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Quantum indeterminacy
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== Measurement == An adequate account of quantum indeterminacy requires a theory of measurement. Many theories have been proposed since the beginning of [[quantum mechanics]] and [[quantum measurement]] continues to be an active research area in both theoretical and experimental physics.<ref>V. Braginski and F. Khalili, ''Quantum Measurements'', Cambridge University Press, 1992.</ref> Possibly the first systematic attempt at a mathematical theory was developed by [[John von Neumann]]. The kinds of measurements he investigated are now called projective measurements. That theory was based in turn on the theory of [[projection-valued measure]]s for [[self-adjoint operator]]s that had been recently developed (by von Neumann and independently by [[Marshall Stone]]) and the [[mathematical formulation of quantum mechanics|Hilbert space formulation of quantum mechanics]] (attributed by von Neumann to [[Paul Dirac]]). In this formulation, the state of a physical system corresponds to a [[Vector (geometry)|vector]] of length 1 in a [[Hilbert space]] ''H'' over the [[complex number]]s. An observable is represented by a self-adjoint (i.e. [[Hermitian operator|Hermitian]]) operator ''A'' on ''H''. If ''H'' is finite [[Vector space dimension|dimensional]], by the [[spectral theorem]], ''A'' has an [[orthonormal basis]] of [[eigenvector]]s. If the system is in state ''ψ'', then immediately after measurement the system will occupy a state that is an eigenvector ''e'' of ''A'' and the observed value ''λ'' will be the corresponding eigenvalue of the equation {{nowrap|1=''Ae'' = ''λe''}}. It is immediate from this that measurement in general will be non-deterministic. Quantum mechanics, moreover, gives a recipe for computing a probability distribution Pr on the possible outcomes given the initial system state is ''ψ''. The probability is <math display="block"> \operatorname{Pr}(\lambda)= \langle \operatorname{E}(\lambda) \psi \mid \psi \rangle </math> where ''E''(''λ'') is the projection onto the space of eigenvectors of ''A'' with eigenvalue ''λ''. === Example === [[Image:PauliSpinStateSpace.png|frame|right|[[Bloch sphere]] showing eigenvectors for Pauli Spin matrices. The Bloch sphere is a two-dimensional surface the points of which correspond to the state space of a spin 1/2 particle. At the state ''ψ'' the values of ''σ''<sub>1</sub> are +1 whereas the values of ''σ''<sub>2</sub> and ''σ''<sub>3</sub> take the values +1, −1 with probability 1/2.]] In this example, we consider a single [[Spin-1/2|spin 1/2]] [[Elementary particle|particle]] (such as an electron) in which we only consider the spin degree of freedom. The corresponding Hilbert space is the two-dimensional complex Hilbert space '''C'''<sup>2</sup>, with each quantum state corresponding to a unit vector in '''C'''<sup>2</sup> (unique up to phase). In this case, the state space can be geometrically represented as the surface of a sphere, as shown in the figure on the right. The [[Pauli matrix|Pauli spin matrices]] <math display="block"> \sigma_1 = \begin{pmatrix} 0&1\\ 1&0 \end{pmatrix}, \quad \sigma_2 = \begin{pmatrix} 0&-i\\ i&0 \end{pmatrix}, \quad \sigma_3 = \begin{pmatrix} 1&0\\ 0&-1 \end{pmatrix} </math> are [[self-adjoint]] and correspond to spin-measurements along the 3 coordinate axes. The Pauli matrices all have the eigenvalues +1, −1. * For ''σ''<sub>1</sub>, these eigenvalues correspond to the eigenvectors <math display="block"> \frac{1}{\sqrt{2}} (1,1), \frac{1}{\sqrt{2}} (1,-1) </math> * For ''σ''<sub>3</sub>, they correspond to the eigenvectors <math display="block"> (1, 0), (0,1) </math> Thus in the state <math display="block"> \psi = \frac{1}{\sqrt{2}} (1,1), </math> ''σ''<sub>1</sub> has the determinate value +1, while measurement of ''σ''<sub>3</sub> can produce either +1, −1 each with probability 1/2. In fact, there is no state in which measurement of both ''σ''<sub>1</sub> and ''σ''<sub>3</sub> have determinate values. There are various questions that can be asked about the above indeterminacy assertion. # Can the apparent indeterminacy be construed as in fact deterministic, but dependent upon quantities not modeled in the current theory, which would therefore be incomplete? More precisely, are there ''hidden variables'' that could account for the statistical indeterminacy in a completely classical way? # Can the indeterminacy be understood as a disturbance of the system being measured? Von Neumann formulated the question 1) and provided an argument why the answer had to be no, ''if'' one accepted the formalism he was proposing. However, according to Bell, von Neumann's formal proof did not justify his informal conclusion.<ref>J.S. Bell, ''Speakable and Unspeakable in Quantum Mechanics'', Cambridge University Press, 2004, pg. 5.</ref> A definitive but partial negative answer to 1) has been established by experiment: because [[Bell's inequalities]] are violated, any such hidden variable(s) cannot be ''local'' (see [[Bell test experiments]]). The answer to 2) depends on how disturbance is understood, particularly since measurement entails disturbance (however note that this is the [[Observer effect (physics)|observer effect]], which is distinct from the uncertainty principle). Still, in the most natural interpretation the answer is also no. To see this, consider two sequences of measurements: (A) that measures exclusively ''σ''<sub>1</sub> and (B) that measures only ''σ''<sub>3</sub> of a spin system in the state ''ψ''. The measurement outcomes of (A) are all +1, while the statistical distribution of the measurements (B) is still divided between +1, −1 with equal probability. === Other examples of indeterminacy === Quantum indeterminacy can also be illustrated in terms of a particle with a definitely measured momentum for which there must be a fundamental limit to how precisely its location can be specified. This quantum [[uncertainty principle]] can be expressed in terms of other variables, for example, a particle with a definitely measured energy has a fundamental limit to how precisely one can specify how long it will have that energy. The magnitude involved in quantum uncertainty is on the order of the [[Planck constant]] ({{physconst|h}}).
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