Saturday, December 1, 2018

Freeing The Schrödinger’s Cat: Solving The Mysteries Of Quantum Mechanics; Part I

Freeing The Schrödinger’s Cat:
Solving The Mysteries Of Quantum Mechanics
Part I

My recent encounter with a new book on the history of quantum mechanics brought back old memories about my student years when I encountered quantum mechanics for the first time in my life.
This piece, coupled with , and Quantum Entanglement pieces, represents a coherent view on the origins of quantum mechanics (it was extended to part II).
After I wrote “Killing The Schrödinger’s Cat” my fellow colleague Nobel Prize in Physics laureate Prof. Sheldon Lee Glashow sent to me his own review of the same book.
Reading his review prompted me to revisit the whole discussion about the meaning of a wave-function, a wave-function collapse, and related concepts.
This process has led me to two discoveries.
First, I realized that the Copenhagen interpretation of quantum mechanics which I was taught when I was a student is not the Copenhagen interpretation of quantum mechanics described in many English-speaking sources.
Wikipedia, for example, says (following many standard textbooks) that: “According to the Copenhagen interpretation, physical systems generally do not have definite properties prior to being measured, and quantum mechanics can only predict the probabilities that measurements will produce certain results. The act of measurement affects the system, causing the set of probabilities to reduce to only one of the possible values immediately after the measurement.”
But according to the Copenhagen interpretation I learned (if my memory serves me right), or at least the one I have been using since I can remember, physical systems always have definite properties (for some quantities) prior to being measured, but, indeed, “quantum mechanics can only predict the probabilities that measurements will produce certain results”. The act of a measurement does affect a system, but it is not “causing the set of probabilities to reduce to only one of the possible values”; a measurement is causing a system to reveal its state it assumes during the interaction, and as the result of the interaction with the measuring device; and possible states which can be revealed in a measurement obey a certain probability distribution; and quantum mechanics is basically a recipe which allows to find that distribution.
The second discovery was much more important than the first one.
I realized that the majority of the mysteries of quantum mechanics have their roots not in quantum mechanics per se, but in misunderstanding of the probabilistic nature of the world in general.
Scientists, even some prominent physicist who can carry out complicated statistical calculations, yet don’t understand the nature of probability (or just denying the probabilistic nature of the nature).
As the result of that misunderstanding, they invent fancy terminology which makes quantum mechanical waters even muddier.
Instead of making things clearer they make things harder to understand by inventing more and more complicated language.
Probably, the desire to position themselves as a unique, original, and deep thinker also may have played some role in that muddling.
In order to develop a “simple” and clear interpretation of quantum mechanics, we need to strip off all the linguistic mysteries created by scientists over a ninety-year period.
There is a “simple” and clear interpretation of quantum mechanics, which I call a “minimal sufficient workable interpretation of quantum mechanics”, or MSWIQM, which is an example of the practical use of the Occam's Rser (from this point forward when I say “we” I mean people who accept MSWIQM).
In order to get to MSWIQM, first we need to review our understanding of probability in the physical world.
Let's take a standard cubic six-faced die numbered from 1 to 6 and place it in a plastic glass, and cover it with our palm, and start shaking the glass.
We know what will happen when we drop the die on a tabletop - eventually it will stop and one of the six numbers will become visible on the top face of the die.
Some quantum mechanics interpreters would say that the die has six independent states, which are revealed when it is dropped on a table, but when the die is being shaken in a glass, the die exists in all six states at the same time. And the act of a measurement, i.e. when the table makes the die stop, makes the die to choose and reveal one of the six states.
This description is based on several assumptions: 1. a die always has a state (the die is always in some state); 2. states may be observed during a measurement; 3. states which are observed during a measurement form the basis for all other possible states; 4. all possible states - visible or not visible for an observer - represent a combination of their observable states.
Since those assumptions sound very reasonable, we accept them, and also we tend to accept all other statements made about this experiment, including: “when the die is being shaken in a glass, the die exists in all six states at the same time”.
But that is only one possible interpretation of the state of a die inside a glass.
There is another one, which does not affect the behavior of the die before or during a measurement, but paints a different picture of its behavior before a measurement, when the die is in the glass.
Instead of saying (that what “interpretation” is – a way of saying) that when being shaken in a glass the die exists in all six states simultaneously, we can say (a new interpretation) that the die always exists in a single state, i.e. at a single instant it is in one of the six specific states, but all those states replace each other randomly, and before the die is being dropped on a table (measured) there is no way to know the state of a die; as long as the die is in a glass, there is no way to know its state and the evolution of the states of the die. As long as the die is in a glass, all we can to say is – there is a die in a glass, and it may be in one of the six states (or we could say that when the die is in a glass it is in a state of a constant change of its states – but we need to understand that when saying that we use word “state” twice but with a different meaning).
To make the experiment less dependent on an observer, we can improve its design, make it look more scientific.
We can take a box, place in it a battery, a switch, a function generator (a vibrator), connect them in a special way, so, when we open the door of a box, a vibrator stops, but when the door is closed, a vibrator vibrates (maybe we could even hear the buzzing).
Then we can attach a plastic glass to the vibrator and place a die in it. When the door is closed, the die in a glass is being shaken by a vibrator. But as soon as we open the door, we can see the die resting in one of the six possible states. We can make hundreds of boxes like that, close all the doors at the same time, wait for one hour, and open all the doors at once. What we will see, is that about one sixth of all dies will have number 1 on their top face; about one sixth of all dies will have number 2 on their top face; about one sixth of all dies will have number 3 on their top face; about one sixth of all dies will have number 4 on their top face; about one sixth of all dies will have number 5 on their top face; and about one sixth of all dies will have number 6 on their top face.
Based on this experiment (and since we know the theory of probability) we make a conclusion that, if we use only one box, the chance that the die in it will have number 1 on its top face when we open the door, is equal to about one-sixth, or about 16.7 %.; and so for numbers 2, 3, 4, 5, and 6.
This is just an example of how science describes probabilistic behavior of the nature.
This example can be easily generalized.
Let's say we have a black box, and that there is a system of any nature inside that box, and we have no ability to look inside a box. However, we can take that system outside of a box, and when it happens we can see that the system can be found in several possible states. Our assumption is that when the system is outside of a box, it has exactly same states as it has when it is inside a box (otherwise the system would be un-study-able). By taking the system outside of a box we reveal some of its properties; the properties which it had when it was inside a box.
Based on our observation of the system when it is outside of a box we make some inferences about the properties of the system when it is inside a box.
Based on our observation of the system when it is outside of a box we make some inferences about what is happening to the system when it is inside a box.
Here is when the human fantasy gives rise to many many many different possibilities.
Many scientists would say that when inside a box the system simultaneously exists in all possible states, those states which can be observed when the system is taken outside of a box.
But for me, much more natural interpretation would be saying that when the system is inside a box, at any given instant it exists in one and only one state from any of its observable states; and those states can change by replacing each other (randomly or not - at this point it does not matter); and there is no law which could tell us in which state the system in the box is at any given instant, and there is no law which could tell us how those states evolve.
The next step is making many black boxes with the same system inside and then taking all the system simultaneously outside of their boxes, and recording the state observed for each system.
Based on the results of this experiment we can derive the state distribution of the states of the system.
This distribution is defined by the properties of the system coupled with the properties of the environment acting on the system when it is outside a box. The next level of reasoning would involve thinking inside a box; we could expect that the final distribution of states could also depend on the properties of the box. This step, however, just makes technical things more complicated, but does not change the basic idea of what a system is and how to study its property when the system is not available for direct observations when it is hidden from an observer in a box.
I'd like to stress the fact that this description (this thought experiment) has nothing to do to the size or the nature of the system; it could have been a microscopic system or a macroscopic system; the description of the properties of a system has nothing to do with a quantum or classical nature of the system.
The description of the properties of a system is solely based on two facts:
1. there are situations when a system is not available to a direct observation.
2. there are situations when a system is available to a direct observation.
All we need to do is to establish a reliable way to infer some statements about the properties of a system when it is not available to a direct observation based on the information received from direct observations.
And to make such inference we may need to employ the probabilistic approach to analyze the natural phenomena (i.e. statistics).
By its nature, a system may be composed from microscopic objects, or from macroscopic objects, or both.
When we study a microscopic system, we assume it is composed of microscopic objects which are not available for a direct observation by a classical observer. That is why we bring a system in a contact with a classical observer, and when I say “a classical observer” I mean any object which behaves according to Newton's laws (i.e. large, macroscopic).
The action of bringing a microscopic system in a contact with a classical observer is named a measurement.
The results of many measurements eventually lead to specific inferences on the properties of a system.
Before moving to the next phase of the discussion, i.e. the meaning of a wave-function and what does “collapse” mean, let’s take one last bow to the Schrödinger’s Cat (the same one discussed in the first piece).
Let’s let the cat out, but instead, let’s place in a box our vibrator with a glass and a die in it.
We will keep radioactive metal, and a counter, and a hammer, but instead of a vial we will place a switch such that when a hammer hits it, the vibrator stops.
Due to an additional random factor (radioactive metal) this is actually not so simple system to study, at least more complicated that our original example, or the Schrödinger’s Cat experiment. But we still can apply to it the same way of reasoning as before. When we open a door, we may find that the die is in a vibrating glass (“alive”), or it is resting in one of six states (“dead 1”, “dead 2”, “dead 3”, “dead 4”, “dead 5”, “dead 6”,). The list of states now is more than just “dead” or “alive”. But there is no reason to think that when the door is closed the die exists is in all possible states simultaneously.
The die is always in some state – one state – and if it is in one state now, it may be in another one state next moment, but we do not know in which state it is until we make a measurement (bring a system in a contact with a classical observer, open a door and look).
BTW: for any system, to study its properties, one of the most important tasks a scientist has to do, is to establish the full list of possible states.
For a small object – a particle – the description of the list of states depends on its size.
For a large and heavy (and slow) “small” particle an elementary state is composed of six numbers/variables/quantities: (x, y , z, vx, vy, vz) – three coordinates and three components of velocity. There are other quantities which can be used to describe the properties of a particle, e.g. kinetic energy, linear momentum, etc. All those quantities can be described in terms of (x, y , z, vx, vy, vz). And all quantities used to describe the properties of a particle can be measured at the same time. A large “small” particle represents a classical (macroscopic) Newtonian object with a negligible size.
For a “small” small particle, i.e. for a microscopic, or quantum particle, a fundamental state is composed of three numbers/variables/quantities: (x, y , z) – three coordinates of a particle.
Why is it three and not six anymore?
No one knows, but some people still try to figure it out.
To describe the behavior of a classical (macroscopic) particle, physics uses the Newton’s laws (plus some extra stuff). Theoretically, with the right amount of information, we can predict the state of any macroscopic particle at any time. We say that an interaction between macroscopic objects (including us, humans) is deterministic.
To describe the behavior of a quantum (microscopic) particle, physics uses the Schrödinger’s equation (plus some extra stuff). Fundamentally, with any amount of available information, we still never can predict the exact state of any microscopic particle at any time; all we can predict is the probability for a particle to be found in one of the states. We say that an interaction involving microscopic objects is not deterministic but probabilistic.
Here we run into the first issue.
A macroscopic particle is composed of many microscopic particles.
As many other scientists (but not all), I believe that that there should be one universal description of the both realms of the universe (large and small).
Yes, scientists also have things to believe in.
In that scientists are not much different from other people. If people believe in something, it is very hard to make them to change their mind.
“I am a strong Trump believer”
“Why?”
“He fights against immigrants flooding our Country.”
“What is wrong with accepting immigrants?”
“They take our jobs, they damage our neighborhoods.”
“But there are no data which support this claim. In fact, data say the exact opposite.”
“I don’t know where you get your “data”, but I don’t believe in them, I believe to President Trump”.
Scientists may have a similar conversation, but about “hidden-variables” v. “decoherence” (or President Trump).
If we believe that there should be one universal description of the both realms of the universe (large and small), that means one of three following options is correct:
1. the Newton’s laws should be sufficient for describing the properties of microscopic particles.
or
2. the Schrödinger’s equation should be sufficient for describing the properties of macroscopic particles.
or
3. there has to be another and very much different law/equation/approach/ which includes in itself both: the Newton’s laws and the Schrödinger’s equation.
The first option was proven to be wrong (which led to the birth of quantum mechanics).
The last option would mean we simply have not yet found the right laws of nature, and the whole discussion is pointless.
So, we accept the second option, i.e. the Schrödinger’s equation should be sufficient for describing the properties of macroscopic particles.
That means, we should be able to apply to a macroscopic particle the law we use for microscopic particles (the Schrödinger’s equation) and get the same results we get from using the Newton’s laws.
For example, if someone drops balls from a top of a tower, we should be able to write the Schrödinger’s equation and solve it and calculate how much time would it take for each ball to reach the ground, or the speed with that it hits the ground. We would have to treat the balls as a composition of huge number of atoms and solve a very complicated Schrödinger’s equation (which, to be as accurate as possible, also should include the Earth in it).
In principle, it is possible; formally, such an equation can be written.
Technically, it is impossible, because no one can ever solve that equation (nine classical planets - I like Pluto -  already represent a problem of an enormous difficulty, and here we talk about billions of billions of quantum particles).
However, there have been work done on finding ways around it, for example, searching for the approximate solutions of the Schrödinger’s equation (including in the form of a Feynman’s path integral) for an object with a significant mass (or wave-packets, or other approaches). In general, most physicists are fine with the transition from a microscopic world description to a macroscopic world description, even if they personally do not know how to derive Newton’s laws from the Schrödinger’s equation (or even don’t believe it is possible to do).
When the exact approach is not doable, we – scientists – do what we always do when we cannot solve the actual problem, i.e. we make a simpler model and solve that new simplified problem. If the solutions for that simplified model are supported by the experiments, we are satisfied with the model.
Since the balls compared with the Earth are very small, we ignore their structure, we ignore that they are composed of many microscopic particles, we considered all balls as classical particles, and use the Newton’s laws, and it perfectly works, and BTW: gives exactly same solutions for all balls. And we just believe (!) that eventually someone will be able to solve that very complicated Schrödinger’s equation which treats the balls and the Earth as a composition of huge number of atoms and will demonstrate that that solution will give the same results as the Newton’s laws.
Now we can concentrate on the description of the microscopic world per se.
According to MSWIQM, at any moment in time a particle exists somewhere in the world, it has certain values of its coordinates (it is in a certain state). But it is impossible to know where it is or where it will be. The genius of the physicists who have developed quantum mechanics was that they were able – from a very limited amount of information and with almost no guidance – come up with an idea of a wave-function.
A wave-function prescribes two real numbers (in a form of one imaginary number) to each location in the universe at each instant in time. And those two real numbers (or one imaginary number) can be mathematically related to the probability of finding a particle in a given place at a given time.
How?
Via the Schrödinger’s equation.
Why?
No one knows.
But the method works, so why bother?
And most scientists don’t.
There are only two seemingly unresolved, but not conceptual and rather technical, issues with the method, which still are being attacked by a small number of scientists.
1. The problem of a collapsing wave-function (a.k.a. the problem of a measurement).
2. The problem of entanglement (a.k.a. the problem of “spooky action at a distance”).
I, personally, see one more problem, namely, the problem of the absence of a reasonable theory of quantum gravity; I believe the only path to that theory is trough “fixing” quantum mechanics; promoting it from a highly effective technical recipe (a prescription of actions) to an actual theory.
Let’s start from the problem of a collapsing wave-function.
When a quantum particle is a part of a quantum system (for example, an electron in a Hydrogen atom), the Schrödinger’s equation describes its wave-function. If we solve the equation we find the wave function (a wave-function is a solution of the Schrödinger’s equation). Turns out, the same equation can have many (even infinitely many) solutions. And - if we combine several solutions, we get again a solution. But we also find that there are solutions which cannot be represented as a combination of any other solutions. Those solutions are special, they have a name “eigenfunctions”. We can innumerate/classify those solutions, eigenfunctions, using some convenient notations (numbers, letters, numbers and letters). Each solution/eigenfunction is to be said describes an electron in a specific state.
And here we run into another problem – not physical, but linguistic one. We said before that a particle (e.g. an electron) has a state described by three coordinates (x, y, z). Now we said that a particle is in a state described by a specific wave-function.
We used the same word – “state” – for two completely different theoretical objects/constructs. Of course, that leads to confusion! Unfortunately, this type of a situation happens more often than you may think (to make things worse, when physicists use for a quantum system term "state" it may have even more meaning than the two we mentioned, but that does not affect the logic of our discussion).
To resolve the issue we two states, we need to refine our terminology.
For example, let us call (x, y ,z) coordinates of a particle “c-state” (“coordinate” + “state”), and a solution of the Schrödinger’s equation “w-state” (“wave” + “state”).
As we said before, if we know a wave-function of a particle, we can calculate the probability to find that particle in a given location at a given time (the way we can do it does not matter - just math).
In other words, if we know w-state we can calculate the probability to find a particle in a given c-state (the second part of this discussion provides much more information on probability).
The importance of eigenfunctions is enormous, because each eigenfunction corresponds to a specific set of numbers (used to innumerate/classify those eigenfunctions), and – for whatever reason – those and only those numbers correspond to measurable values which appear as results of all measurements.
For example, if we measure the energy of an electron in a Hydrogen atom, we only get a set of specific values. Each of those values has a direct connection/correspondence with a specific eigenfunction of an electron in an atom. Each of those values comes as a result of finding the solution of the Schrödinger’s equation for an electron in a Hydrogen atom. Other values for the energy of an electron in a Hydrogen atom have never been measured, and it is believed (!), will never be.
This is one of the important differences between quantum mechanics and classical mechanics. In classical mechanics, any values are allowed. In quantum mechanics only certain values are allowed, but most of the values are actually restricted.
The second important difference is that in classical mechanics we can measure values of any quantities at the same time. In quantum mechanics that is not always a case. For example, while we succeed to measure the energy of an electron in a Hydrogen atom, we will never be able to measure its exact location. Even for a free particle the further measurement can measure only its location or only its velocity, but not both. This is just a peculiar feature of quantum mechanics which disappears in classical mechanics. However, we believe – again – that sometime in the future someone will be able to solve a hugely complicated Schrödinger’s equation and will prove once and for all that by adding in a system more and more atoms which strongly interact with each other the system will eventually behave as a classical particle. So far we just have examples which show that when a mass of a particle increases, one trajectory - the classical - becomes more and more probable, and the average values for such quantities as force, acceleration, etc. begin being governed by equations which are practically identical to the classical ones.
Now we can describe what the collapse of a wave-function means.
An electron in a Hydrogen atom does not have to be only in one  w-state described by a specific eigenfunction with a specific value of energy. An electron in a Hydrogen atom can be described by any general wave-function which represents a complicated combination of all possible eigenfunctions. But when we try to measure the energy of an electron, all but one eigenfunctions seem to disappear, and only one eigenfunction with a specific value of energy survives – the one which corresponds to the measured energy.
Another classical (pun intended) example of a wave-function collapse is an electron diffraction experiment. An electron is traveling through a crystal and then falling on a photo-plate. It hits a plate at a specific location. However, it is impossible to predict where, at the what location, an electron will hit that photo-plate. Before an electron hits a photo-plate the wave-function, the w-state of an electron describes non-zero probability to find that electron anywhere in a space. However, as soon as an electron approaches a photo-plate, only one specific wave-function which corresponds to a specific location of an electron survives (or, in other words, only one specific w-state corresponding to one specific c-state survives; a wave-function which describes that state is named Dirac delta-function; it is equal to zero everywhere in space, except one location where an electron is being registered by a photo-plate).
This seemingly instantaneous transition from a wave-function which represents a combination of the infinite number of eigenfunctions to a single eigenfunction is named “the collapse”.
To understand what is really happening during the “collapse” we need to turn again to the discussion of a “measurement”.
First, what we need to understand is that the nature does not need measurements, and the nature does not have measurements. The world is filled with object interacting with each other and they act/behave/evolve according to the laws governing those interactions (even if we do not yet know all of the laws, we believe they exist).
We, humans, need measurements.
We need measurements to make an imprint of the world, and to make that imprint in our mind (and texts, and pictures, and equations, and even songs, and dances).
Measurements allow us to classify the world around us and to develop a symbolic description of the world in such a way, so we would not have to physically act on objects every time when we need to find out what will happen to them; instead we could operate with symbols representing those objects and predict what would happen to them. Then, based on our prediction we would be able to select our own acting/living strategy which would optimize our survival (“don’t worry, be happy”).
Why does the world need us, humans?
My answer is that the world needs us to optimize its own survival.
Without intelligent spices the world would eventually reached thermal equilibrium and stopped evolving. The existence of intelligent spices at least gives the world a chance to keep evolving forever.
The main purpose, the mission of intelligent spices – the only anti-entropial force in the universe – is to keep the world evolving.
And for that, they (intelligent spices) have to be able to build - in their own minds - the picture, the description, the model of the world.
And for that they invented sciences; the mission of science is making reliable and testable predictions.
And for that scientists invented measurements – the instruments and tools for describing the world in terms which allow making reliable and testable predictions (the mission of a scientist is discovering truth about world and represent it in a clear form which allows making testable predictions).
For the world, a measurement is just an interaction between two systems.
But for us, a measurement is an interaction between a system which properties we need to describe (and hence we study that system) and a system which properties we already know. The latter system automatically has to be a classical system, because as classical object ourselves, we cannot perceive quantum objects, we can only perceive other large classical objects. We, humans, were first measuring devices. Then we started devising devices specifically designed to make measurement. But we still use ourselves as a measuring device to read the measurements registered by the measuring devices we devised/designed/developed to study properties of other systems.
Scientific community needs measurements because they need to be able to reproduce experiments done by different scientists. That can only be done because scientists have the same fundamental measuring devices: eyes, ears, hands. That divides all knowledge into two large groups: observable knowledge - the one which has been obtain via direct observations, and abstract knowledge - the one which is based on inferences from the observable knowledge (a better name would be "imaginative", or "theoretical", "descriptive" because for many people "abstract" means "mathematical" or "non-descriptive"; mathematical knowledge is an abstract knowledge, but that knowledge not about the world, it is about abstract logical relationships which could be used for describing some properties of the world, as part of the "descriptive" knowledge). The abstract knowledge is the subject of interpretation. Everything, anything, which is beyond direct measurements is the result of interpretation (e.g. any description of what may be happening in an atom, or in any microscopic system, is only interpretation). Quantum mechanics is a good example of that even a well working mathematical model (the Schrödinger's equation; Feynman's path integrals) may be based on different interpretations, or a fundamentally wrong interpenetration may lead to a correct experimental results
When scientists begin study a new phenomenon, they generate many possible interpretations (there are two post on scientific thinking: in general, in physics). Eventually, one of those interpretation is accepted by the majority of scientific community and becomes "a paradigm". The history of science demonstrates, though, that paradigms are not absolute and may change.
Science development is a consensus-building processes (an example of which we all just have witnessed!): first, scientists reach a consensus on what they perceive, then on the language the use for the description of what they perceive, then on the reasons for the happening of what they perceive (in reality, all three processes interfere and intertwine).
When we measure some property of a classical system, we use an interaction between two macroscopic systems: the system which we study and the system we developed as our extension to study other systems (a measuring device).
When we measure some property of a quantum system, we use an interaction between a macroscopic system (a measuring device) and a microscopic system (the system we study).
But our measuring device (the macroscopic system the properties of which we know) and our system under study (the microscopic system we study) together compose one larger system which has two interacting parts, a classical part and a quantum part.
To be as exact as possible, we have to write the Schrödinger’s equation for this whole super-system and solve it, and then extract from the solution the information about the properties of the quantum subsystem.
In principle, it is possible; formally, such an equation can be written.
Technically, it is impossible, because no one can ever solve that equation (nine classical planets already represent a problem of an enormous difficulty, and here we talk about a quantum particle which interacts with   billions of billions of quantum particles – our classical measuring system).
For example, let’s take the classical (in a cultural sense, i.e. well-known, established) “electron diffraction” experiment.  Electrons are being emitted from a heated electrode, accelerated by an electric field, aimed at a crystal, travel through the crystals and fall on a screen. The exact equations for this experiment should include all those elements. And strictly speaking, instead of the Schrödinger’s equation we would need to use equations of quantum electrodynamics (because electric field is composed of photons, and also due to other reasons).
If such approach would have been used – and doable – the solution would include all observable situations, including the final collusion of an electron with a screen. We would simply did not have a need for a “collapsing” wave-function.
Of course, there may be people who would disagree with the last statement.
And I would ask those people:
"Why?"
And they would say:
"Because I do not want to study how to apply the Schrödinger’s equation to a complicated system composed of a microscopic and a macroscopic parts".
"So, you agree that the Newton's laws correctly describe macroscopic systems and cannot be applied to microscopic systems, that the Schrödinger’s equation correctly describes at least some of the properties of microscopic systems, but you think it cannot correctly describe properties of combined microscopic and macroscopic systems?"
"Yes"
"Well, in that case I am sorry to say that you have wasted your time on reading this piece, because you are selecting the third option from the three options listed before, and this piece is written for people who select the option two".
So, we believe that the Schrödinger’s equation should correctly describe properties of combined microscopic and macroscopic systems.
However, since such approach is not doable, we – scientists – do what we always do when we cannot solve the actual problem, i.e. we make a simpler model and solve that new simplified problem (again, and again, and again).
In our simplified model, we do not even consider any interaction between an electron and a screen (and forget about photons; BTW: there is another classical experiment, i.e. measuring e/m ratio of an electron, where we also forget about photons and do not analyze it using quantum electrodynamics, as we should have for the exact analysis). We study an electron traveling in a classical electric field (or even a free electron traveling on its own), which then encounters a crystal (BTW: to keep the model as simple as possible we also do not write any equations to describe actual interactions between an electron and the lattice of a crystal; we find a way around). But this approach let us solve the Schrödinger’s equation for an electron and find all possible eigenfunctions. We then “know” (meaning, believe) that when an electron hits a screen, its actual wave-function eventually will be transformed into the one with the observable measurable value. And the experiment supports our belief.
I want to stress the fact that so far we have no exact mathematical proof that if we would have started from an exact equation for the whole system (quantum plus classical) we would be able to derive the results predicted by our “collapsing” model; i.e. find all eigenfunctions for an electron not touching a screen, but then observe how one of theses eigenfunctions is being revealed during touching a screen. There are some people who work on bridging the gap between a system free from interaction with a measuring devices and the system measured by the device (they call it quantum decohirence), but from my point of view the whole issue is not fundamental but just technical.
The good thing about physics is that in physics we don’t have to have an exact logical mathematical proof for all our models; instead we can use experiments – if our model allows us to predict the results of our experiments, we can live with that.
And we can definitely live without agonizing about collapsing wave-functions, because there is nothing very special about it; the fact that we use this simple model is just the result of our mathematical insufficiencies; if we could solve the equation for the whole system (quantum plus a measuring device) the whole problem would disappear – and I believe that eventually that will happen.
Some people may disagree with me, of course. But in order to disprove me they would have to first solve  the equation for the whole system (quantum plus a measuring device) and to demonstrate that the solution does not explain a collapse of a wave-function.
Sure, I’ll wait.
The last “mystical” issue of quantum mechanics is entanglement.
I had a post on that matter and so far, I have not much to add to that post.
I don’t see an entanglement as an issue.
From my point of view, the real mystery of quantum physics is why to describe a quantum mechanical system we cannot use classical probabilities (in form of distributions), but instead we need to use a wave-function, and then use it to find all the probabilities we need. I believe that solving this mystery will automatically solve the mystery of entanglement.
I also believe that solving this mystery will automatically answer question why a quantum mechanical system has many values restricted and only some allowed; I think it is because out of all possible wave-functions only eigenfunctions represent w-states stable in time, and all other wave-functions quickly decay (at least when they start interacting with macroscopic objects).
Some people say that real mystery of entanglement is that it leads to “spooky action at a distance”.
I am with Einstein on this, and I think there is a physical explanation which has nothing to do with “spooky action at a distance”.
If I was a physicist, I would explore different models for describing the behavior of quantum particles.
The models I would try to explore would all be describing our world. I think that trying to explain what is happening in our world by employing infinite number of other possible existing worlds does not help at all, and only demonstrates our inability to developing the correct understanding of our phenomena.
I would not disregard completely hidden-variable theories, especially if those hidden variables are exotic like hidden dimensions.
But I believe that all currently observed physics can be understood on a basis of currently observable physics.
From my view, the best approach, in that venue has been offered by the "pilot wave theory" (but the wave which guides particles should be quantized as well).
For example, when an electron travels through a crystal, it constantly emits virtual photons, those photons travel in all directions, they may reflect from a crystal back to the electron and affect its motion (this example is borrowed from another post).
And when two entangled electrons travel away from each other, they also keep interacting via virtual photons (which is the realm of quantum electrodynamics, which is beyond the classical Schrödinger’s equation commonly used to describe entanglement; although, explaining entanglement of two photons via exchange of virtual photons seems less plausible). This picture simply eliminates “spooky action at a distance” but leaves an actual interaction at a distance via virtual photons. (which, again, are not presented in the standard version of the Schrödinger’s equation used to discuss EPR paradox).
There is another possible explanation, which may solve all the mysteries - the matter we know also interacts via particles which cannot travel slower than the speed of light: tachyons - those hidden variables that establish faster than light "spooky" interaction. However, there is no yet any model which could offer a reasonable mathematical description of that interaction.
I would like to finish this piece with a reminder that all we try to do in quantum physics is to infer some information about properties of a system which cannot be observed via direct measurements (our eyes, ears, fingers) using a system which properties can be observed via direct measurements (our eyes, ears, fingers). That is why we need to use specifically designed measuring devices (large classical objects which properties we know).
However, even in classical physics we may have to infer some information about properties of a system based on other available information. And when we do that, often there may be more than one explanation (interpretation) of what has been observed, and why.
When a stone hits a ground at 20 m/s, it does not necessarily mean that it was released from rest 20 m above the ground. It might mean that it was shot down from a 15-meter height starting moving down at 10 m/s. Or, there are infinitely more initial situations leading to the same observation.
The observable final state does not define the unique initial conditions.
We know that our universe is expanding. The expansion is consistent with the solution of the Albert Einstein’s General Relativity equation. If the size of the universe increases, it means that if we go back in time its size should be decreasing. But it does not mean its size should go straight to zero (to singularity). The fact that the universe expends does not necessarily mean that the universe expands from a singularity, i.e. from a dimensionless dot/point. The solutions when the universe begins from being “a ball” also satisfy the Einstein’s equation. What if the Big Bang created a "ball" of our universe? If our whole universe would have been created at once one second ago in exactly same state as it was one second ago (the key word is "exactly"), we, people, would never had a way to know it. Although, quantum mechanical effects (fluctuations) probably would prevent this option from happening. But if a universe could start from being “a ball”, could it have been a huge ball, maybe even with the size of the whole universe – and still expanding right after it was created?
I don’t know the answers to these questions.
I don’t even know if those questions make any sense.
But I do know that those questions have the right to be asked.
And maybe, they even have the right to be answered.
Gravity intrinsically connected to space and time (thanks, Einstein!). To quantize gravity, we would need to know how to quantize space and time. And for that we need to know what does it really mean “to quantize” (an actual explanation of why the recipe works)?
That – what does “to quantize” really mean –  is the true mystery of quantum mechanics.
And to answer that question we need to understand why does the world need to use a wave-function?