Is the spatial spread of the wave function over large distances the rule or the exception in quantum mechanics?

Question

In quantum mechanics, I've read that wave functions for individual particles technically extend over all space, though the probability density tends to decrease with distance. This suggests that a particle's wave function can theoretically "spread" across large distances. However, I also understand that entangled particles share a single wave function that spans the entire system, even across vast distances, allowing for non-local correlations upon measurement.

Does this mean that wave functions generally spread across large distances as a rule, or is this only the case in special situations like entanglement? In other words, how should I understand the "spatial spread" of wave functions across large distances? Are entangled wave functions and independent (non-entangled) particle wave functions fundamentally different in their spatial extent, or are they just different manifestations of the same principle?

Answer 1

I've read that wave functions for individual particles technically extend over all space, though the probability density tends to decrease with distance

It depends. We could have well localized wave functions, for example the Gaussian state $$ \Psi(x) = {1\over \sqrt[4]{2\pi(\Delta x)^2}}e^{-(x-x_0)^2/4(\Delta x)^2}e^{ip_0x} $$

This suggests that a particle's wave function can theoretically "spread" across large distances.

Formally, a particle's wave function could spread to infinity, for example the plane waves $$ \psi_p(x) = {1\over 2\pi\hbar}e^{ipx/\hbar} $$ but such states are not physically attainable, since they have no momentum uncertainty (we would need a process that could prepare a system with infinite precision in momentum).

entangled particles share a single wave function that spans the entire system, even across vast distances, allowing for non-local correlations upon measurement.

The wave function don't need to be spread along all the space between the particles. We could have, for example for two particles with positions $x_1$ and $x_2$, $$ \Psi(x_1,x_2) = {1\over \sqrt[4]{2\pi(\Delta x)^2}}e^{-(x_1-x_2)^2/4(\Delta x)^2} $$ with is a entangled wave function in $x_1$ and $x_2$ basis, but localized in the relative position of the particles.

how should I understand the "spatial spread" of wave functions across large distances?

Position is not a special observable in quantum mechanics and it can be treat like any other observable. The spread, or uncertainty $\Delta x$ are non null for all quantum states that are not an eigenstate of such observable. Since the position eigenstates are not physically attainable (for the same reason of the plane wave above), we should expect position uncertainty for any quantum state describing a physically attainable system.

Answer 2

There are a lot of ideas here that need to be unpacked.

First, if you are talking basic quantum mechanics there are some excellent answers already provided regarding infinite plane waves convolving with wave packets and spreading. Basically, every particle has some spatial probability function regarding their detection which evolves over time.

Particles that are entangled can be translated as meaning they share a quantum property because they were co-located in the same system either at their creation or as the result of some interaction within a system of particles. With the case of quantum teleportation, that property is in some sense transferable to other particles via interaction with other particles at some later distance and time.

In an uncontrolled system the property shared between the intial entangled pairs can spread through interaction across a larger set of particles to the point where knowledge of which particles where originally entangled is lost. The shared quantum property is thus dispersed so that when a state is finally observed in a particle it is unknown exactly which other particle has a state which becomes definite. In this sense you can define a spreading of entanglement within highly interactive multiparticle systems.

The last point is that in a modern sense particles themselves are excitations of a field which spreads across all space and time. If the universe had only one type of particle there would be only one field. A quantum property shared by an original entangled pair of particles is actually a property of the field which has been excited enough to produce a pair of particles. While the particles and their entanglement are highly localized at creation, their exact whereabouts will evolve over time. If you introduce different types of particles which can interact with the original type of particle, then you are actually introducing additional fields, each filling all space and time, which are effectively interacting through the action of their related particles. In principle, an entangled property can be effectively transferred between fields. This further complicates the spreading of entanglement within a highly interactive set of fields.

It is arguable that our perception of definiteness of objects at great distance is the result of this mixing of entangled properties where we have made definite observations of the state of particles in our localized region.

Answer 3

The answer to this question depends on what interpretation of quantum theory you adopt. In the Copenhagen and statistical interpretations there is no answer and asking the question is a sign of moral turpitude. If you find that satisfactory you can stop reading.

If we take the equations of motion of quantum theory without any modifications such as collapse as a description of how reality works it is possible to answer this question.

In quantum theory the probability to find a particle in a state in general depends on what happens to all of the states the particle could be found in. for an example see Section 2 of this paper:

https://arxiv.org/abs/math/9911150

If we have an isolated particle it will be described by a state with a non-zero spread of position and momentum and that state will in general spread out over time. So even if it starts out very heavily concentrated in a particular region it will in general spread over time so that the probability of finding it will increase.

Real systems are in general not isolated. They interact with other systems and those interactions copy information out of a system. This information copying prevents interference: a process called decoherence:

https://arxiv.org/abs/1911.06282

In particular the objects you see around you in everyday life have information copied out of them on scales of space and time much smaller than those over which they change significantly so they don't undergo much interference. For objects in everyday life the resulting states tend to be narrowly peaked in position and momentum compared to the size and momentum of the objects concerned. As a result for such objects the world according to quantum theory looks a bit like a bunch of versions of those objects evolving autonomously from one another:

https://arxiv.org/abs/1111.2189

https://arxiv.org/abs/quant-ph/0104033

This is commonly called the many worlds interpretation but it is just an implication of applying quantum theory to macroscopic systems and taking decoherence into account. Smaller systems like photons may spread out significantly without being decohered depending on how strongly they interact with the environment. There are experiments describing precise control of the degree of decoherence in the first paper I linked above.

Entanglement isn't directly related to the spreading. Under some circumstances spatially separated quantum systems can exhibit correlations that are stronger than what would be allowed by classical physics: entanglement. The gist of the issue goes like this. You have two systems $S_1$ and $S_2$. If you measure the quantity $X$ on $S_1$ you get the result $+1$ with probability 1/2 and $-1$ with probability 1/2 and the same is true if you measure the quantity $Z$ on $S_1$. If you measure $X$ or $Z$ on $S_2$ the possible results and probabilities are the same. Nothing you do so $S_1$ affects the set of possible outcomes or the probabilities of the results on $S_2$ and vice versa. But if you measure $Z$ on both and compare the result they are the same. If you measure $X$ on both and compare the result they are the same. But if you measure $X$ on one and $Z$ on the other they match with probability 1/2. So the probability of a match depends on whether you mentioned the same quantity on both systems. This result creates a problem precisely because the systems are clearly spatially separated from one another and yet the probability of a match depends on whether you measure the same quantity on each of them.

The standard approach to this in textbooks and many papers is to repeat the prediction and give no explanation or model of how the correlations are produced and vaguely mumble something about non-locality.

The only model I have seen that actually gives an explanation claims that the correlations between the results are produced when the results are compared. The information required to produce those correlations is carried as quantum information in decoherent systems that is protected from decoherence because it can only be revealed by comparing the measurement results: locally inaccessible information

https://arxiv.org/abs/quant-ph/9906007

Using quantum theory to explain the results of quantum experiments allows you to understand what's happening. Refusing to apply the theory and take it seriously prevents you from understanding it.