Abstract
We have seen in the previous chapter that the analysis of von Neumann has little impact on the question of whether a viable hidden variables theory may be constructed.
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Notes
- 1.
- 2.
See Albert [7].
- 3.
The original form presented by A.M. Gleason referred to a probability measure on the subspaces of a Hilbert space, but the equivalence of such a construction with a value map on the projection operators is simple and immediate. This may be seen by considering that there is a one-to-one correspondence between the subspaces and projections of a Hilbert space and that the values taken by the projections are 1 and 0, so that a function mapping projections to their eigenvalues is a special case of a probability measure on these operators.
- 4.
See Bell [5]. Bell proves that any function \(E(P)\) satisfying the conditions of Gleason’s theorem cannot map the projection operators to their eigenvalues.
- 5.
The set of projections on a three-dimensional space is actually a larger class of observables.
- 6.
We follow here the argument given in Belinfante [8, p. 38]
- 7.
- 8.
See [11].
- 9.
As we have mentioned, since the observables are formally equivalent to projections on a three-dimensional Hilbert space, this theorem is actually a special case of Gleason’s. Therefore, Bell’s argument essentially addresses the Kochen and Specker theorem as well as Gleason’s.
- 10.
One can derive the analogous first-order perturbation term arising for a charged particle of orbital angular momentum \(L=1\) in such an electric field using the fact that the joint-eigenstates of \(L^{2}_{x},L^{2}_{y},L^{2}_{y}\) are the eigenstates of the potential energy due to the field. This latter result is shown in Kittel [15, p. 427].
- 11.
Orthohelium and parahelium are two species of helium which are distinguished by the total spin S of the two electrons: for the former we have \(S=1,\) and for the latter \(S=0.\) There is a rule of atomic spectroscopy which prohibits atomic transitions for which \(\Updelta S=1,\) so that no transitions from one form to the other can occur spontaneously.
- 12.
Using spectroscopic notation, this state would be written as the ‘\(2^{3}\)S’ state of orthohelium. The ‘2’ refers to the fact that the principal quantum number n of the state equals 2, ‘S’ denotes that the total orbital angular momentum is zero, and the ‘3’ superscript means that it is a spin triplet state. Orthohelium has no state of principal quantum number \(n=1,\) since the Pauli exclusion principle forbids the ‘\(1^{3}\)S’ state.
- 13.
Orthorhombic symmetry is defined by the criterion that rotation about either the x or y axis by \(180^{\circ}\) would bring such a field back to itself.
- 14.
A straightforward way to see this is by analogy with a charged particle of orbital angular momentum \(L=1.\) The effects of an electric or magnetic field on a charged particle of spin 1 are analogous to the effects of the same field on a charged particle of orbital angular momentum 1. To calculate the first-order effects of an electric field of orthorhombic symmetry for such a particle, one can examine the spatial dependence of the \(L_{z}=1,0,-1\) states \(\psi_{-1},\psi_{0},\psi_{+1},\) together with the spatial dependence of the perturbation potential \(V({\mathbf r}),\) to show that the states \(1/\sqrt(\psi_{1}-\psi_{-1}),\) \(1/\sqrt{2}(\psi_{1}+ \psi_{-1}),\) and \(\psi_{0}\) are the eigenstates of such a perturbation. A convenient choice of V for this purpose is \(V=Ax^{2}+By^{2}+Cz^{2}.\) See Kittel in [15, p. 427].
- 15.
- 16.
- 17.
As is usual in discussions of Stern–Gerlach experiments, we consider only those effects relating to the interaction of the magnetic field with the magnetic moment of the particle. We consider the electric charge of the particle to be zero.
- 18.
The term added to the particle’s Hamiltonian to account for a magnetic field is \(g{\mathbf s} \cdot {\mathbf B},\) where \({\mathbf s}\) is the spin, \({\mathbf B}\) is the magnetic field and g is the gyromagnetic ratio. To determine the form of this term in the case of a Stern–Gerlach apparatus, we require the configuration of the magnetic field. A Stern–Gerlach magnet apparatus has a “long axis” which for the example of Fig. 2.1 lies along the x-axis. Since the component of the magnetic field along this axis will vanish except within a small region before and after the apparatus, the effects of \(B_{x}\) may be neglected. Furthermore, \(B_{y}\) and \(B_{z}\) within the apparatus may be regarded as being independent of x. The magnetic field in the x, z plane between the magnets lies in the z-direction, i.e., \({\mathbf B}(x,0,z)=B_{z}(z)\hat{k}.\) Over the region of incidence of the particle, the field is such that \({\frac{\partial B_{z}}{\partial z}}\) is constant. See for example, Weidner and Sells [19] for a more detailed discussion of the Stern–Gerlach apparatus. The motion of the particle in the y-direction is of no importance to us, and so we do not consider the effects of any Hamiltonian terms involving only y dependence. The results we discuss in the present section are those which arise from taking account of the magnetic field by adding to the Hamiltonian term of the form \(g\sigma_{z}B_{z}(z=0)+g\sigma_{z}\left({\frac{\partial B_{z}}{\partial z}}(z=0)\right)z.\)
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Hemmick, D.L., Shakur, A.M. (2012). Contextuality. In: Bell's Theorem and Quantum Realism. SpringerBriefs in Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-23468-2_2
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