What's the significance of HUP if Ψ collapses?

In summary, the uncertainty principle states that a particle can exist in a superposition of multiple positions and momenta. When we measure one of these quantities, the wavefunction collapses and the particle's state is changed. This means that any subsequent measurements will always yield the same result. The uncertainty principle also tells us that the spread of positions is related to the spread of momenta, meaning that we cannot have both quantities simultaneously known with arbitrary precision.
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Phys12
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If I understand it correctly, a particle doesn't have a definite momentum and a definite position, but is in a superposition of multiple positions and momenta. And when we measure either of the two quantities, say, position, the wavefunction collapses to tell us where the particle is. Now when we measure the particle again, it will always be in that spot since we have changed its state. If that's true, then what exactly is the use of the uncertainty principle? If the measurement changes the state, wouldn't measurement of the other quantity be irrelevant? Or does the uncertainty principle tell us that the spread of the various positions of a particle is related to the spread of the various momenta that a particle can have?
 
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Phys12 said:
Or does the uncertainty principle tell us that the spread of the various positions of a particle is related to the spread of the various momenta that a particle can have?
Yes.

We can prepare the particle in any state that we please, and having done so we can measure either the position or the momentum. Suppose we prepare the particle in such a way that the result of a position measurement will be somewhere between ##x## and ##x+\Delta{x}##; for example we might put the particle in a box of width ##\Delta{x}##. In principle we can make ##\Delta{x}## arbitrarily small, so we can measure the position out to as many digits as our lab equipment is capable of.

But suppose that after having prepared the particle in such a state, we choose to measure the momentum instead? In principle we can measure the momentum to as many decimal places as our lab equipment is capable of, and we can always buy better lab equipment if we want even more accuracy.
If we do this experiment over and over again, we will find that our super-accurate momentum measurements are spread out between ##p## and ##p+\Delta{p}##. The uncertainty principle is a relationship between ##\Delta{p}## and ##\Delta{x}##; we can prepare a particle in which one of them is arbitrarily small but then the other will be commensurately larger.
 
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What is the Heisenberg Uncertainty Principle (HUP)?

The Heisenberg Uncertainty Principle (HUP) is a fundamental concept 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 accurately we know the position of a particle, the less accurately we can know its momentum, and vice versa.

How does the HUP relate to Ψ collapse?

The HUP is closely related to the concept of Ψ collapse, also known as wave function collapse. This is the phenomenon in which the wave function of a quantum system collapses into a definite state when it is observed or measured. The HUP explains why we cannot simultaneously know the exact position and momentum of a particle; when we measure one, we disrupt the other, causing the wave function to collapse.

What is the significance of the HUP?

The HUP has significant implications for our understanding of the behavior of particles at the quantum level. It shows that there are inherent limitations to our ability to measure and predict the behavior of particles, and that the act of measurement itself can affect the state of a particle. It also highlights the probabilistic nature of quantum mechanics, where we can only predict the likelihood of a particle being in a certain state rather than its exact state.

How does the HUP impact scientific research and technology?

The HUP has had a profound impact on scientific research and technology, particularly in fields such as quantum computing and cryptography. It has also led to the development of new technologies, such as scanning tunneling microscopes, which allow us to observe and manipulate particles at the quantum level. The HUP has also influenced our understanding of the fundamental nature of reality and has sparked ongoing debates and discussions in the scientific community.

Are there any exceptions to the HUP?

While the HUP holds true for most situations, there are a few exceptions. One example is the case of entangled particles, where measuring one particle can actually provide information about the other particle with greater accuracy than the HUP would suggest. However, these exceptions do not undermine the overall significance and validity of the HUP in our understanding of quantum mechanics.

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