In the following, I want to consider both photons in a sharply focussed, monochromatic beam of light (''type P'') and electrons in an electron beam (''type E'') on the same footing. In the following, X is either P or E. If we only concentrate on the internal degrees of freedom, both kinds of particles can be described as a 2-level system, with a 2-dimensional Hilbert space.(adsbygoogle = window.adsbygoogle || []).push({});

Consider a stationary thermal source that emits particles of type X in a sharply focussed, monochromatic beam. Due to the thermal nature of the source, one has to use statistical mechanics to model its statistical behavior. Thus a ##2\times 2## density matrix ##\rho## gives a complete account of the statistics of the ensemble of particles emitted by the source. The density matrix is Hermitian and positive semidefinite of trace 1, such that ##\langle A\rangle=\mbox{tr} \rho A## gives the mean response of a particle received in a detector measuring ##A##. In our particular case, the density matrix is given by ##\rho=\pmatrix{0.5 & 0\cr 0 &0.5}## since the thermal setting implies that no transversal direction is preferred. We say that the beam is unpolarized. (For the electron, see spin polarization.)

By passing the beam through various equipment (filters) we may prepare modified states. I am considering only perfect filters that project to a 1-dimensional subspace in direction of a normalized state vector ##\phi##, which is adjustable by the experimenter. Such a filter can be realized for photons (type X=P) by a polarizer, and for electrons (X=E) by deflecting the beam by a magnet and passing the two resulting daughter beams through a slit that absorbs the electrons in one of the daughter beams.

If we denote by ##P=\phi\phi^*## the associated orthogonal projector, the effect on the filtering is that particles pass the filter with a probability of ##\langle P\rangle=\mbox{tr} \rho P =\phi^*\rho\phi##, and the density matrix of the ensemble of particles that passed the beam is ##P=\phi\phi^*##. This corresponds to a pure state, and implies that the modified beam is completely polarized, with polarization (if X=P) resp. spin direction (if X=E) determined by the filter setting.

The above is theshut-up-and-calculate descriptionof the experiment. It only refers to macroscopic objects and to properties that can be directly checked by statistical measurements. The latter consist of counting sufficiently many detection events of a particle counter placed after the filter, using enough filter vectors ##\phi## to be able to determine the density matrix of an arbitrary beam.

I'd like to know what some interpretations of quantum mechanics assert about states and observables of individual particlesand about which properties they are silent,

In particular, I invite the currently most outspoken proponents on PF of three interpretations (

- (a) direcly after emission,

- (b) directly after having passed the filter,

- (c) directly after having been measured.
minimal ensemble, @vanhees71;Copenhagen, @atyy;Bohmian, @Demystifier) to state their views about individual particles regarding the above experimental setting.

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# I Particles from a thermal source

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