OK. This raises the (certainly naïve) question: suppose we refer to the momentum of particle at a determined point. Since the point is precise, you have the infinite spectrum of momentum. However, beyond a certain momentum the particle has enough mass-energy at that point you end up with a black hole, making the point lose its exact value, which would mean that the spectrum of the momentum would not be able to go beyond that limit. Hence not infinite. There is obviously a basic flaw in my reasoning here, but what?A. Neumaier said:Canonically conjugate observables always have all reals in the spectrum, hence are unbounded and have no limits.
Canonical commutation rules for interacting particles are valid in flat space-time only. This excludes black holes.nomadreid said:OK. This raises the (certainly naïve) question: suppose we refer to the momentum of particle at a determined point. Since the point is precise, you have the infinite spectrum of momentum. However, beyond a certain momentum the particle has enough mass-energy at that point you end up with a black hole, making the point lose its exact value, which would mean that the spectrum of the momentum would not be able to go beyond that limit. Hence not infinite. There is obviously a basic flaw in my reasoning here, but what?
The Heisenberg Uncertainty Principle (HUP) is a fundamental principle in quantum mechanics that states that it is impossible to simultaneously know the exact position and momentum of a particle. This means that the more precisely we know the position of a particle, the less precisely we can know its momentum, and vice versa.
HUP is related to the standard deviation in that it sets a limit on how precisely we can measure certain physical properties of a particle. Standard deviation is a measure of the spread of a set of data, and in the case of HUP, it represents the uncertainty in our measurements due to the principles of quantum mechanics.
When the standard deviation in HUP is infinite or undefined, it means that the uncertainty in our measurements is so large that it cannot be accurately quantified. This can occur when trying to measure both the position and momentum of a particle with extreme precision, as the uncertainty in one measurement will cause the uncertainty in the other to increase.
The uncertainty in HUP is a fundamental principle in quantum mechanics and cannot be reduced or eliminated. However, by using advanced techniques and technology, scientists can minimize the uncertainty and make more precise measurements, but it will never be completely eliminated.
HUP has significant implications in various fields of science, including quantum mechanics, particle physics, and even technology. It limits the precision of measurements, which can affect the accuracy and reliability of scientific experiments and technological devices. However, it also opened the doors to new discoveries and advancements in fields such as quantum computing and cryptography.