Advanced models for the Stern-Gerlach experiment

[A. Neumaier]

i cannot find much advanced stuff on Stern-Gerlach. The basic modelling of the experiment in QM you were referring to is in itself pretty simple and i do understand it. And I also understand how the resulting superposition of spin up/down states maps to the experimental outcome using the classic QM interpretation/measurement mechanics. [...]
But when it comes to talking about the collapse of the wave function this simplistic handling is at the range of validity. To begin within that framework the measurement is not well defined from a mathematical perspective – or at least I couldn’t find any strict axiom set from which one could derive that the measurement (or rather collapse or equivalent) does or does not happen directly at the SG device (please correct me if I’m wrong).

A detailed discussion of measurement in the Stern-Gerlach experiment can be given in the POVM setting; see this post.

 

[AndyW]

First of all, I understand your frustration with the lack of advanced material on the Stern-Gerlach experiment. It is a fundamental experiment in quantum mechanics, but it tends to be taught at a basic level in introductory courses. However, there is a wealth of literature on the topic if you know where to look.

You mentioned the classic interpretation/measurement mechanics, which is often referred to as the Copenhagen interpretation. This interpretation does provide a good understanding of the experimental outcome in terms of the superposition of spin states. However, as you correctly pointed out, it does not provide a rigorous mathematical framework for understanding the collapse of the wave function during measurement.

In order to address this issue, the POVM (Positive Operator Valued Measure) formalism was developed. This framework allows for a more general description of measurement, including the Stern-Gerlach experiment. In this formalism, the measurement process is described as a series of operations on the quantum state, rather than a collapse of the wave function.

To understand the measurement process in the Stern-Gerlach experiment within the POVM framework, we need to consider the measurement apparatus as well. The SG device itself can be described by a set of positive operators, which correspond to the possible measurement outcomes (spin up or spin down). These operators act on the quantum state, and the resulting state after measurement is given by the projection of the original state onto the corresponding eigenstate of the measurement operator.

In this way, the POVM formalism provides a more comprehensive understanding of measurement in the Stern-Gerlach experiment. It also allows for a more general treatment of measurement in other quantum systems.

I hope this helps to clarify the mathematical perspective of measurement in the Stern-Gerlach experiment. If you are interested in exploring this topic further, I recommend looking into the POVM formalism and its applications in quantum measurement theory.