Measurement in quantum mechanics

The framework of quantum mechanics requires a careful definition of measurement. The issue of measurement lies at the heart of the problem of the interpretation of quantum mechanics, for which there is currently no consensus. The question of how the operational process measurement affects the ontological state of the observed system is unresolved, and called the measurement problem.A measurement always causes the system to jump into an eigenstate of the dynamical variable that is being measured, the eigenvalue this eigenstate belongs to being equal to the result of the measurement. The framework of quantum mechanics requires a careful definition of measurement. The issue of measurement lies at the heart of the problem of the interpretation of quantum mechanics, for which there is currently no consensus. The question of how the operational process measurement affects the ontological state of the observed system is unresolved, and called the measurement problem. Measurement plays an important role in quantum mechanics, and it is viewed in different ways among various interpretations of quantum mechanics. In spite of considerable philosophical differences, different views of measurement almost universally agree on the practical question of what results from a routine quantum-physics laboratory measurement. To understand this, the Copenhagen interpretation, which has been commonly used, is employed in this article. In classical mechanics, a simple system consisting of only one single particle is fully described by the position x → ( t ) {displaystyle {vec {x}}(t)} and momentum p → ( t ) {displaystyle {vec {p}}(t)} of the particle. As an analogue, in quantum mechanics a system is described by its quantum state or wave function, which contains the probabilities of possible positions and momenta. In mathematical language, all possible pure states of a system form an abstract vector space called Hilbert space, which is typically infinite-dimensional. A pure state is represented by a state vector in the Hilbert space. Once a quantum system has been prepared in laboratory, some measurable quantity such as position or energy is measured. For pedagogic reasons, the measurement is usually assumed to be ideally accurate. The state of a system after measurement is assumed to 'collapse' into an eigenstate of the operator corresponding to the measurement. Repeating the same measurement without any evolution of the quantum state will lead to the same result. If the preparation is repeated, subsequent measurements will likely lead to different results. The predicted values of the measurement are described by a probability distribution, or an 'average' (or 'expectation') of the measurement operator based on the quantum state of the prepared system. The probability distribution is either continuous (such as position and momentum) or discrete (such as spin), depending on the quantity being measured. Average values are important. If the rule for calculating the average is known, also the rule for calculating the probabilities. The measurement process is often considered as random and indeterministic. Nonetheless, there is considerable dispute over this issue. In some interpretations of quantum mechanics, the result merely appears random and indeterministic , whereas in other interpretations the indeterminism is core and irreducible. A significant element in this disagreement is the issue of 'collapse of the wave function' associated with the change in state following measurement. There are many philosophical issues and stances (and some mathematical variations) taken—and near universal agreement that we do not yet fully understand quantum reality. In any case, our descriptions of dynamics involve probabilities, not certainties. The mathematical relationship between the quantum state and the probability distribution is, again, widely accepted among physicists, and has been experimentally confirmed countless times. This section summarizes this relationship, which is stated in terms of the mathematical formulation of quantum mechanics. It is a postulate of quantum mechanics that all measurements have an associated operator (called an observable operator, or just an observable), with the following properties: