What is a position measurement?

[nonequilibrium]

Hello,

To let you know where I stand, I am aware of the formal quantum theory (Hilbert space formulation).

As you know, it states that an observable is an operator, and that's fine.
My question is the following: what is the concrete realization of a measuring apparatus? The theory says "physically we're looking at the concept of position, this means you have to use this certain operator" and it is silent about the construction of the measurement apparatus. Can somebody tell me about a (relatively simple) position measuring apparatus? If possible, also for measuring impulse?

One often hears about the following primitive position measurement apparatus (which, I presume, one has to describe on a classical level, otherwise one gets into an endless generalization of the wavy quantum system):
one fires of a photon from a certain position, it bumps into the electron (which I take as my system) and bounces back to a certain final position
Now, how exactly does this give you information about position? If I understand correctly, the apparatus now has three numbers: the place the photon was fired from, the time it was underway, and the place where the photon came back. How do we combine these numbers to give a rough estimate of position (if even only conceptually)? Or is this a bad example of a position measurement? What is it lacking?

Furthermore, am I allowed to say that the wave function of the electron has thus collapsed? (I understand the ontological status of the psi is still up for discussion, but all I'm interested in at the moment is in knowing if I can talk about the collapse [in this situation] in a consistent way) So shortly after this photon-electron interaction, the psi (more exactly: its resulting probability distribution) is a delta function? This would mean that at that same moment, nothing is known about velocity (property of Fourier analysis).
I often hear as a physical argument for that latter: ah yes, the photon has interacted with the particle and has given it some impulse and energy.
Is this latter way of thinking consistent? Is it made up? How can any speed be possible, given that the photon had only a finite energy to begin with? Surely the system (= electron) cannot have gained more energy than the photon had to begin with.

I welcome all replies,
thank you.

 

[eljose79]

Thank you for your question regarding the concrete realization of a measuring apparatus in quantum theory. This is a common question and one that has been the subject of much discussion and research in the scientific community.

Firstly, it is important to note that the formal quantum theory, as you mentioned, does not provide a detailed description of the construction of a measuring apparatus. This is because the theory focuses on the mathematical framework and formalism of quantum mechanics, rather than the physical implementation of experiments. However, there have been many studies and experiments that have shed light on the nature of measurement in quantum theory.

One of the main challenges in quantum mechanics is the issue of measurement and how it affects the state of a system. As you mentioned, the theory states that an observable is represented by an operator, and the measurement of that observable causes the state of the system to collapse to one of the eigenstates of the operator. This is known as the "measurement problem" and has been a topic of debate for many years.

To answer your question about the concrete realization of a measuring apparatus, there is no one specific apparatus that is used universally in all quantum experiments. The design and construction of a measuring apparatus depends on the specific system being studied and the type of measurement being performed. However, there are some general principles that can be applied to understand the process of measurement in quantum theory.

In your example of a position measuring apparatus, the photon-electron interaction is a common method used to measure the position of an electron. This is because the photon carries information about its position and can transfer this information to the electron upon interaction. The three numbers you mentioned (place of photon emission, time of travel, and place of photon detection) are all important in determining the position of the electron. By analyzing this information, we can estimate the position of the electron with some degree of accuracy.

Regarding your question about the collapse of the wave function, it is important to note that the concept of collapse is still a topic of debate in quantum mechanics. Some interpretations of the theory suggest that the collapse is a real physical process, while others argue that it is simply a mathematical tool used to describe the measurement process. Therefore, it is difficult to say definitively whether the wave function collapses in this situation. However, it is safe to say that the measurement process does affect the state of the system in some way.

In terms of the energy and momentum transfer in the photon-electron interaction, it is important to