An interpretation of quantum mechanics that rejects hidden variables has to say something about the way measurement can be understood as a transformation on states of individual systems, and that leads to the core of the interpretive problems posed by Luders' projection rule: What, if any, is its physical content? In this paper I explore one suggestion which is implicit in usual interpretations of the rule and show that this view does not stand on solid ground. In the process, important aspects of the role played by the projection postulate in the conceptual structure of quantum mechanics will be clarified. It will be shown in particular that serious objections can be raised against the (often implicit) view that identifies the physical relation of compatibility preserved by Luders' rule with the relation of simultaneous measurability. Preparatory measurements are measurements for which we can con- clude on the basis of the result of measurement that the system is (immediately after measurement) in an eigenstate of the eigenvalue corresponding to the given result. Preparatory measurements are first of all experimental processes that select an ensemble to which a quantum (statistical) state can be assigned. Statistical interpretations of the state vector take as fundamental this assignment of a quantum state to (sub)ensembles selected by experimental procedures. But if the exist- ence of hidden variables is denied, then it seems that it should be possible to interpret preparatory measurements in terms of individual state transformations. This can be done by thinking of these measure- ments as idealized measurement transformations satisfying the so-called first kind condition: an immediate repetition of the same measurement gives the same result as the initial measurement. This idealization (taken here for the sake of simplicity) will be assumed here to be non-controversial. Furthermore, it will be assumed here that every measurement can be represented as a first kind measurement: thus, a 'measurement' in this investigation is always to be a 'first kind measure- ment'. For measurements of maximal magnitudes (i.e., magnitudes with no degenerate eigenvalue) Von Neumann argued (1955) that the final state is given by the (unique) eigenstate of the eigenvalue representing the
An interpretation of quantum mechanics that rejects hidden variables has to say something about the way measurement can be understood as a transformation on states of individual systems, and that leads to the core of the interpretive problems posed by Luders' projection rule: What, if any, is its physical content? In this paper I explore one suggestion which is implicit in usual interpretations of the rule and show that this view does not stand on solid ground. In the process, important aspects of the role played by the projection postulate in the conceptual structure of quantum mechanics will be clarified. It will be shown in particular that serious objections can be raised against the (often implicit) view that identifies the physical relation of compatibility preserved by Luders' rule with the relation of simultaneous measurability. Preparatory measurements are measurements for which we can con- clude on the basis of the result of measurement that the system is (immediately after measurement) in an eigenstate of the eigenvalue corresponding to the given result. Preparatory measurements are first of all experimental processes that select an ensemble to which a quantum (statistical) state can be assigned. Statistical interpretations of the state vector take as fundamental this assignment of a quantum state to (sub)ensembles selected by experimental procedures. But if the exist- ence of hidden variables is denied, then it seems that it should be possible to interpret preparatory measurements in terms of individual state transformations. This can be done by thinking of these measure- ments as idealized measurement transformations satisfying the so-called first kind condition: an immediate repetition of the same measurement gives the same result as the initial measurement. This idealization (taken here for the sake of simplicity) will be assumed here to be non-controversial. Furthermore, it will be assumed here that every measurement can be represented as a first kind measurement: thus, a 'measurement' in this investigation is always to be a 'first kind measure- ment'. For measurements of maximal magnitudes (i.e., magnitudes with no degenerate eigenvalue) Von Neumann argued (1955) that the final state is given by the (unique) eigenstate of the eigenvalue representing the
An interpretation of quantum mechanics that rejects hidden variables has to say something about the way measurement can be understood as a transformation on states of individual systems, and that leads to the core of the interpretive problems posed by Luders' projection rule: What, if any, is its physical content? In this paper I explore one suggestion which is implicit in usual interpretations of the rule and show that this view does not stand on solid ground. In the process, important aspects of the role played by the projection postulate in the conceptual structure of quantum mechanics will be clarified. It will be shown in particular that serious objections can be raised against the (often implicit) view that identifies the physical relation of compatibility preserved by Luders' rule with the relation of simultaneous measurability.
In this paper, we shall show how the theory of measurements is to be understood from the point of view of a physical interpretation of the quantum theory in terms of "hidden" variables, developed in a previous paper. We find that in principle, these "hidden" variables determine the precise results of each individual measurement process. In practice, however, in measurements that we now know how to carry out, the observing apparatus disturbs the observed system in an unpredictable and uncontrollable way, so that the uncertainty principle is obtained as a practical limitation on the possible precision of measurements. This limitation is not, however, inherent in the conceptual structure of our interpretation. We shall see, for example, that simultaneous measurements of position and momentum having unlimited precision would in principle be possible if, as suggested in the previous paper, the mathematical formulation of the quantum theory needs to be modified at very short distances in certain ways that are consistent with our interpretation but not with the usual interpretation.We give a simple explanation of the origin of quantum-mechanical correlations of distant objects in the hypothetical experiment of Einstein, Podolsky, and Rosen, which was suggested by these authors as a criticism of the usual interpretation.Finally, we show that von Neumann's proof that quantum theory is not consistent with hidden variables does not apply to our interpretation, because the hidden variables contemplated here depend both on the state of the measuring apparatus and the observed system and therefore go beyond certain of von Neumann's assumptions.In two appendixes, we treat the problem of the electromagnetic field in our interpretation and answer certain additional objections which have arisen in the attempt to give a precise description for an individual system at the quantum level.
The usual interpretation of the quantum theory is self-consistent, but it involves an assumption that cannot be tested experimentally, viz., that the most complete possible specification of an individual system is in terms of a wave function that determines only probable results of actual measurement processes. The only way of investigating the truth of this assumption is by trying to find some other interpretation of the quantum theory in terms of at present "hidden" variables, which in principle determine the precise behavior of an individual system, but which are in practice averaged over in measurements of the types that can now be carried out. In this paper and in a subsequent paper, an interpretation of the quantum theory in terms of just such "hidden" variables is suggested. It is shown that as long as the mathematical theory retains its present general form, this suggested interpretation leads to precisely the same results for all physical processes as does the usual interpretation. Nevertheless, the suggested interpretation provides a broader conceptual framework than the usual interpretation, because it makes possible a precise and continuous description of all processes, even at the quantum level. This broader conceptual framework allows more general mathematical formulations of the theory than those allowed by the usual interpretation. Now, the usual mathematical formulation seems to lead to insoluble difficulties when it is extrapolated into the domain of distances of the order of ${10}^{\ensuremath{-}13}$ cm or less. It is therefore entirely possible that the interpretation suggested here may be needed for the resolution of these difficulties. In any case, the mere possibility of such an interpretation proves that it is not necessary for us to give up a precise, rational, and objective description of individual systems at a quantum level of accuracy.
Some interpretations of quantum mechanics try to describe the probabilistic nature of quantum measurements in a way that allows a system in a pure state to always have precise (if not simultaneously knowable) values for all its observable variables. In what might be called a simple ensemble interpretation, the quantum mechanical state vector is assumed to define a statistical ensemble of identically prepared systems, each of which has precise values for all its observable variables, and the act of measurement is equivalenced to a straightforward sampling of that ensemble. It is shown here, in a manner that should be suitable for a first course in quantum mechanics, that a simple ensemble interpretation is not possible for some quantum systems, and hence is untenable as a general premise. The analysis is essentially a repackaging of the Bell argument against hidden variables, although hidden variables are never invoked here.
Extracting information from a quantum system inevitably requires the performance of a measurement, and it is no surprise that the theory of measurement plays a central role in our subject. The physical nature of the measurement process remains one of the great philosophical problems in the formulation of quantum theory. Fortunately, however, it is sufficient for us to take a pragmatic view by asking what measurements are possible and how the theory describes them, without addressing the physical mechanism of the measurement process. This is the approach we shall adopt. We shall find that it leads us to a powerful and general description of both the probabilities associated with measurement outcomes and the manner in which the observation transforms the quantum state of the measured system. The simplest form of measurement was given a mathematical formulation by von Neumann, and we shall refer to measurements of this type as von Neumann measurements or projective measurements. It is this description of measurements that is usually introduced in elementary quantum theory courses. We start with an observable quantity A represented by a Hermitian operator Â, the eigenvalues of which are the possible results of the measurement of A. The relationship between the operator, its eigenstates {|λnñ}, and its (real) eigenvalues {λn} is expressed by the eigenvalue equation . . . Â |λn_ = λn|λn_. (4.1) . . .
The Statistical Interpretation of quantum theory is formulated for the purpose of providing a sound interpretation using a minimum of assumptions. Several arguments are advanced in favor of considering the quantum state description to apply only to an ensemble of similarily prepared systems, rather than supposing, as is often done, that it exhaustively represents an individual physical system. Most of the problems associated with the quantum theory of measurement are artifacts of the attempt to maintain the latter interpretation. The introduction of hidden variables to determine the outcome of individual events is fully compatible with the statistical predictions of quantum theory. However, a theorem due to Bell seems to require that any such hidden-variable theory which reproduces all of quantum mechanics exactly (i.e., not merely in some limiting case) must possess a rather pathological character with respect to correlated, but spacially separated, systems.