Uncertainty principle and Einstein's thought experiments

[peterpang1994]

Recently, I have a look of the Einstein's thought experiments challenging the uncertainty principle.
One of the experiments is the Einstein's slit.
I found the description about it :

"Consider a particle passing through a slit of width d. The slit introduces an uncertainty in momentum of approximately h/d because the particle passes through the wall. But let us determine the momentum of the particle by measuring the recoil of the wall. In doing so, we find the momentum of the particle to arbitrary accuracy by conservation of momentum.
Bohr's response was that the wall is quantum mechanical as well, and that to measure the recoil to accuracy Δp the momentum of the wall must be known to this accuracy before the particle passes through. This introduces an uncertainty in the position of the wall and therefore the position of the slit equal to h / Δp, and if the wall's momentum is known precisely enough to measure the recoil, the slit's position is uncertain enough to disallow a position measurement.A similar analysis with particles diffracting through multiple slits is given by Richard Feynman."

I want to ask the wall's position uncertainty is the uncertainty of the position of the silt , the width of the silt on the wall or the uncertainty of the position where the particle collides on the wall?

 

[peterpang1994]

Can anyone help? I am confused!

 

[DrChinese]

It's not really a well defined question, nor is the example a correct analysis of the interaction described. Einstein was simply wrong on this point, as we now know.

 

[dx]

The Copenhagen 'Interpretation' - Complementarity Theory

There are several central points of quantum mechanics illustrated by this example. The idea of a photon introduced by Einstein contains the following famous feature called wave-particle duality. The energy and momentum of a photon, which refer to the concept of point particle, are defined by the relation P = hK which refers to the idealization of an infinite plane wave. Thus, the two notions of 'particle' and 'wave' are both inherent in the idea of a photon. Observation of the properties of a photon can therefore only mean the comparison of the idealizations of a particle and a plane wave with the effects obtained in an experimental situation, which are described in ordinary language supplemented by the concepts of classical physics.

There can be no distinction between the behavior the the atomic objects and their interaction with the measuring instruments. Thus, we reserve the word phenomenon to refer to effects obtained under a given experimental situation, described using ordinary language and the concepts of classical physics.

The ultimate reason for the uncertainty principle is therefore the complementary nature of the concepts used in the description of different experimental situations.

The state of a system may be given only in terms of the statistics of measurements performed on the system.
Only the totality of phenomena exhausts the possible information about the objects.​

In the experiment under discussion, there is a diaphragm with a slit, and a photographic plate behind it. In a situation in which the diaphragm and the photographic plate are fixed, we assume them to be accurately coordinated with the reference frame. In such a situation, a control of the momentum exchange between the photon and the diaphragm is excluded, since the weight of diaphragm is assumed to be large enough that exchange of momentum with the photons does not appreciable affect its velocity, so that it may be assumed to be accurately coordinate with the reference frame. Thus, these objects are placed outside the quantum mechanical description.

The heavy measuring apparatus is outside the description. All unambiguous use of space-time concepts refer to their use in the description of their state, such as the positions of marks on a photographic plate. The description of the experimental situation is necessarily done using the concepts of ordinary language supplemented by the concepts of classical physics.​

In a situation in which the diaphragm is not 'fixed', it is no longer outside the description, and the uncertainty relation applies to it too. Thus, if we wanted to determine the path of a photon by measuring the momentum transfer, the interference effect is destroyed. Thus we are presented with a choice of either tracing the path of the photon or observing interference effects.

In the arrangement suited to study an exchange of momentum between the diaphragm and the photons, the diaphragm together with the photons constitute the system to which the quantum mechanical formalism is to be applied.

In the quantum mechanical way of description, there exists a fundamental distinction between the measuring instruments and the atomic objects.

 

[peterpang1994]

Re:dx Can I summarise the content you written in one simple sentence. The instruments which is used to "measure" the quantum phenomenon would take part into it and follows the uncertainty principle. Is that a correct description??

 

[DrChinese]

...In the quantum mechanical way of description, there exists a fundamental distinction between the measuring instruments and the atomic objects.

I don't disagree with this at all. However, the OP may misinterpret to mean that the HUP is a consequence of limitations on measurement precision in some way. And that is clearly not the case, as the EPR paradox shows us.

 

[dx]

Re:dx Can I summarise the content you written in one simple sentence. The instruments which is used to "measure" the quantum phenomenon would take part into it and follows the uncertainty principle. Is that a correct description??

The heavy objects are assumed to be accurately coordinated with the reference frame and therefore are outside the quantum mechanical description. The account of their behavior is given as in classical physics. The weight of the measuring bodies defining the space-time frame ensures that exchange of momentum with the atomic objects does not affect the velocities of the measuring bodies.

On the other hand, if a control of exchange of momentum between the atomic objects and such a body were attempted, then that body would no longer be treated as in classical physics (i.e. it is no longer outside the description), and the uncertainty relation would apply to it too.