusion in situations like Young's double-slit experiment, which seem to imply action-at-a-distance. Such difficulties are resolved when the wavefunction is understood to be a function of the entire experiment.
Another class of paradoxes, of which the Einstein-Podolsky-Rosen experiment is the most famous, appears to demonstrate that one's choice of measurement variable can physically change the system. First, we should clarify what is meant by a quantum measurement. Typically, one pictures an identically prepared experiment repeated many times, and due to intrinsic randomness, there may be varied results. Equivalently, a measurement can be done with many identical particles simultaneously subjected to the same experiment. This latter method is how most experiments are done in practice, though experiments with individual particles have also been done to clarify questions of quantum indeterminacy.
For simplicity, we shall consider the two-state spin problem, since all other problems are mere mathematical complications of the same paradox. When "measuring" spin, we test how particles are deflected by a magnetic field. Employing Popper's "propensity" interpretation, an electron chosen at random will have equal propensity for being deflected up or down. Assume it is deflected up along the z direction. We repeat the experiment with the same electron in the x direction, and again along the z axis. In this last step, there is a 50% chance the electron will be deflected down. This Stern-Gerlach paradox suggests that the act of observation changes the particle. If we only measured the z-spin repeatedly, we would always have the same result. But by interposing an x measurement, the z state can actually be reversed. Mathematically, this is a consequence of the non- commutativity of the spin operator matrices involved. Also, the positive x state can be expressed as a mathematical superposition of the two z states. It seems we have found a physical basis for the principle of superposition, and the dependence of reality upon observation, have we not?
Of course we have not. We should be alerted to this fact by noting that this paradox is a consequence of linear algebra, and so the premises upon which the calculus is based must be upheld. One is that the eigenstates are mutually exclusive with probability zero. This eliminates the interpretation of the x state being simultaneously positive and negative z. All we can measure are the eigenvalues, and no z eigenvalue can be measured by a magnetic field gradient in the x direction. We do not need to actually observe the x measurement to make this experiment work. We could just skip to looking at the last result, and we would get (after repeated iterations) a fifty-fifty distribution of particles with up and down spin. So we have not reintroduced subjectivism into physics.
Nonetheless, it seems that the very act of "measurement" has forced the particle into an eigenstate of x, and erased any information regarding z-spin. Here lies the crux of the matter: our "measurement" is not really a measurement in the purest sense, but an act of filtration that re-orients the system in a manner somewhat analogous to a linear polarization filter. The wavefunction transforms from being a z- eigenstate to one of the x-eigenstates in a probabilistic fashion. The electron is always in an eigenstate, never a superposition, except in a mathematical sense.
It is meaningless to ask what has become of the z- spin, because there is only one spin, which may be oriented in different directions by magnetic field gradients. The phenomenon is as remarkable as a polarizer "forcing" light to align itself with it or not pass through at all, but no more. That there is in reality only one spin is mathematically acknowledged by the fact that only one of the spin operators is necessary to determine the system. Two non- commuting operators may not be simultaneously diagonalized; that is, they have no common eigenstates. That two things can not be simultaneously measured is no surprise when they are actually different aspects of the same thing. If we placed equal magnetic gradients in both the x and z directions, we would simply align the spin along a forty-five degree angle, so this non-commutativity does not prevent us from conducting any experiment. It does prevent us from collecting contradictory information. Eigenstates are mutually exclusive, and a particle cannot simultaneously be in an x and z eigenstate. Similar arguments hold for angular momentum, quark color, and other systems with such paradoxes. With this analysis in mind, let us look at the Einstein-Podolsky-Rosen paradox anew.
We consider two particles created with necessarily opposite spin as a result of conservation of angular momentum. After they have traveled some distance from each other, we may choose to measure the spin along an arbitrary direction. If the other particle has its spin measured along the same direction, the exact opposite result would occur. One might ask how the other particle "knows" to orient itself in the opposite direction. We could either accept action at a distance, which runs into some problems with relativity, or acknowledge that we are actually measuring the whole system while we seem to be measuring a part. None of this contradicts the supposition that the system is always in an eigenstate, although seeming action at a distance may be intuitively unsettling. There is no violation of logic here, nor of objective reality, nor of the notion that the particle always has a well-defined physical state at every point in time.
This last point is perhaps the most controversial, since it strikes directly at the common interpretation of Heisenberg's uncertainty principle. For mathematical reasons, I have deliberately avoided using position and momentum as exemplary observables. This is because they are continuous variables and cannot be expressed as typical matrices and vectors. The same Dirac notation of bras and kets may be used, but they have slightly different meaning here. There is now a continuous basis of eigenstates |x> and |p> which may range over all real numbers, and have eigenvalues x and p, position and momentum. The operators can no longer be represented as matrices, but they are still linear and Hermitian, and probabilities may be computed by integrating the wavefunctions as a function of x or p. There exist proofs of the mathematical equivalence of the matrix and integral formalisms, but they would be much too cumbersome to include here.
The position-momentum uncertainty principle imposes a constraint on the accuracy of measurement: the more accurately one measures position, the less accurately one can measure momentum, and vice versa. This qualitative assertion comes from the non-commutativity of the momentum and position operators (which is mathematically unsurprising, since momentum is proportional to the derivative of position, and an operator does not commute with its derivative). From the commutation relation of these operators, we may derive that ΔxΔp > ħ/2. Due to the extreme smallness of Planck's constant, h, this relation imposes no practical constraint on measurements of position and momentum, so it cannot be subjected to a direct experimental test. We should articulate precisely what this mathematical statement means, and how it differs from most qualitative assertions about the uncertainty principle. The deltas in the expression represent standard deviation widths of distributions. Recall that the wavefunction tells us the probabilities of occupying various final states given an initial state. The uncertainty principle implies that it is impossible, in principle, to determine both position and momentum with infinite precision. This may seem like a new paradox, but we have actually dealt with it earlier.
In the two-state spin system, we could have computed standard deviation widths for the spin values prior to each measurement. Had we done so, we would have found uncertainties as large as the magnitude of the spin; in other words, "spin x" would have been totally "washed out" prior to the x-measurement, leading to an equal probability of states. We dealt with that situation by noting that there is really only one spin. Are we to say here that position and momentum are actually different aspects of the same quantity? This supposition may seem dimensionally problematic, but in its favor is the fact that momentum is proportional to the derivative of position, hence the two are inextricably linked. Continuing our analysis, we could say that "measuring" momentum actually forces the "position-momentum" to align itself in a certain way, and we must then take leave of physical intuition.
A simpler, more constructive approach would avoid assuming that mathematical similarity implies similarity of physical interpretation. First, what do these supposed "uncertainties" Δx and Δp really mean? The wavefunction may be a superposition of eigenstates with different amplitudes such that, if graphed, the overall waveform has a width of what we call the uncertainty. It is a measure of the distribution, or spread, of eigenstates. As a postulate of quantum mechanics, we can only measure a particle to be in an eigenstate (the only physically meaningful states), giving a real eigenvalue as the value of the observable. Hence, whenever we make a measurement in the formal sense, the uncertainty is infinitesimal. The measured uncertainty in the complementary variable is not infinite, so the product of uncertainties must be much less than Planck's constant! The uncertainty principle is a statement regarding distributions (of many particles, or the same particle in a repeated experiment); it does not refute the idea of a particle having a well-defined position.
In fact, if we hold this continuous variable formalism to the same rigidity we did in the discrete case, we may not be able to observe the particle to have anything but a well defined position. Remember that each different position corresponds to a different eigenstate, and all eigenstates are mutually exclusive. This is directly implied by the normalization of the wavefunction to have an integral sum of one. If the eigenstates were not mutually exclusive, the integral in probability space would be less than one. As for the wrongheaded objection that the probability of being at a specific spot is numerically zero, this is merely a consequence of classical measure theory and is fully consistent with the supposition of well-defined trajectories. In fact, one may derive directly from the commutation relation [as Cohen-Tannoudji does] the fact that there is an infinite continuum of eigenstates. A particle, therefore, can not be "smeared" in space (unless we radically alter our physical interpretation of an eigenstate). It may have a probabilistic propensity to be in a given spot if placed in the same initial conditions repeatedly, but that is all. The same holds true for momentum, so how can there be no simultaneous measurement of both?
To be precise, the uncertainty relation holds only for those components of position and momentum along the same direction. If we consider that momentum measurements can be reduced to making successive position measurements at different times, we approach a solution to the problem. Since this is a thought experiment, we may suppose the position measurements to be made with infinite accuracy, so the only uncertainty would come from the measurement of time. (Alternatively, we might have supposed the time measurement to be accurate, but we have not discussed time enough to justify such an assumption.) Earlier, we puzzled over the possibility of position and momentum being aspects of the same thing. Now, it is reduced to position and time. That space and time are different aspects of the same thing is the most elementary relativistic concept, and here it is encoded in the uncertainty principle. The particle exists in space- time, and attempting to fix it spatially or temporally aligns it in a specific four-dimensional direction which may not agree with its previous alignment, so its new direction is determined probabilistically. The analogy with the two-state spin system will hold, after all.
It is important to re-emphasize that the position-momentum uncertainty relation is of miniscule magnitude of order 10-34. No combination of position-momentum measurements even approach this scale. If we could, we would find there is an uncertainty in position or time because of the interdependence of the two. This minute uncertainty has virtually nothing to do with electron orbital structure. The scatter plot distributions of electrons bound to atoms are simply collections of many measurements: at each point the electron is found to have a well defined position and momentum. The electron is not smeared throughout the orbital, only to conveniently collapse into a nice point when it is measured. Besides, at what point would it "collapse"? When its spatial wavefunction is "touched" by the probing particle? The wavefunction extends to macroscopic distances! The dimensions of the orbital are of the order of 10-10, much bigger (even when combined with error in the momentum) than any constraints required by the uncertainty principle.
At the risk of sounding repetitive, the wavefunction simply gives the probability of the particle appearing at some point at a later time, given an initial state. This is true for the free particle, the hydrogen atom, and more exotic examples. When the wavefunction is represented without time dependence, what is really meant is what the probability distribution would be like an extremely long time after a measurement. In free space, this would be an ever-spreading wave packet. The particle is not smearing out over space, but only its probability distribution is spreading. In a bound atom, depending on the angular momenta, the electron's probability distribution will be defined by well-shaped orbitals. This seems much like classical probability, in that the particle doesn't smear, but only our knowledge of the system does. However, it is important to point out that this does not imply a classical deterministic trajectory. The trajectory can be non-deterministic, yet still continuous. The particle does not smear or blink out of existence between measurements, making a mockery of the empiricism we claim to uphold.
The nature of particle trajectories merits serious discussion, since it leads us into the business of quantum leaps, tunneling and other popular science fare. Again, let us begin with what the experimentally verified mathematical theory actually says, and then derive an interpretation consistent with its logical foundation. Schrödinger's equation is a differential equation with a continuous time variable. It holds for all time, during and between measurements. The equation tells us the probability of finding the system in a particular state at some later time, given an initial condition. The probabilistic nature of the wavefunction comes from causal indeterminacy or randomness. It is not a refutation of the notion that a particle does in fact occupy an eigenstate at every point in time. Making discrete transitions in energy levels does not imply instantaneous motion or blinking on the part of the particle. What changes is the particle's propensity for being in a certain location, so its subsequent behavior will be modeled by a new probability distribution, the wavefunction corresponding to that energy level. Quantum mechanics may not circumvent the limiting velocity of light by instantaneously teleporting particles to different locations. Consistent treatment of Schrödinger's equation as an expression of conditional probability precludes this.
Barrier penetration, while certainly a non-classical result due to the probabilistic element, provides no immediate philosophical difficulty. Giving a satisfying explanation of quantum mechanics does not mean eliminating anything non-intuitive, but neither must we fall into the trap of mistaking the absurd for the profound. Aristotelian logic might seem gauche to some, but it is the basis for the linear algebraic formalism upon which quantum mechanics rests. Similarly, the experimental evidence for the theory rests upon a foundation of empiricism. The abandonment of these principles in the explication of quantum theory would be foolishly self-contradictory. Empirically, we can only observe eigenstates, so we only have reason to believe in the physical existence of eigenstates. Logically, a system cannot physically occupy two mutually exclusive states. The idea of penetrating a finite potential barrier does not contest these principles.
In the case of quantum tunneling, where moving from one potential well to another appears to involve transit where the kinetic energy is negative, confusion arises. Some texts and teachers maintain that the particle instantaneously jumps from one well to another, yet experimental observation has shown that tunneling electrons can be observed in transit, with positive kinetic energy. Texts which recognize this fact generally attribute the positive kinetic energy to the act of measurement. In the act of measurement, of course, there is the usual Copenhagen ambiguity: the interaction of the physical apparatus versus the mere act of knowing. We should address this energy problem, but first let us note that here there is no issue that contradicts the idea of particles having a well-defined, though non- deterministic, trajectory. The issue that lies before us is whether the kinetic energy is well-defined at all times.
It is asking far too much to expect that an apparatus would always affect a particle in just such a way to give it a positive kinetic energy. Empiricism would tell us that particles in the "classically forbidden" region always have positive kinetic energy, since they have always been observed as such. The idea that they suddenly acquire this quality as the result of observation, while technically irrefutable, has no basis in experimental observation, nor in mathematical theory.
The idea that the "collapse of the wavefunction", or observation, confines the particle, thereby increasing the spread in its momentum to yield a positive kinetic energy, is a physical misinterpretation of a mathematical fact. Detecting a particle at a certain position imposes a condition on the probability distribution, or "collapse of the wavefunction". There is necessarily an ambiguity in the position- momentum due to the mathematical interdependence of the two variables. This ambiguity persists regardless of whether the particle is being observed.
We could plot a momentum wavefunction of a particle, and it would show the momentum to be everywhere non-negative. Thus the kinetic energy of the particle is always non-negative. Quantum tunneling implies a non-zero probability of having a positive kinetic energy in an area where it would be classically impossible. That the total energy eigenvalue is less than the nominal potential would be a contradiction classically, but we know that the classical potential is a rather arbitrary and illusory concept (with no meaning, for example, at relativistic energies), so we should, in retrospect, perhaps not be too surprised that only infinite potential barriers are impenetrable.
We may similarly examine the time-energy uncertainty relation, often regarded as "non-canonical" because it does not result directly from the non-commutativity of operators, since time is merely a parameter in quantum mechanics. Instead, it is a direct consequence of the form of the time evolution operator: eiEt, which results in an uncertainty relation of the form ΔEΔt > ħ. Here, the uncertainty in E refers to the standard deviation of a statistical distribution of identically prepared systems. The only energies that can be measured are, as always, the eigenvalues. Δt is the characteristic time which defines the frequency with which the system changes eigenstates, on average. The relation is purely mathematical and does not imply a fundamental ambiguity between energy and time analogous to position-momentum. Thus it is inaccurate to make statements such as "the more accurately you know the time, the less accurately you know the energy". Quantum states are probabilistic distributions of eigenstates; the only measurable quantities are eigenvalues.
I repeat that the eigenstates are the only physical states as if it were a mantra, for it is on this point that Copenhagen theorists make their fatal blunder and stumble into self-contradiction. Dirac, for example, in his discussion of position-momentum uncertainty, astonishingly maintains that a system may never be in an eigenstate of position, nor of momentum. We surely acknowledge that they cannot be simultaneous eigenstates, but Dirac makes the additional unfounded claim that the system can not be in an eigenstate of one or the other observable. On what basis is this claim made? The non-commutativity of x-spin and y-spin does not contradict the principle that we can always measure the x-spin to be in an eigenstate or the y-spin similarly, just not both at the same time. So the non-commutativity of the operators can not be invoked as a justification for Dirac's interpretation. The only other consideration would be an inexplicable abandonment of the full equivalence between continuous and discrete systems, made possible only by the fact that we can not, of course, measure anything with infinite precision, so we are unable to test whether or not systems occur in eigenstates. Yet we would have a gross incongruity, interpreting the continuous case as never being in an eigenstate, while the discrete system is always in an eigenstate, even though they are fully mathematically equivalent.
Moreover, if an individual particle does exist in a "superposed" state, what is its value? We would then delve into philosophical absurdity, saying it simultaneously has different values with different probabilities, feebly attempting to reconcile blatant logical contradiction with the defense of the mathematical formalism. Indeed, there is no flaw in the formalism, only in its interpreters. We might further note that the only difference between the discrete and continuous case is often the mere presence or absence of a potential. We have seen earlier that the spread in an atomic wavefunction is not a measure of the actual uncertainty in an individual position measurement; there is no reason to assume that such is the case for a free particle.
Physics suffers from a self-defeating contempt of philosophy, mathematics, metaphysics, and all things abstract and ethereal, despite its utter dependence upon them. As such, physicists are wont to take an all-or-none approach to the interpretation of mathematical formalisms: either something is non-physical and therefore has no physical implications, or it has physical implications and is therefore physical. The quantum wavefunction is a non-physical object with real physical implications, so physicists are totally at a loss in coping with it, either choosing one of the two extreme positions, or more commonly, oscillating between the two positions depending on context. The specialness of the quantum wavefunction is that it forces upon physics its underlying dependence upon metaphysics and mathematics in a very real way.
In the case of classical probability, we have seen that outcome is dependent not upon ignorance or knowledge, but the actual distribution of initial conditions. Determinism is simply the special case where a certain initial condition produces the same result, or eigenstate, with probability one. We might say that all eigenstates are stationary in a deterministic system. If we understand probability more generally, as a distribution of initial states leading to a distribution of final states, without assuming anything else, we have a framework in which all probabilistic phenomena can be explained with full equivalence. The distinction between quantum and classical probability is a shibboleth invented by twentieth-century physicists who, having assumed strong determinism as essential fact, felt certain that only a fundamentally new concept could overturn their view of the world. In fact, they might have altered that view before the advent of quantum mechanics, had they understood classical probability properly instead of regarding it as a mere computational device. We see, rather, that all probability distributions are a measure of an actual propensity of the system to reach a final state given a distribution of initial states.
The physicist's inability to treat metaphysical phenomena consistently is further illustrated by the concept of virtual particles. These are also mathematical constructs with real physical implications. The physicist who takes an "all-or-none" approach is forced to indulge in anti-empirical sophistries such as the notion that the virtual particle conveniently blinks in and out of existence before it can be measured. Of course, the formalism dictates that it can not be measured in principle, since they are not eigenstates of the full Hamiltonian; the formalism cares not for the nature of the measuring apparatus. The supposed proof that these objects have real physical existence is that the endproducts of interactions are accurately predicted by models postulating their intercession. Indeed, this is no accident, for the mathematical theory is sound, but the philosophy is mediocre; it assumes that anything with physical implications must itself be physical. As for the blinking, this is defended by the fact that the characteristic time of the transition is extremely small, so there is not enough time to observe the virtual particle. This is a terrible misunderstanding of the characteristic time, which is a statistical quantity that measures the expected amount of time, on average, a system will remain in a certain state before changing to another state. The change itself is instantaneous, as it merely means adopting a new wavefunction, and behaving accordingly thenceforth. The characteristic time is something which can be measured only by the consideration of many identically prepared systems; it is not a contradiction of the idea that eigenstate transitions are instantaneous, no more than eigenstate transitions require "quantum leaps", as discussed earlier.
As a final clarification of the principles outlined, let us consider Young's double-slit experiment. We begin with a single photon, a corpuscular entity associated with an electromagnetic field. This field propagates in a wavelike manner, but this is only an incidental similarity with the wavefunction of the photon. We could just as well use an electron to obtain similar results. The wavefunction is a function in probability (more accurately, Hilbert) space which gives the likelihood of various outcomes given specified initial conditions. First, let us specify that the photon has a specific position at a given time. This necessarily implies ambiguity in the momentum, but that is only because position and momentum are different aspects of the same quantity. One cannot specify momentum at an instant since momentum requires the passage of time and space for it to be defined. The Δx of the uncertainty principle is a statistical spread obtained from measurements of many identically prepared systems. As the photon passes through a single slit, the uncertainty of its final destination on a photographic plate (placed at some large distance from the slit) is much greater than the localized mark that the individual photon will leave. The Δx of the photon after passing the slit is not indicative of a smearing of the photon; there is no reason to presume such is the case for the free photon, simply because the uncertainty is too small to be experimentally confirmed.
So we repeat our preliminary single-slit experiment with many identically-prepared photons, and achieve a statistical distribution on the photographic plate, from which the wavefunction might be inferred or its computation confirmed. Each individual photon behaves in a corpuscular manner, albeit in a probabilistic way which is different from the billiard-ball determinism of classical mechanics. No matter where we place the plate, before or after the slit, each photon leaves a local imprint. It is an utterly unscientific and anti-empirical position to maintain that the photon conveniently "delocalizes" when it is not being measured. It is akin to saying there is a fairy on your shoulder which disappears every time you turn to look at it; the proof of the fairy being the pixie dust remaining on your shoulder. One may indeed be compelled to believe in the pixie dust, but the existence of fairies has by no means been established.
The inability of many physicists to philosophically grasp the double-slit experiment forces them to abandon the empiricism they normally extol. When there are two slits, the resulting distribution resembles that of interference fringes in electromagnetism. We should note the important distinction that, in the case of interference fringes, we are measuring the amplitudes of actual electromagnetic fields that constructively or destructively interfere with each other, but in Young's experiment, we are measuring a probability amplitude through a statistical ensemble of photon intensities. Rather than physical electromagnetic fields interfering with each other, Young's experiment shows a mathematical "interference" of wavefunctions. What disturbs the physicist is that this interference pattern does not come from the mere summation, or overlap, of two single-slit patterns. If the photon were passing through one slit or the other, it is difficult to see how its behavior should be affected by the presence or absence of another slit some distance away.
Here it is necessary to reiterate that a wavefunction, like any other probability function, is meaningful only if initial conditions are specified. For the double slit experiment, we have a random distribution of photons, most of which will not pass through either slit, some of which will pass through either of the slits. The wavefunction is not some ghostly companion of the photon, but a function of the experiment being considered; failure to specify conditions can change the function. So the configuration of the slits is relevant to the computation of the wavefunction of the experiment. No individual photon physically interferes with any other photon, for the experiment can be conducted one photon at a time. Each photon passes through one or the other slit, but its probability of final destination is a function of the system as a whole, not just the part it, the individual photon, happens to interact with. The same is true of a ball falling down a pinboard; if a pin is removed, the ball's probability distribution instantly changes, even if it does not pass through the area of the missing pin. The change in distribution will be measured through repetition, as some balls will be affected by the missing pin. Yet we are still left with an apparent discrepancy: why isn't the double-slit wavefunction the mere mathematical summation of two single-slit wavefunctions?
Recall that for the double-slit experiment, we may only specify at best some narrow range of initial positions/momenta for the photons. If we made them, hypothetically, nearly exactly the same (an experimental impossibility), they could only pass through one or the other slit, and we would essentially have a single slit experiment. As it is arranged, the total distribution of incident photons will be some cone of momenta approaching the region of interest; a distribution which may be treated mathematically as a wave. That Schrödinger's equation is mathematically similar to the wave equation is at once a pleasant and lamentable discovery. Its virtue of mathematical elegance is somewhat mitigated by the fact that it is liable to confusion with the electromagnetic wave or a tangible mechanical wave. It is the mathematics of the Schrödinger wavefunction distribution which leads to the interference-style fringes. The ergodic principle of probability makes it irrelevant whether this experiment is done with many photons or one photon at a time. Before we conclude, it should be pointed out that the similarity between Schrödinger's equation and the wave equation need not be entirely accidental; in fact the latter may be a specific application of the former. Nonetheless, they are distinct concepts; one exists in Hilbert space, the other in real space.
It is easy to see how one might fall astray in interpreting the Young experiment, for it is tempting to regard the infinite dimensional Hilbert space of position eigenvectors as simply real space. Obviously, there is a correlation between the spaces, but the two are not identical. Errors such as these invariably force physicists to adopt a somewhat subjectivist view of physical reality, a view that is in sharp contrast with all other results of experimental science. Schrödinger pointed out this absurdity with his cat paradox: at what point does the cat cease to be both alive and dead? That which is logically absurd at the macroscopic level must also be so at the microscopic level, regardless of whatever other differences may abound. Unfortunately, physicists are loath to regard philosophical problems as serious considerations, so it has been necessary to point out their inconsistencies.
Most prominent among these has been the failure to physically interpret the continuous position/momentum wavefunction in the same way as the discrete case with respect to eigenstates. Mathematically, the discrete and continuous systems can be shown to be fully equivalent, and they are both functions whose magnitude squared gives a probability of occupying a certain eigenstate. Both discrete and continuous systems have uncertainty principles, but it is only in the latter case where the argument is advanced against ever being measured in an eigenstate; in the former, we can only measure eigenstates. While this incongruity is permitted, strong identification is made between things which are not identical: the wavefunction and a physical wave. This equation is drawn because of a more serious mental obstacle prevalent among physicists: the notion that anything which affects experimental results must be physical. The stubborn refusal to abandon this position, which is not demanded by logic, forces the adoption of completely absurd interpretations which are in contradiction with logic. A mental alarm should sound immediately when absurd results are derived from a theory whose basis is linear matrix algebra, a most ordinary branch of mathematics. The claim that the vindication of the mathematical theory is a vindication of the Copenhagen interpretation is most soundly refuted by the wide disparity of interpretations among supposed Copenhagen theorists themselves. Part of the reason for the scheme's continued popularity, aside from a general disinterest in philosophical matters, is that most attacks on the interpretation have come from the standpoint of strong determinism. More cogent interpretations, such as that advanced by Popper, arise only when we understand the probabilistic nature of the wavefunction, and apply it consistently in all experiments.
The Copenhagen interpretation forces one to accept one "paradox" after another, many of them different in quality. A more proper understanding would begin with a recognition of the interdependence of physical and metaphysical concepts, from which all other things may be understood. A similar situation arises in relativity: once one accepts the invariance of the speed of light, all else follows logically. Instead, quantum mechanics is replete with ad hoc interpretations which follow no general scheme: either the act of measurement perturbs the system, or reality is intrinsically ill- defined, or the question has no meaning. Countless physics students have muddled along, supposedly learning to "think quantum mechanically", while in fact accepting the least offensive absurdity, and applying the formalism mechanistically to yield the correct results. Richard Feynman at least was honest enough to admit, "No one understands quantum mechanics." However, this does not mean that it can not be made understandable. It is my hope that a comprehensible basis for interpretation has here been outlined. Most with the expertise to fully appreciate what has been discussed will also be burdened with preconceived notions of superpositions and the like which will be difficult to overcome. However, it should be evident that a subjectivist interpretation of reality is not a consequence of quantum mechanical formalism, much less can it claim to be an experimentally verified fact.
Physics has proceeded admirably in spite of its poor qualitative interpretations; the real danger in Copenhagen sophistry is its application to other fields of thought. Innumerable books and articles have made reference to quantum theory as evidence of the fact that reality is subjective, or that the act of measurement changes physical reality, where measurement is often considered to mean sentient perception. The Copenhagen theorists have sown fertile fields for solipsists and nihilists of all varieties, as well as for their more temperate agnostic brethren. Lately, there have been efforts to undo the damage within the physics community, but the misinterpretation of quantum mechanics by the outside world is likely to persist much longer. Perhaps, nonetheless, it is reasonable to hope that someday the subjectivist quantum quagmire will be perceived as an historical curiosity, much as we may view those philosophically innocent mechanists of the nineteenth century who thought they had dealt a death blow to theism with the law of conservation of energy.
© 2000, 2006 Daniel J. Castellano. All rights reserved. http://www.arcaneknowledge.org
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