# Does the collapse of the wave function happen immediately everywhere?

It is usually taught that when we measure some measurable value the wave function collapses immediately everywhere. This idea sounds like a simplification of some more complicated mechanism.

1. Are there any theories that suggest it?

2. Are these theories considered by physicists to be the real way in which the wave function collapses?

• Commented Mar 18, 2021 at 22:38
• I'll just point out that there are many ways people interpret the wave function. Some think of the wave function as having some sort of ontological status (sometimes, those who like MWI). Some (including myself) think of the wave function as representing (in some way) a subjective state of knowledge of the user of the theory. In that case, the collapse happens only locally because it's my state of knowledge that is being updated, and not the physical state of the system (whatever that is). We use math to describe the universe. The universe is not made of math. Commented Mar 19, 2021 at 3:37
• Similar questions about the measurement problem, the nature of "collapse", and the speed of light: physics.stackexchange.com/q/154640/50583, physics.stackexchange.com/q/497374/50583, physics.stackexchange.com/q/137809/50583 Commented Mar 21, 2021 at 11:34
• It doesn't "happen immediately everywhere" because it doesn't "happen" at all. It's part of the understanding of the person interpreting QM. Commented Mar 21, 2021 at 16:51
• This is related to the measurement problem en.wikipedia.org/wiki/Measurement_problem and the honest answer to the question is that "we not know." You will notice that in discussions like this, there are always many answers which are convinced that there is no problem, yet they propose radically different solutions to "the problem" Commented Dec 13, 2022 at 19:54

That isn't really the right question to ask. We never measure wave functions. We measure properties like position, momentum, energy of an electron. Whether the electron is spin up or spin down. The behavior of these properties doesn't match what you would expect from classical physics. Wave functions are a mathematical construct that help predict what measurements we can expect.

In classical physics, an electron is a small point-like particle. It follows a trajectory. A force acts smoothly to change the trajectory. You could measure position and momentum at any time you like to arbitrarily good precision without disturbing the trajectory.

By contrast, in quantum mechanics, the effect of the outside world on an electron is often better described by discrete interactions. We may know a measured value before hand. We can measure it again afterward. But we don't see what happens during an interaction. These kinds of interaction change the state of the electron, but they can tell us information about the the electron. We can use them to make measurements of the electron.

A typical such interaction would be to shine a light on an electron and learn things by how it is reflected. But light comes in lumps called photons. Bouncing a photon off an electron can tell you about the electron, but it also changes the electron's energy and momentum.

You can use a short wavelength photon, which can be well localized. This will tell you more precisely where the reflection occurred. But a short wavelength photon is high energy. It gives the electron a strong kick that can't be very precisely determined. So you don't know the electron's momentum or energy very well afterward.

You can use a long wavelength to make the kick as gentle as you like. But you don't know very accurately where this photon is. You can't learn as much about where the electron is.

This inverse relationship between accurately knowing position and accurately knowing momentum turns out to be fundamental, not just a limitation in measurement. It is one of the reasons why waves are used to describe reality. An electron does not have a precise position or momentum. It always has a range of possible positions and momenta.

These ranges are different from anything classical. A bag of gas doesn't have a definite size or shape, and is always spread out to some degree. Some parts of it can go fast and others slow. But everywhere inside the bag has some definite amount of gas and that part has some definite momentum. An electron is not like this.

You can shoot an electron through vacuum to a phosphor covered glass screen. If you prepare the electron in a spread out state, the electron has some presence everywhere in the vacuum chamber. You know this because it can hit anywhere on the screen with equal probability. But it is wrong to think that everyplace in the chamber has some piece of electron. When the electron hits the screen, it hits one phosphorus atom and make it give off light. The other atoms are not disturbed. If you repeat the experiment, you will find spots of light uniformly distributed over the screen.

It is also wrong to think that the electron just is a particle, and you learn where it is when it hit the screen. In the spread out state, it does not have a position. There is no way to predict which atom will be hit. If you put two slits in the way, the electron would go through both slits at once and would interfere with itself like a wave on the other side. You would still find one electron lights up one atom. But the distribution would would be concentrated where interference added and less where it cancelled.

An electron always has a range of possible positions. That range can be as big as a vacuum chamber or as small as an atom. It can be far smaller. No experiment has found a limit as to how small. But it cannot be $$0$$. If the position range is small, the momentum range is necessarily big. The electron in this state does not have a speed. There is no way to predict how long it will take to travel somewhere. It has a range of speeds and you can predict a range of times.

If you want to do physics with such an electron, you need to describe its properties and behavior with math.

Since the electron has some presence over an extended region of space, you describe it with a function over that region. The value of the function describes the "amount" of presence. The electron interferes with itself like a wave because it has a phase like a wave. So the value must have both a magnitude and a phase. Complex numbers fit. This function tells you all there is to know about the position of the election. The Fourier transform of it tells you all there is to know about the momentum. The function completely describes the state of the electron.

Further considerations about conservation of energy lead to the Schrödinger equation. This allows you to predict the form of the wave function in the presence of an electric field, or how it evolves in time. The Schrödinger equation is a wave equation. The state evolves like a wave, and is called a wave function.

This only works between interactions. An interaction replaces the state with a new one. Once the interaction is done, the Schrödinger equation will tell you how it will evolve in time.

The Schrödinger equation predicts how the state of the electron will evolve as the electron crosses a vacuum chamber. It predicts that the electron will have a uniform presence across the screen. Since the electron only has a position spread out over the screen, neither the Schrödinger equation nor anything else can predict which atom will be hit. Once an atom has been hit and the electron has a new state, the Schrödinger equation predicts how the new state will evolve.

Quantum mechanics works very well, and has been experimentally verified many times many ways. But there are some obvious problems with it.

There is a perfectly good law that works as long as the electron doesn't interact with anything else. Something else must describe the interaction. And then back to the first law.

This just doesn't smell right. Quantum mechanics leaves room for "interpretations", which are mechanisms that explain the parts of the theory that can't be measured. The description so far has been the Copenhagen interpretation. Nobody has defined just exactly what an interaction is, or exactly what goes on when the unobservable wave function collapses.

People have thought about how to fix this. One way is the Many Worlds interpretation. In it, the wave function never collapses. It just continues to evolve.

When an electron hits a screen, the wave function encounters many atoms. Each atom has strong electric fields which affect the evolution of the part of the wave function near it. There are many states that the electron could enter, each describing the electron after encountering a different atom. The true electron state is a superposition of all of them.

The atoms all have wave functions. They have ground states and excited states. Each atom is strongly affected by the part of the electron wave function near it and weakly affected by the rest. Each atom enters a superposition of ground and excited states. There is a small amplitude that it is excited and a large amplitude that it is in the ground state.

The world splits into a superposition of many states. Each state evolves according to the Schrödinger equation and never interacts with the other states. In effect, the world has split into many worlds. Each world is complete in itself and is unaware of the others.

The Many Worlds interpretation has the advantage of being the most straightforward mathematical interpretation of the wave function. The drawback is having to accept that the world is continually splitting at an unimaginable rate, and we are just unaware of it.

• This seems to me like a really long answer that doesn't really answer the question. I read through the whole thing. But I guess the general take is the first sentence "That isn't really the right question to ask. We never measure wave functions". Well, if we don't measure them, it's not because we can't, it's because we don't. It is possible in principle. And so the relevance of us most-of-the-time-not-measuring-wave-functions is definitely questionable in this context. Of course doing so takes a statistical ensemble and not just one measurement. Yet our description of the w.f. is still Commented Feb 28, 2023 at 19:30
• ...a matter of concern, since getting the w.f. wrong means getting experimental predictions wrong in general. Commented Feb 28, 2023 at 19:31

What is a wave function? It is a mathematical function depending on energy and momentum or space and time,$$Ψ(p_x,p_y,p_z)$$ or $$Ψ(x,y,z,t)$$ ( in its simple form). This function is a solution of a wave equation, a second order differential equation.

Mathematical functions are a billion, what is the wave function's connection with measurable physical quantities? The connection is postulated axiomatically in the postulates of quantum mechanics¨, $$Ψ^*Ψ$$ is the probability distribution of the products of the interaction; this means that a number of measurements have to be done with the exact boundary conditions to get an experimental distribution to compare with the theory that has calculated the wavefunction. So the mathematical function itself is not directly attributed to a given event, so it cannot be measurable.

How is a measurement performed?by interaction. Each interaction changes the boundary conditions, and thus the specific mathematical $$Ψ$$ is different before or after the measurement. That is what the collapse is, change of the specific wavefunction.

You state:

the wave function collapses immediately everywhere.

If one has to choose a different wavefunction, since it is a mathematical construct of course it can change immediately everywhere.

I have found it useful in understanding the difference between probability distributions and the need for a different wavefunction, to contemplate the single electron double slit experiment.

Each little dot seems random, but it is one materialization of the "collapse of the wavefunction" (btw I think the term collapse is not very smart, the wavefunction is not a balloon).

Before the electron hits the screen it has one $$Ψ$$, after it hits the screen it has another given by new boundary conditions:electron interacting with atoms in the screen, a completely different function.

Does it take time for the transform? Of course as the interaction is electromagnetic , nothing is "immediate" in that sense, it takes time to change experimental boundary conditions. Note though that the result of one electron is a single dot, not a function in space and time all over the place. The probability distribution responsible for the accumulation of many many dots, is bounded by the boundary conditions of the original problem "electron scattering off two slits given distance apart , given width".

• Comments are not for extended discussion; this conversation has been moved to chat. Commented Mar 21, 2021 at 11:26
• vixra.org/pdf/1905.0208v1.pdf Commented Mar 25, 2021 at 18:47

There is no way to tell if wavefunction collapse is immediately everywhere (whatever that might mean in a relativistic universe), because wavefunction collapse has no observable consequences.

The Everett Interpretation of quantum mechanics (also misleadingly known as the 'Many Worlds Interpretation') explains all observations without requiring wavefunction collapse. Since the Copenhagen Interpretation (with wavefunction collapse) and the Everett Interpretation (without wavefunction collapse) make exactly the same predictions about what you would observe, there is no way to distinguish them experimentally. And since wavefunction collapse can't be observed, there is no way to tell if it's instantaneous.

However, if you do choose to interpret observations using the Copenhagen Interpretation, then that requires that the collapse be instantaneous, faster than light, backwards in time, and non-reversibly reversible.

If you separate a pair of entangled particles, measurements on one are correlated with measurements on the other even when the particles are spacelike separated (meaning you have to move faster than light to get from one event to the other). If the collapse and choice of outcome occurs at the time of measurement, then the influence of one has to move faster than light to tell the other measurement how to answer. The decision can't be baked into the particles before they separate, either - this is called a 'hidden variables' theory and violates Bell's inequalities, which has been experimentally shown not to happen. Any pair of space-like separated events are ambiguous as to time order, so to an observer flying past them in one direction A happens before B, but to another observer flying past them in the other direction B happens before A. Hence faster-than-light implies backwards-in-time. And because changing your velocity changes the order of events, if you walk across the room one way and then turn round and walk back, your definition of 'now' across the universe swings backwards and forwards in time. At the distance of the Andromeda galaxy, 'now' shifts by about a day depending on which direction you walk. So if wavefunction collapse propagates instantaneously to the Andromeda galaxy 'now', then when you turn round and walk the other way the 'now' clock in Andromeda jumps back by a day and the collapse unhappens.

This makes little sense, and if the Copenhagen Interpretation didn't have such an established and venerable history and instead was proposed today, it would likely be rejected as unphysical and incoherent on these grounds. However, it is still very popular and widely taught.

The Everett Interpretation, on the other hand, asserts that the quantum superposition of states that applies at the microscopic level also applies at the macroscopic level of daily experience, but because the superposed states are orthogonal to one another, they don't interact and therefore we don't see them. (This can be seen in classical physics, too. When two oscillators interact, they shift into a superposition of 'normal modes of vibration' which are related to the eigenvectors of the matrix equation governing the interaction and thus orthogonal. The normal modes act independently of one another.) This is like saying that when a single electron passes through two slits, the electron passing through one slit does not electrostatically repel the alternate version of itself passing through the other slit. (If it did, the interference pattern would shift.) They cannot 'see' each other. It is as if they existed in separate worlds. Or if you send two sets of ripples moving across a pond, they pass through one another without either affecting the other, as if the other didn't exist.

If you toss a pair of entangled quantum coins at two locations, each changes to a superposition of heads and tails, and the scientists observing them each change to a superposition of a scientist observing heads and a scientist observing tails. When the scientists later return to base to compare notes, the scientist from A who saw heads can only see/interact with the scientist from B who saw tails, and vice versa. Thus the two observations are found to be correlated without anything having to move faster than light.

Since there is no way to tell if the alternative outcomes of a quantum measurement simply disappear from future reality as we pass them (collapse), or are just mutually unobservable (a superposition of orthogonal states), there is no way to tell if the collapse is faster-than-light instantaneous. But if you believe the collapse interpretation, then yes, it is.

Everett's original dissertation explaining in much more detail how his interpretation works can be found here.

• re: first line: In physics we have a technical term for "has no observable consequences". It's "doesn't exist". Commented Mar 25, 2021 at 4:19

There are no theories describing wave function collapse. The concept is pure interpretation and in my opinion problematic. Wave functions are fully specified as solutions of a wave equation, such as the Schrödinger or the Dirac equation. These equations do not allow for a collapse.

The ensemble or statistical interpretation is among others does not require wave function collapse.

• Comments are not for extended discussion; this conversation has been moved to chat. Commented Mar 21, 2021 at 11:27

when we measure some measurable value the wave function collapses immediately everywhere.

When I studied physics (30 years ago), this concept stopped me from believing what was taught; because it is clearly wrong. Professors couldn't even answer the question "then what IS a measurement"? At the time it was even thought to have to do with humans observing. At the same time, a universe wide instantaneous change because a human observes something was as indigestible for me as it was for Einstein.

Ever since I have given this deep thought and the only satisfactory answer that I could come up with is the following (which turns out to already exist as the Many Worlds interpretation):

Consider the simplest of cases where two particles have just two possible eigen states (classical states, or measurable states): spin up or spin down. This as opposed to a continuous wave function, but the idea is the same. Furthermore, assume these particles are entangled so that the only possible states of the combined system of the two is {up,down} and {down,up}. Then, before measurement there are two possible future realities: either you measure up for the first and down for the second, or you measure down for the first and up for the second. You don't know which of the two cases that will be, but you know (knowing the super position state and entanglement since this is how the system of two particles were created) that you will never measure {up,up} or {down,down}.

My interpretation was then that these two (future) realities are indeed two realities: two real "time-lines", which I called "realities". A reality however is purely linked to an observer: the reality is what you perceive as such. Other people perceive different realities (even just in the context of general relativity), quantum "realities" however are orthogonal: they do not interact.

So, the state of the system is a superposition, but you can think of it as a vector that can be projected onto two axis to get two coordinates, where the axis are the (two) possible realities.

This solves two things: 1) there is no collapse of wave functions necessary anymore at all. 2) there is no need for anything to go faster than light.

Consider sending one of the particles away to a friend who is a light year away. And then "at the same time" you measure each your particle. In both cases you become entangled with the particle you measure (a process bound by the speed of light) and your reality seems to split into two: there is now a you who measured up and a you who measured down. This is simply propagation of the Schrodinger equation, without talking about "measurement" or "wave function collapse". It is much more simple and intuitively correct. However, these two realities are orthogonal and can not interact. You yourself will only be conscious of one of them, and to you it seems you either measured up or down, with a 50% chance to find yourself in either of the two realities.

Your friend does the same and also finds himself in one of two possible realities. The fun part is: there were only two realities to begin with (with respect to the eigen states of these particles that is): {up,down} and {down,up}. One of the realities that your friend is in CAN interact with one of your realities, and the other two realities too. Your friend when traveling slower than light to you in either or both of his realities will meet you in a reality where he measure up and you down, or in a (now shared) reality where he measured down and you up.

The full realization of how this works came to me only 25 years later, when I wrote a program to simulate a quantum computer: in this program I had to write out the exact formulas and mathematics; and what turned out? I could reuse the code for entanglement for the measurements: they are EXACTLY the same thing!

Measurement == entanglement.

There is no wavefunction collapse and nothing going faster than light. We'll just have to accept that what we perceive as "reality" is just that what propagates in a consistent way over larger distances and time (ie, macro scopic) and therefore is limited to eigen values, but that there is in fact a universe-wide quantum wave that is evolving according to the Schrodinger equation that describes many many such realities at once.

PS My quantum computer simulation can be found here: https://github.com/CarloWood/quantum and a measurement node is the same (just slightly changed to see the difference in the output) as a controlled-NOT gate: https://github.com/CarloWood/quantum/blob/master/src/Circuit.h#L122

• If there is a universe-wide quantum wave that is evolving according to the Schroedinger equation, why is that theory completely useless in practice, and the imperfect theory where the psi function is atom/molecule-wide, with the Born rule, so useful? How do you calculate hydrogen emission spectrum using a universe-wide quantum wave? Commented Sep 15, 2023 at 22:46

It's pity that none of the answers (though great ones have been given!) mentions an interpretation of quantum mechanics which assumes a physical reality of the wavefunction: the hidden variables interpretation. Louis-Victor-Pierre-Raymond de Broglie initially proposed the pilot wave, which is a physical wave corresponding to the wavefunction, and David Bohm, later on, devoted much of his work to the construction of a consistent (whatever that means) theory including such a wave. I think that Einstein would have liked the theory, as he couldn't stand the thought that god plays dies. If this direction of thought was given more attention at the beginning of the quantum mechanics tale then maybe that would be the prevalent picture today. Instead of the Copenhagen interpretation.
What does it entail? In the HV interpretation, the wavefunction is not just a mathematical entity (construct, object, device) to describe physical observations and calculate the probabilities to find these. The wavefunction is built out of hidden variables in a way reminiscent of the way that a gas or liquid influences the motion of a small particle (Brownian motion) suspended in it. The gas or liquid represents the wavefunction "surrounding" a particle (quantum field theory makes use of Brownian motion too: look here).
This physical wavefunction affects the particle continuously. The particle corresponding to the wavefunction is corresponding in a literal sense. The particle's position and velocity are well defined at any moment. They are continuously changing though, in accordance with the wave.
What exactly are these variables? Who knows? They are hidden! They are not made out of the same stuff like the stuff out of which the medium surrounding the Brownian particle is made. I'm not sure what Bohm (or van 't Hooft, a Dutch physicist well known in the realm of quantum field theory and advocate of hidden variables) has (had) in mind.
The existence of local hidden variables was ruled out by Bell's experiment, which is an experiment involving quantum entanglement. Not surprisingly! After all, quantum entanglement is a non-local phenomenon. When a measurement is made, the physical correlate of the wavefunction influences a faraway particle (entangled with a particle on which an observation or measurement is made) instantaneously. This only goes to show how weird these hidden variables (particles?) behave. Note that the instantaneous influence on a faraway particle doesn't mean that the speed of light is superseded. No information is transmitted faster than this speed, though you might think that information about one particle is received by the other particle instantaneously. I remember I had a hard time explaining this to a professor of the philosophy of the sciences, who had even studied physics. I took a number of copies of a popular article to college about the 1982 experiment conducted by Alain aspect, another experiment in the entanglement arena (obviously, quantum entanglement is very sexy!). It's tempting to view the two specially separated particles as a whole (Bohm's most notable book is somewhat new-age-like called Wholeness and Implicate Order.
So, in the light of hidden variables, the wavefunction collapses all over space instantaneously indeed. One can even argue that spacetime itself constitutes hidden variables surrounding particles, though the concept will always be problematic due to the hidden quality.
One last thing. It was once said that the very act of observing makes the wavefunction collapse. If this were the case, then life never could have evolved. Everything in existence (before the arrival of humans) would have been in an evolving superposition, meaning that it would be impossible for whatever to evolve.

It doesn't "happen immediately everywhere" because it doesn't "happen" at all. It's part of the understanding of the person interpreting QM.

In particular, since you've tagged this question quantum-information, faster-than-light, and causality, "wave function collapse" does not convey any information, so it's not subject to any of the associated limitations on these things. There is a lot of misinformation suggesting that wave function collapse and interaction with entanglement provide some sort of information/communication channel. This is not the case.

x,y,z,t are mind variables. They exist as physical variables in the physical system of our mind. Our mind also has memory. We measure x,y,z,t of a particle to some precision and then summarize our memory or the memory of the photographic plate or instrument. This measurement procedure is detached from the actual events. We compare with our expectations. If they fit the wave function is OK.

The wave function uses x,y,z,t. This makes us think that this is the actual x,y,z,t of the particle. It is not. Any value of such a mind variable (x,y,z,t) is a summary over many possible actual states and times of the particle, which do not have a coordinate system and are also not Cartesian.

Specifically about time: The particle has its own changes of state and thus its own time. Our clock is quite detached from the physical system measured/observed. Many independent changes/times are summarized in a Δt of our clock of choice.

The state mapped to a value combination of the Cartesian x,y,z or some other coordinate system is just a mapping, also quite detached, and also summarizes many actual physical states.

As every particle has its own time and states, it has its own world. This would be the Many-World-Interpretation. Our world, the Ψ(x,y,z,t), is a summary over the possible world lines of particles. Many-World and Ensemble Interpretations do not conflict.

Particle world lines are not infinitely precise. That is the basic statement of quantum theory. Thus the development from the setup of the particle to the actual measurement sums up the uncertainty. This uncertainty is physical.

In this context, the wave function is our best possible and verified description that also considers the physical uncertainty. It does not collapse.

To understand this, you have to give up your classical intuition and think about quantum entanglement and measurement.

The "collapse of the wavefunction" is just an interpretation and different people think about it differently, and some even say it is not physical and not even real.

Entanglement is just a correlation - one that can potentially affect all combinations of quantities (that are expressed as operators, so the room for the size and types of correlations is greater than in classical physics). In all cases in the real world, however, the correlation between the particles originated from their common origin - some proximity that existed in the past.

Why is quantum entanglement considered to be an active link between particles?

When we make a quantum measurement of one of the entangled particles, we select (by decoherence, information leaks to the environment and causes this) an eigenstate and a pair of eigenvalues from a set of allowed pairs of eigenvalues. The emphasis is on allowed pairs. Entanglement is a phenomenon, where you upfront set the allowed sets of eigenvalues. This is the trick.

Viewed this way the interaction with the CCD collapses the wavefunction, and wavefunction collapse happens at arbitrarily high speeds. The superluminal collapse doesn't violate special relativity because it cannot transmit any information.

You say the "collapse of the wavefunction" happens everywhere immediately. I say it (the whole set of allowed pairs) already was set at the time of creation of the entanglement. This is how you create them.

If you make a measurement you effectively add a condition that the system obeys so reducing the the possibilities and so you are now considering a subset of the original list. This is what the collapse of the wavefunction is. This is why a measurement can collapse the wavefunction everywhere instantaneously rather than propagating out from the measurement location at the speed of light as it would if the wavefunction were some sort of material thing.

Why does observation collapse the wave function?

Our universe has a screenplay that was written so that the roles of these entangled particles was written (when the entanglement was created), and when you make a measurement you realize this, and you wonder why these particles act so suspiciously (and you and many others feel like the collapse happens everywhere immediately). Both particles read the same screenplay before the act (of measurement). This is the answer to your question.

Taking a measurement could be seen as "continuous" process per this article. You can see that the particle measured becomes fixed in a certain eigenstate the more you (weakly) measure it/it gets more entangled/information piles up (note: I kind of guess that a "weaker" measurement corresponds to lessened interactions).

"A deeper mechanism" can be possibly drawn from from Sydney Coleman's QM in your face-- measurement is an entanglement of your consciousness/some measurement machine/some observer's wavefunction with that of your system. The "certainty" of the measurement corresponds to your confidence that you measured spin up (for example).

*I "assume" here that the more macroscopic the measuring device, the more interactions it'll have with the system, which speeds up what happens in the first paragraph.

Note: I am an undergrad, and would welcome feedback from those with greater experience.

There is no collapse of the wave function. Most physicists agree on this today. Although it wasn't the same only 20 years ago. There cannot be such a process, if the Schrödinger equation is to represent a universal pattern. And I will, of course, adopt the stance that it does.

The collapse of the wave function is an convention adopted by the so-called Copenhagen school, which says that, after a measurement of observable $$A$$ with result $$a$$, a pure quantum state changes from $$\left|\psi\right\rangle$$ to, $$\left|\psi\right\rangle \mapsto\frac{\left|a\right\rangle \left\langle a\left|\psi\right.\right\rangle }{\left\Vert \left|a\right\rangle \left\langle a\left|\psi\right.\right\rangle \right\Vert } \tag{1}\label{1}$$ If the measured observable is position (1-spatial-dim for simplicity), $$\left|\psi\right\rangle \mapsto\frac{\left|x_{0}\right\rangle \left\langle x_{0}\left|\psi\right.\right\rangle }{\left\Vert \left|x_{0}\right\rangle \left\langle x_{0}\left|\psi\right.\right\rangle \right\Vert }=\frac{\left\langle x_{0}\left|\psi\right.\right\rangle }{\left|\left\langle x_{0}\left|\psi\right.\right\rangle \right|}\left|x_{0}\right\rangle$$

By calling, $$\frac{\left\langle x_{0}\left|\psi\right.\right\rangle }{\left|\left\langle x_{0}\left|\psi\right.\right\rangle \right|}=e^{i\alpha}$$ $$\left|\psi\right\rangle \mapsto e^{i\alpha}\left|x_{0}\right\rangle$$ In the position representation: $$\psi\left(x\right)\mapsto e^{i\alpha}\delta\left(x-x_{0}\right) \tag{2}\label{2}$$ This is the reason why people called this "collapse" or "reduction of the wave packet." Bell called this prescription FAPP ("for all practical purposes; Bell 1990 Against Measurement.")

On the other hand, Schrödinger's equation tells us, that quantum states always change according to some, $$\left|\psi\right\rangle \mapsto U\left(t\right)\left|\psi\right\rangle$$ with $$U$$ being a time-dependent linear and unitary (isometric) operator.

Now, the first problem is that change proposed in $$\eqref{1}$$ in not linear in $$\left|\psi\right\rangle$$. If you wanted to remove the offending factor, $$\frac{1}{\left\Vert \left|a\right\rangle \left\langle a\left|\psi\right.\right\rangle \right\Vert }$$ that normalises the salient state, you'd find yourself faced with a non-isometric change, $$\left|\psi\right\rangle \mapsto\left|a\right\rangle \left\langle a\left|\psi\right.\right\rangle$$ Considering that the system is an open system, IMHO, doesn't make things much better (the reason being that a time-varying Hamiltonian and the ordered-product solution can always be decomposed into an infinite sequence of linear and unitary steps. Never mind the system being open.)

Many interpretations are open from here. A summary of the most important is:

1. Copenhagen is right: "Shut up and calculate"
2. Double solution: DeBroglie-Bohm interpretation
3. Transactional interpretation (advanced waves + retarded waves à la Feynman-Wheeler solution
4. Many-worlds (Everett) / consistent histories (Gell-Mann-Hartle);(similarly inspired) etc.

But another very important one, I think, is:

1. Ensemble interpretation of quantum mechanics (Leslie E. Ballentine) If I understand it correctly, it's something like:

A quantum state (even a pure one) does not represent this or that electron. In fact, if you think about it, this or that electron doesn't make much sense, once you understand QM. It's all the electrons in the world that enter my experimental setup by responding coherently to my experimental "summoning" (filtering). Thereby, acting as one big coherent entangled state. This big coherent state comes out as a strict mixture state when I perform the measurement.

To understand all of this, some comments by other users that came before, are essential. I.e., and e.g.: The wave function is not an observable. The subject is vast, as of today.

https://en.wikipedia.org/wiki/Wave_function_collapse

Heisenberg did not try to specify exactly what the collapse of the wavefunction meant. However, he emphasised that it should not be understood as a physical process. Niels Bohr also repeatedly cautioned that we must give up a "pictorial representation", and perhaps also interpreted collapse as a formal, not physical, process.

https://en.wikipedia.org/wiki/Wave_function

https://en.wikipedia.org/wiki/Wave_equation

The above links of Wikipedia give a general introduction to the concepts, and where they are used.

I hope that helps.

The wave function doesn't collapse. A measurement is just an interaction that transfer or copies information about some observable from one system to another:

https://arxiv.org/abs/1212.3245

Measurement interactions are local and the information about observables spreads locally from one system to another as a result of measurements. You may need to update the relative state of a system after measuring it, but since the measurements are local changes to relative states also spread locally:

https://arxiv.org/abs/2008.02328

The "wave-function collapse" is an illusion, caused by mis-interpretation of the wave function as being a physical wave, which is not! It is only a mathematical tool, only defined in the complex Hilbert space (and not in the physical configuration space) to calculate, according the Born-rule, the chance to measure a certain position or momentum of a particle. The physics is not in the wave function but only in the observable to be measured. The wave function is just a mathematical solution of the Schrödinger equation in a certain environment with some boundary restrictions. The physical measurement is just the expectation value of the operator connected with the observable, and nothing more than that!