Interpreting Quantum Mechanics

[sayak83]

There has been much discussion on quantum mechanics in this forum. This thread is an attempt to understand, in a scientifically accurate fashion, some of the ideas and perplexities behind quantum mechanics. Then we can have a more informed discussion on what quantum mechanics implies philosophically, metaphysically or spiritually.
The mathematics of quantum mechanics is relatively simple (far simpler than General Relativity in my view). The results you get out of the mathematics is also accurate when we tally it to what is observed in experiments or in the natural world. The problem comes in interpreting the mathematics in a physically meaningful manner.

In the mathematics of quantum mechanics we have mathematical entities called wavefunctions (or state-vectors), that in some sense incorporate all possible measurement outcomes of the given quantum system state. When we do an experiment, we are trying to measure a specific property of the system (like say location or say momentum or say energy etc.). Depending on which property we want to measure, there exists a corresponding mathematical function called the operator function. When the wavefunction is "fed" into that specific operator as an "input", the mathematical outcome is a set of property values and the individual probability that the experiment will measure each of these particular property values. Thus,

Location-operator{System Wavefunction} = Set of location values and their individual probability of realization.
or
Energy-operator {System Wavefunction} = Set of Energy values and their individual probability of realization.

The issue comes is how to understand the physics behind these mathematical operations, and that is where the debate is. I will try to discuss this over the next posts.

 

[Polymath257]

I would also point out that the modern versions have quantum 'fields' for each type of particle. And, among the operators that can be used is a 'number' operator that gives the number of possible particles and the probability of finding that number of particles when observed. This is particularly important when we consider antimatter, where particles can appear and disappear in the course of an interaction.

 

[LegionOnomaMoi]

In the mathematics of quantum mechanics we have mathematical entities called wavefunctions (or state-vectors), that in some sense incorporate all possible measurement outcomes of the given quantum system state. When we do an experiment, we are trying to measure a specific property of the system (like say location or say momentum or say energy etc.). Depending on which property we want to measure, there exists a corresponding mathematical function called the operator function. When the wavefunction is "fed" into that specific operator as an "input", the mathematical outcome is a set of property values and the individual probability that the experiment will measure each of these particular property values.

This is not what happens even in the special case of pure states and even in the Schrödinger picture (in the Heisenberg picture, much of information encapsulated by the operator valued measures for wavefunctions is instead encoded in the system's state itself, which is an infinite dimensional matrix). No matter the scheme or where (or whether) one describes the time evolution of the system, the general formalisms for pure states share some basic ingredients. The specification of the state in general encodes information about the experimental arrangement including measurements that are interpreted or taken to be preparations of the state of the system. This is connected to the results of measurement by (positive)-operator valued transformations that are subsets of the relevant noncommutative algebra of observables.
Importantly, there is not in general an operator corresponding to "which property we want to measure" but a set of operators corresponding to the manner of preparation and limited by the manner and the nature of the system that correspond to the specifications of system preparation and the manner in which we can measure the prepared system. Thus, for example, while we can "prepare" an electron with a specified spin direction, this measurement cannot be carried out on a photon because it will destroy the system (the photon will be absorbed).
Also, since for the most part the distributions of possible outcomes are mixed (that is, they are neither absolutely continuous nor are they discrete), for most of the possible outcomes the probability is 0, just as in the case of elementary probability theory with density functions. In any event, it is not in general true that when we do an experiment the possible outcomes are encoded in the observable operator which corresponds toe the "property we want to measure". The specification of the system determines in a large part the manner of measurement and the possible outcomes as the experimental setting is (at least partially- again, it depends on the formalism scheme chosen) encoded in the state of the system. Most importantly, we cannot associate in general any ontological status to the system beyond that of a mathematical entity that doesn't really exist, nor can we say that the observable operators in some sense are representations of measurements or the measured properties or the possible measured properties.

 

[Brickjectivity]

BASICS as I understand them?

At the quantum level 'Particle' loses its meaning. Even 'Packet' loses its meaning. A quantum particle is more like an excited hamster in a room full of hamster wheels. Its like a twin children let loose in a theme park.
---> This may have implications for humanity. What does 'Solid' mean? What does 'Real' mean?

Ok, so to start break it down from atoms. Are all quantum particles part of an atom? Surprisingly, no. Not all quantum particles are part of nor are contained within an atom, and some never have been or will be part of atoms -- as far as is known. A lot of quantum particles never impact our physical lives as far as anyone can tell and would not even interact with atoms, no matter what.
---> This may have implications for humanity. Why aren't all quantum particles serving us? Are we irrelevant to them, or does it merely look that way?

An atom is thought to be composed of Protons, Neutrons and Electrons. These are, it is thought, composed of particular kinds of quantum particles; but particles aren't particles at the scales we are speaking of. They don't occupy just one place necessarily.
--> The existence of atoms has implications for humanity. Matter is not composed of only 5 things (as many people once thought). You really cannot turn lead into gold using chemistry or alchemy. You really can have something alive that doesn't have an active principle in it. There really is a sense to how things function, and an order to transformation.

A quantum particle can't be seen with light. Its either because its too small, too fast or doesn't have a specific location. It is seen only by theoretical models and detected with equipment. It is only partly understood using formulas that partially describe its behavior, and these formulas always, always involve Calculus. To truly understand these formulas you need to be able to deal with partial differential equations, matrices and some statistics. Otherwise just leave the room.
--> This, too, has implications for humanity. We are at the point in which some of the particles which make up our universe can only be understood after much study. Quantum particles are like nothing in our daily experience. Its difficult to believe in them and to understand how they behave or why they do. You can't even do Chemistry with them like you can with atoms. They're as far away from most of us as the stars.

 

[sayak83]

This is not what happens even in the special case of pure states and even in the Schrödinger picture (in the Heisenberg picture, much of information encapsulated by the operator valued measures for wavefunctions is instead encoded in the system's state itself, which is an infinite dimensional matrix). No matter the scheme or where (or whether) one describes the time evolution of the system, the general formalisms for pure states share some basic ingredients. The specification of the state in general encodes information about the experimental arrangement including measurements that are interpreted or taken to be preparations of the state of the system. This is connected to the results of measurement by (positive)-operator valued transformations that are subsets of the relevant noncommutative algebra of observables.
Importantly, there is not in general an operator corresponding to "which property we want to measure" but a set of operators corresponding to the manner of preparation and limited by the manner and the nature of the system that correspond to the specifications of system preparation and the manner in which we can measure the prepared system. Thus, for example, while we can "prepare" an electron with a specified spin direction, this measurement cannot be carried out on a photon because it will destroy the system (the photon will be absorbed).
Also, since for the most part the distributions of possible outcomes are mixed (that is, they are neither absolutely continuous nor are they discrete), for most of the possible outcomes the probability is 0, just as in the case of elementary probability theory with density functions. In any event, it is not in general true that when we do an experiment the possible outcomes are encoded in the observable operator which corresponds toe the "property we want to measure". The specification of the system determines in a large part the manner of measurement and the possible outcomes as the experimental setting is (at least partially- again, it depends on the formalism scheme chosen) encoded in the state of the system. Most importantly, we cannot associate in general any ontological status to the system beyond that of a mathematical entity that doesn't really exist, nor can we say that the observable operators in some sense are representations of measurements or the measured properties or the possible measured properties.

Good to see you after a long time.

 

[sayak83]

There has been much discussion on quantum mechanics in this forum. This thread is an attempt to understand, in a scientifically accurate fashion, some of the ideas and perplexities behind quantum mechanics. Then we can have a more informed discussion on what quantum mechanics implies philosophically, metaphysically or spiritually.
The mathematics of quantum mechanics is relatively simple (far simpler than General Relativity in my view). The results you get out of the mathematics is also accurate when we tally it to what is observed in experiments or in the natural world. The problem comes in interpreting the mathematics in a physically meaningful manner.

In the mathematics of quantum mechanics we have mathematical entities called wavefunctions (or state-vectors), that in some sense incorporate all possible measurement outcomes of the given quantum system state. When we do an experiment, we are trying to measure a specific property of the system (like say location or say momentum or say energy etc.). Depending on which property we want to measure, there exists a corresponding mathematical function called the operator function. When the wavefunction is "fed" into that specific operator as an "input", the mathematical outcome is a set of property values and the individual probability that the experiment will measure each of these particular property values. Thus,

Location-operator{System Wavefunction} = Set of location values and their individual probability of realization.
or
Energy-operator {System Wavefunction} = Set of Energy values and their individual probability of realization.

The issue comes is how to understand the physics behind these mathematical operations, and that is where the debate is. I will try to discuss this over the next posts.

Let me flesh out the mathematics a bit more to show how (very crudely) the wavefunction, when fed into an operator, gives the property value and its probabilities.
Let us assume that we have a quantum system and we want to measure a specific property, say color, of the system. Let us further say that we can observe only three colors: Red (R), Blue (B) or Yellow (Y). Let us also suppose that the experiments show that we get Red 40% of times, Yellow 25% of time and Blue 35% of time. Then the quantum wavefunction describing this quantum system would be (roughly) of the form,
System 1 Color WaveFunction = Sqrt(0.4)*Red + Sqrt(0.35)*Blue + Sqrt(0.25)*Yellow.
Where Sqrt means "the square root of". Thus each of the possible property values has a factor in front that is the square root of the probability of observing that specific property value during the experiment and then they are combined (added or subtracted) together to give the total wavefunction.

Now, suppose, you have another quantum system whose state is slightly different such that its wavefunction is given by,
System 2 Color Wavefunction = Sqrt(0.4)*Red + Sqrt(0.35)*Blue - Sqrt(0.25)*Yellow
Note that the probability of getting each of the colors is the same as before (40%, 35% an 25%) but the Yellow property value is combined through a subtraction. (This does not make sense for color, but makes sense for properties that have directions as well as magnitude...say the orientation of a magnetic field etc.)

Now suppose the two quantum systems are allowed to interact with each other (interfere) during the experiment before we measure them. Then we get a combined system whose wavefunction is the probability-weighted linear combination of the two. Thus,
Combined System Color Wavefunction = [(System 1 Color Wavefunction) + (System 2 Color Wavefunction)] / (Probability Weights) = 0.73*Red + 0.68*Blue = sqrt (0.51)*Red + Sqrt(0.49)*Blue

Note that the positive and negative Yellow wavefunction components cancel out by the addition. So we only get the Red and Blue terms in the Combined Wave. This is an example of destructive interference. Even though, individually each system had a chance of 25% outcome as Yellow, when combined, the yellow color is never observed, and we get a roughly equal chance of observing Red or Blue. (The exact calculation of Red and Blue probability is unimportant here. Don't worry about it).

This gives you a clue as to why the function is called a wavefunction, as it can combine and interfere like a wave...but puzzlingly the thing that is interfering is the square root of the probability associated with the measurable property values. The issue then is how to physically interpret such a strange type of mathematics.

 

[LegionOnomaMoi]

Let me flesh out the mathematics a bit more to show how (very crudely) the wavefunction, when fed into an operator, gives the property value and its probabilities.

Your example doesn't have operators. Nor does it address one of the absolutely central issues in quantum theory: probabilities aren't calculated as you do or as mathematicians and everybody else does but are derived even in the simplest cases in a manner unique to quantum theory. In the textbook QM form, probabilities are calculated from the (complex-valued) amplitude, whence the interference terms.

Finally, there is already a very well-known, intuitive explanation for quantum effects in terms of colors by David Albert, beginning with superposition:
Quantum Mechanics and Experience Chapter 1: Superposition
Albert also uses hardness, but the point is that he illustrates in a very simple way something of the counter-intuitive nature of quantum mechanics. For visual learners I should point out that the first lecture of MIT OCWs course on QM begins using Alberts' example (after the course intro preamble) :

 

[sayak83]

Your example doesn't have operators. Nor does it address one of the absolutely central issues in quantum theory: probabilities aren't calculated as you do or as mathematicians and everybody else does but are derived even in the simplest cases in a manner unique to quantum theory. In the textbook QM form, probabilities are calculated from the (complex-valued) amplitude, whence the interference terms.

Finally, there is already a very well-known, intuitive explanation for quantum effects in terms of colors by David Albert, beginning with superposition:
Quantum Mechanics and Experience Chapter 1: Superposition
Albert also uses hardness, but the point is that he illustrates in a very simple way something of the counter-intuitive nature of quantum mechanics. For visual learners I should point out that the first lecture of MIT OCWs course on QM begins using Alberts' example (after the course intro preamble) :

I am not providing an exact mathematical exposition of Quantum Mechanics, but a simplified "toy" maths that fleshes out some of the features of quantum mechanics. Anybody can get at the math by reading a textbook. So that's not my intention.

 

[dad]

In the mathematics of quantum mechanics we have mathematical entities called wavefunctions (or state-vectors), that in some sense incorporate all possible measurement outcomes of the given quantum system state. .

I suspect that outcomes depend on man and God. Science does not use this in it's math. They do seem to realize that a lot seems to depend on someone observing things. They also seem to suspect that time and space seem to possibly be invalidated in entanglement.

 

[leroy]

My prediction is that people from the future (say in 20 years) will laugh at us for inventing (and taking seriously) so many weird interpretations of QM

1 cats are alive and dead at the same time

2 new universes simply branch and come in to existance all the time.

3 stuff travels faster than light

4 observers play a role

Etc.

My prediction is that some day a boring coherent and simple interpretation will be proven to be correct, and Einstein will be proven to be correct all along

 

[LegionOnomaMoi]

I am not providing an exact mathematical exposition of Quantum Mechanics, but a simplified "toy" maths that fleshes out some of the features of quantum mechanics. Anybody can get at the math by reading a textbook. So that's not my intention.

I get that. The textbooks also use toy models as well. The issue is whether or not the toy model (regardless of how simple or how complex) fleshes out something of the conceptual nature of quantum theory, particularly when the goal concerns interpretation. Probability theory is one way to get at the heart conceptual and interpretative issues in quantum mechanics. Consider the two-slit experiment. We have two mutually exclusive possibilities: some photon (or electron) can arrive at a detection plate/film via one of two routes we will call slit A and slit B. These are two mutually exclusive possibilities. We are interested in using QM to predict where the "particles" will land, and we use the formalism to encode information about the experimental arrangement as well as what we wish to predict (namely, the probabilities associated with position). According to probability, theory, if something can happen one of two ways (either a "particle" arrives at the detection plate via route A or it does via route B), then we add the probabilities. In QM, this will lead to the wrong answer. Instead, we must calculate probabilities using the amplitudes, which means that we end up adding the probability that a "particle" arrives via route A plus the probability that it arrives via route B plus the interference term. This is also where complex numbers are shown to be crucial (another departure from both classical physics and probability theory).
With respect to observables, because the set of observables associated with a given system consist of a noncommutative operator algebra yielding statistical predictions, there is no common measure space (nor corresponding joint probability density) for incommensurable observables.
The bottom line, though, is that a toy model should highlight key aspects of the theory. A fundamental aspect of QM is that probabilities can't typically be calculated as you do above nor in accordance with probability theory.

 

[LegionOnomaMoi]

My prediction is that people from the future (say in 20 years) will laugh at us for inventing (and taking seriously) so many weird interpretations of QM

1 cats are alive and dead at the same time

This isn't taken seriously, nor was it intended to. Schrödinger began his example by calling it ridiculous.

2 new universes simply branch and come in to existance all the time.

Yep. This one is nonsense. It didn't start out that way, but current many-worlds interpretations are to me quite desperate attempts.

3 stuff travels faster than light

No part of QM predicts this. Nothing travels faster than light in quantum theory.

4 observers play a role

This isn't actually that bizarre. In classical physics, we made belief we could paradoxically perfectly isolate a system and somehow observe it. Perfect isolation, if it were possible, would preclude observation/measurement. QM just forced us to confront the fact that we cannot forever both extrapolate from our physical theories and also continue to pretend they need not apply to us as the ones observing. We aren't gods, and classical physics is based on a theological model in which the laws come from an omniscient entity outside of the universe (originally God, but then immutable mathematical laws and principles).

My prediction is that some day a boring coherent and simple interpretation will be proven to be correct, and Einstein will be proven to be correct all along

Einstein was shown to be correct. In fact, his contribution to nonlocality was key. The issue is that he left out a key assumption he was making that he took for granted, and he didn't want to be correct in the manner in which he turned out to be.

 

[dad]

No part of QM predicts this. Nothing travels faster than light in quantum theory..

I see one site explain it like this

"So, while the process of disentanglement happens instantaneously, the revelation of it does not. We have to use good old-fashioned no-faster-than-light communication methods to piece together the correlations that quantum entanglement demands. Thus, Einstein's universal speed limit is preserved, and so is the fundamentally quantum worldview."
Quantum Weirdness May Seem to Outrun Light — Here's Why It Can't | Space

So if a process happens to two particles far apart in the universe instantly, who really cares how long it takes some slow poke to measure it or see it?

 

[ChristineM]

1 cats are alive and dead at the same time

A real world example

 

[LegionOnomaMoi]

So if a process happens to two particles far apart in the universe instantly, who really cares how long it takes some slow poke to measure it or see it?

The process in question only happens via measurement. For instance, in the typical examples/tests of entanglement/Bell inequalities/EPR/etc. one has e.g., paired photons or a singlet state of anti-correlated "particles" from a decayed atom with total spin determined by the atom they decayed from.
So, going back to your rhetorical question, before one slow poke measures the polarization of one of the paired photons or the spin of one of the "particles", there is no such property. Once said slow poke measures the polarization of one of the paired photons or the spin of one of the "particles", then the state of the system is determined as a result of measurement. This means the entire state, which includes the spin/polarization of the other "particle"/photon.
Before the slow poke performs a measurement, no definite process or property exists. If it did, this would just be trivial correlation, not nonlocality.

 

[dad]

The process in question only happens via measurement. For instance, in the typical examples/tests of entanglement/Bell inequalities/EPR/etc. one has e.g., paired photons or a singlet state of anti-correlated "particles" from a decayed atom with total spin determined by the atom they decayed from.
So, going back to your rhetorical question, before one slow poke measures the polarization of one of the paired photons or the spin of one of the "particles", there is no such property. Once said slow poke measures the polarization of one of the paired photons or the spin of one of the "particles", then the state of the system is determined as a result of measurement. This means the entire state, which includes the spin/polarization of the other "particle"/photon.
Before the slow poke performs a measurement, no definite process or property exists. If it did, this would just be trivial correlation, not nonlocality.

Hmm. Well, if it did exist then well, it did exist. If the particles were far apart, that would mean light speed would take a certain time to go the distance. Let's say that particles are a billion light years apart. If the particles were entangled, the change in both particles would be instant. Now whether you could see the change or not doesn't matter. In any case light would take a billion years to get there! It seems to me instant is faster.

Now you claim that unless someone was out there to look at the far away entangled particle, that "no definite process or property exists". How would you know since you were not there?

Let's say we had an observer stationed a billion light years away where the entangled particle was. The earth observer reports that a change happened, and you now see the change also in this far away entangled particle. So now we have a confirmed observation of an entangled particle a billion light years away instantly changing. You could not say that no definite property existed. Even if the observer was not there, maybe that far away particle that was entangled still instantly changed. You could not say it did not if you were not there.

You could not say that the change did not happen faster than C.

 

[LegionOnomaMoi]

Hmm. Well, if it did exist then well, it did exist.

If a property such as spin (or any other observable) existed in some sense independently of measurement, then it wouldn’t be nonlocal and would be classical in nature. For example, we could easily explain how a spin-0 particle decays into a pair of anticorrelated spin-1/2 particles if we could say that the direction of their spins existed independently of measurement. We might imagine how the original atom decays into two component pieces that are oriented in opposite directions due to their common origin. As they separate, they remain oppositely oriented indefinitely so long as they aren’t disturbed. Once we measure one, we would instantly know the state of the other without any nonlocality or anything non-classical happening.

The problem is that we can show (and indeed this is central to understanding the Bell inequality) that no such common, local cause underlying entangled states is possible. Before we measure the spin of one of the particles, it possesses no definite spin. Afterwards, we determine the spin state of the system which (due to the nonseperable nature of the pair) determines not only the spin of the measured particle but also that of its “partner”.

However, the same formalism that allows us to predict that this entanglement happens and to confirm that it does also allows us to prove that it cannot involve in any manner whatsoever so much as a single bit of information (let alone some other physical process) travelling at any speed, let alone faster than light.

How would you know since you were not there?

The same way in which we can know there is such a thing as entanglement at all (or anything else related to quantum mechanics or physics more generally). The formalism. Einstein and co-authors first wrote about this phenomenon in 1935, and Bohm formulated the most frequently used form in his 1951 textbook. This was long before it could be empirically confirmed. Just as we have now confirmed experimentally what the formalism predicted with respect to nonlocality or entanglement, so too have we confirmed what the formalism dictates concerning the impossibility of any superluminal processes underlying it or anything else in quantum theory.

Let's say we had an observer stationed a billion light years away where the entangled particle was. The earth observer reports that a change happened, and you now see the change also in this far away entangled particle. So now we have a confirmed observation of an entangled particle a billion light years away instantly changing.

There is no change. If two observers independently try to measure the system then it will not be entangled. The observers have to agree to measure it in a particular manner before time in order to observe the correlations. It is only through the quantum formalism and through repeated experiments that we can say that the correlations in fact exist in the manner predicted and then attribute them to the properties of the entangled system, but this same formalism shows that there is no faster-than-light travel nor is it possible to say that what we observe existed before we observe it. We can never observe anything like the change you speak of. It was in an attempt (one that ultimately failed) to partially circumvent this limit that EPR was written and entanglement first described, but even then it was known that such observations of changes were impossible.

 

[dad]

If a property such as spin (or any other observable) existed in some sense independently of measurement, then it wouldn’t be nonlocal and would be classical in nature.

So what?

For example, we could easily explain how a spin-0 particle decays into a pair of anticorrelated spin-1/2 particles if we could say that the direction of their spins existed independently of measurement.

Right, so your powers of prediction are limited.

We might imagine how the original atom decays into two component pieces that are oriented in opposite directions due to their common origin. As they separate, they remain oppositely oriented indefinitely so long as they aren’t disturbed. Once we measure one, we would instantly know the state of the other without any nonlocality or anything non-classical happening.

That is one possible parable, but you can't really check the universe a billion light years away and check can you?

The problem is that we can show (and indeed this is central to understanding the Bell inequality) that no such common, local cause underlying entangled states is possible. Before we measure the spin of one of the particles, it possesses no definite spin. Afterwards, we determine the spin state of the system which (due to the nonseperable nature of the pair) determines not only the spin of the measured particle but also that of its “partner”.

In earth labs, yes. So can you demonstrate that the same is true in all the universe?

However, the same formalism that allows us to predict that this entanglement happens and to confirm that it does also allows us to prove that it cannot involve in any manner whatsoever so much as a single bit of information (let alone some other physical process) travelling at any speed, let alone faster than light.

As long as we are talking about very local experiments, this may be true. But I do not think it explains all that much. (like why it is probable)
For example, if man was created to be able to affect the particles and make them do certain things, then it would not be random. It would have been dependent on what he wanted. So your experiments, if you had been in Eden, would not show anything different than what we see now. However, what we would not have seen is that there was no real probability involved so much as a flexibility for particles to obey the observer Adam! To you, it would appear as random probability. Or, if God willed that particles do something, they would then start acting accordingly, such as possibly when He created the world. Nothing really random or probable about it. The reason it looks that way to you is because you don't know how and why it works and exists.

The same way in which we can know there is such a thing as entanglement at all (or anything else related to quantum mechanics or physics more generally). The formalism. Einstein and co-authors first wrote about this phenomenon in 1935, and Bohm formulated the most frequently used form in his 1951 textbook. This was long before it could be empirically confirmed. Just as we have now confirmed experimentally what the formalism predicted with respect to nonlocality or entanglement, so too have we confirmed what the formalism dictates concerning the impossibility of any superluminal processes underlying it or anything else in quantum theory.

In your lab, yes. Anywhere else?

There is no change. If two observers independently try to measure the system then it will not be entangled.

You can't say that if you have not been a billion light years out there and observed. You can say that for earth labs. You have only one observer and he lives on and observes here on earth.

The observers have to agree to measure it in a particular manner before time in order to observe the correlations. It is only through the quantum formalism and through repeated experiments that we can say that the correlations in fact exist in the manner predicted and then attribute them to the properties of the entangled system, but this same formalism shows that there is no faster-than-light travel nor is it possible to say that what we observe existed before we observe it. We can never observe anything like the change you speak of. It was in an attempt (one that ultimately failed) to partially circumvent this limit that EPR was written and entanglement first described, but even then it was known that such observations of changes were impossible.

The idea of 'formalism' should also be looked at when attempting to apply it to things quantum.

But thanks for the well thought out reply.

 

[Polymath257]

Let me flesh out the mathematics a bit more to show how (very crudely) the wavefunction, when fed into an operator, gives the property value and its probabilities.
Let us assume that we have a quantum system and we want to measure a specific property, say color, of the system. Let us further say that we can observe only three colors: Red (R), Blue (B) or Yellow (Y). Let us also suppose that the experiments show that we get Red 40% of times, Yellow 25% of time and Blue 35% of time. Then the quantum wavefunction describing this quantum system would be (roughly) of the form,
System 1 Color WaveFunction = Sqrt(0.4)*Red + Sqrt(0.35)*Blue + Sqrt(0.25)*Yellow.
Where Sqrt means "the square root of". Thus each of the possible property values has a factor in front that is the square root of the probability of observing that specific property value during the experiment and then they are combined (added or subtracted) together to give the total wavefunction.

Now, suppose, you have another quantum system whose state is slightly different such that its wavefunction is given by,
System 2 Color Wavefunction = Sqrt(0.4)*Red + Sqrt(0.35)*Blue - Sqrt(0.25)*Yellow
Note that the probability of getting each of the colors is the same as before (40%, 35% an 25%) but the Yellow property value is combined through a subtraction. (This does not make sense for color, but makes sense for properties that have directions as well as magnitude...say the orientation of a magnetic field etc.)

Now suppose the two quantum systems are allowed to interact with each other (interfere) during the experiment before we measure them. Then we get a combined system whose wavefunction is the probability-weighted linear combination of the two. Thus,
Combined System Color Wavefunction = [(System 1 Color Wavefunction) + (System 2 Color Wavefunction)] / (Probability Weights) = 0.73*Red + 0.68*Blue = sqrt (0.51)*Red + Sqrt(0.49)*Blue

Note that the positive and negative Yellow wavefunction components cancel out by the addition. So we only get the Red and Blue terms in the Combined Wave. This is an example of destructive interference. Even though, individually each system had a chance of 25% outcome as Yellow, when combined, the yellow color is never observed, and we get a roughly equal chance of observing Red or Blue. (The exact calculation of Red and Blue probability is unimportant here. Don't worry about it).

This gives you a clue as to why the function is called a wavefunction, as it can combine and interfere like a wave...but puzzlingly the thing that is interfering is the square root of the probability associated with the measurable property values. The issue then is how to physically interpret such a strange type of mathematics.

A couple of comments. First, it should be made clear that this is the decomposition of the wave function *for color*. The Red, Blue, and Yellow of your examples are known as the 'color eigenvectors' and we can decompose *any* wave function into those.

But, if we want to measure strength (say), we have to write that same wave function in *strength* eigenvectors, so we have to write each of the color eigenvectors, Red, Blue, Yellow in terms of the strength eigenvectors, Strong, Weak, Puny.

So, we may have Red=sqrt{.2} Strong -sqrt{.5} Weak +sqrt(.3) Puny, with similar expressions for Blue and Yellow. Then we *rewrite* Our wave functions above in terms of these to get the probabilities for that wave function for Strength.

This is where a LOT of the strangeness of QM comes from. There is more than one way to write the *same* wave function, depending on what you want to measure. And, in fact, for each observable operator, we get a different set of such eigenvectors.

For observables that "don't commute", there is NEVER a complete list of eigenvectors for both at the same time. This leads to the uncertainty principle.

And that gets to my second point. The decomposition chosen depends on the observable you want to measure. Those coefficients of the eigenvectors give the probabilities for that observable. But if you want to measure a different observable, you need to rewrite in terms of the eigenvectors for *that* new observable to get its probabilities.

There is a special place for the observable that corresponds to energy, which is known as the Hamiltonian. Schrodinger's equation is the equation for the eigenvectors of the energy operator.

 

[gnostic]

I could almost agree with everything you have said except this part here, especially what I highlighted in bold:

Then we can have a more informed discussion on what quantum mechanics implies philosophically, metaphysically or spiritually.

Since there are real world scientific application in Quantum Mechanics, so I don’t think QM in terms of philosophy (including metaphysics) or spirituality.