#### PeterDonis

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The experiment is the Stern-Gerlach experiment. For concreteness, I will specify that we are considering a beam consisting of a large number of electrons all prepared with spin-z up. The beam then passes through a Stern-Gerlach device oriented in the x direction. Experimentally we know that the beam splits in two: there is a left beam and a right beam (corresponding to the two possible spin-x eigenstates), with a low intensity region in the middle.

Part III of the series of papers by @A. Neumaier on the thermal interpretation deals with measurement:

On p. 4 of this paper, it is stated:

Applying this to the Stern-Gerlach experiment, we would view the observed result as an uncertain measurement of the q-expectation of the spin-x operator as applied to the electrons in the beam. This q-expectation is easily seen to be zero. For this case we can use the usual formalism of state vectors and matrix operators: in the spin-z basis, the state of the beam after preparation is the vector ##\psi = \begin{bmatrix} 1 \\ 0 \end{bmatrix}##, the spin-x operator ##\sigma_x = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}## flips the state to ##\begin{bmatrix} 0 \\ 1 \end{bmatrix}##, and the expectation ##\langle \psi \vert \sigma_x \vert \psi \rangle## is therefore the inner product of ##\psi = \begin{bmatrix} 1 \\ 0 \end{bmatrix}## and ##\begin{bmatrix} 0 \\ 1 \end{bmatrix}##, which is ##0##.[T]he measurement of a Hermitian quantity ##A## is regarded as giving an uncertain value approximating the q-expectation ##\langle A \rangle## rather than (as tradition wanted to have it) as an exact revelation of an eigenvalue of ##A##.

So the thermal interpretation is telling us to interpret the experimental result, of a left beam and a right beam with low intensity in between, as an uncertain measurement of the q-expectation of ##0##, which would correspond to a point in the center. But this does not seem right: it seems like an uncertain measurement of a quantity whose q-expectation is ##0## would be a normal distribution about the value ##0##, i.e., an intensity

*peak*in the center and decreasing to the left and right. (This is, of course, the classical EM prediction which was famously falsified by Stern and Gerlach.) So my question is, in the light of this seeming inconsistency, how does the thermal interpretation explain the Stern-Gerlach experiment?