# Wavefunction possibilities

Do wavefunctions have to have every conceivable possibility? Say for instance you have a chair. Does the wavefunction of the chair necessarily have a possibility where the chair breaks apart spontaneously? Or a set of worlds where the chair breaks apart if MWI is true? Or can the wavefunction simply consist of possibilities where the chair does not splinter apart?

Does the wavefunction of a being have to have a possibility where the being changes their mind about something or can all possibilities of the wavefunction simply be ones where the being's mind stays the same?

ZapperZ
Staff Emeritus
Do wavefunctions have to have every conceivable possibility? Say for instance you have a chair. Does the wavefunction of the chair necessarily have a possibility where the chair breaks apart spontaneously? Or a set of worlds where the chair breaks apart if MWI is true? Or can the wavefunction simply consist of possibilities where the chair does not splinter apart?

Does the wavefunction of a being have to have a possibility where the being changes their mind about something or can all possibilities of the wavefunction simply be ones where the being's mind stays the same?

When we solve for the equation of motion using Newtonian mechanics, what we first do is account for all the forces acting on the system, i.e. we do F=ma. So say you have an object falling to the ground, you then have

$$F_g = ma$$

where $F_g$ is the force due to gravity. But if you want to include more realistic situation, you put in other facts, such as frictional force due to air friction $F_f$, and maybe the object itself has its own propulsion $F_p$. Then you write

$$F_g + F_f + F_p = ma$$

and then you solve (if you can) for the equation of motion.

The same thing occurs for the wavefunction. You first start with the Hamiltonian/Schrodinger equation. You need to know all of the potential landscape that the system has. This may or may not be trivial. In one of the simplest case, say for an infinite square well potential (which every student in intro QM classes should know), you write down the kinetic term and then the potential representing that square well. That's the whole system! So the wavefunction that you solve describes the system fully based on what you have given as the starting point, i.e. what you wrote for the kinetic and potential term.

But here's where it can get complicated, especially when you start adding complexity to the system.

1. You don't know what the exact Hamiltonian is, and so you have to make either an estimate or an approximation. This is true when you are dealing with a gazillion particles, as in condensed matter physics. It is impossible to write an exact Hamiltonian for a many-body system. So in such a case, you make some clever approximation for the potential, such as using the mean-field approximation. You say that, even though a particle in the system sees all the potential from other particles, we can simply make the approximation that, on average, it sees a constant "mean field" from all of the particles.

So your Hamiltonian will consist of the kinetic term, and a mean-field approximation of the potential term. Therefore, your wavefunction can only be as good as what you have done in the beginning. It cannot predict or describe something beyond that. In many situations, the mean-field approximation is perfectly valid and can account for a large number of phenomena. But in other situations, this approximation breaks down. It is not because the wavefunction is inadequate, it is rather our starting point and our knowledge of the system is inadequate.​

2. You know the exact Hamiltonian, but you cannot get a full, exact wavefunction. In many instances, you can write the exact wavefunction, but solving the differential equation is often a major problem. One also encounters this in classical newtonian mechanics (try to find exact, closed solution for the 3-body or more problem). This is where you either do numerical solutions, or in other cases, you make an approximate solution as a simplification, or even only consider special cases that gives you nice, analytical solutions. So obviously, it is not inconceivable that the solution could miss something when such simplifications are applied.​

So in principle, the wavefunction should be able to describe ALL of the observables as described in the Hamiltonian. It depends on how well you can construct a Hamiltonian that accurately and fully describe the system you are looking at, and how well you can arrive as the wavefunction solution.

Zz.

Do wavefunctions have to have every conceivable possibility? Say for instance you have a chair. Does the wavefunction of the chair necessarily have a possibility where the chair breaks apart spontaneously? Or a set of worlds where the chair breaks apart if MWI is true? Or can the wavefunction simply consist of possibilities where the chair does not splinter apart?

Does the wavefunction of a being have to have a possibility where the being changes their mind about something or can all possibilities of the wavefunction simply be ones where the being's mind stays the same?

QM does imply that one thing could in principle be in two different places at the same time, even macroscopic objects. so there are theories that try to explain away such possibility, like GRW

http://en.wikipedia.org/wiki/Ghirardi–Rimini–Weber_theory

from

Phys. Rev. D 34, 470–491 (1986)
Unified dynamics for microscopic and macroscopic systems

An explicit model allowing a unified description of microscopic and macroscopic systems is exhibited. First, a modified quantum dynamics for the description of macroscopic objects is constructed and it is shown that it forbids the occurrence of linear superpositions of states localized in far-away spatial regions and induces an evolution agreeing with classical mechanics. This dynamics also allows a description of the evolution in terms of trajectories. To set up a unified description of all physical phenomena, a modification of the dynamics, with respect to the standard Hamiltonian one, is then postulated also for microscopic systems. It is shown that one can consistently deduce from it the previously considered dynamics for the center of mass of macroscopic systems. Choosing in an appropriate way the parameters of the so-obtained model one can show that both the standard quantum theory for microscopic objects and the classical behavior for macroscopic objects can all be derived in a consistent way. In the case of a macroscopic system one can obtain, by means of appropriate approximations, a description of the evolution in terms of a phase-space density distribution obeying a Fokker-Planck diffusion equation. The model also provides the basis for a conceptually appealing description of quantum measurement.

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Do wavefunctions have to have every conceivable possibility? Say for instance you have a chair. Does the wavefunction of the chair necessarily have a possibility where the chair breaks apart spontaneously? Or a set of worlds where the chair breaks apart if MWI is true? Or can the wavefunction simply consist of possibilities where the chair does not splinter apart?

Yes, I think for for all reasonable physical systems every possible configuration will have a nonzero probability associated with it. For instance if you have a hydrogen atom the electron has a probability to be found literally anywhere in relation to the proton. However the probability is only non-negligible inside a very small volume of size about 10^-10 meters, and decays exponentially outside this volume. You can appreciate that while the probability of finding the electron on the other side of the room from the proton is nonzero, it is vanishingly small.

Similarly for a more complicated system like a chair the wave function should assign a nonzero probability to all possible configurations of the particles that make up the chair. However it is vanishingly unlikely that you will actually observe the particles of the chair adopt some configuration that is radically different from their current one, i.e. your chair is not going to spontaneously fall apart (absent some outside force like a sledgehammer).

Yes, I think for for all reasonable physical systems every possible configuration will have a nonzero probability associated with it. For instance if you have a hydrogen atom the electron has a probability to be found literally anywhere in relation to the proton. However the probability is only non-negligible inside a very small volume of size about 10^-10 meters, and decays exponentially outside this volume. You can appreciate that while the probability of finding the electron on the other side of the room from the proton is nonzero, it is vanishingly small.

Similarly for a more complicated system like a chair the wave function should assign a nonzero probability to all possible configurations of the particles that make up the chair. However it is vanishingly unlikely that you will actually observe the particles of the chair adopt some configuration that is radically different from their current one, i.e. your chair is not going to spontaneously fall apart (absent some outside force like a sledgehammer).

So basically there is a chance or if MWI is true there are an infinite number of universes where an army of 100 story tall pie eating sumo robots spontaneously materializes in NYC?

So basically there is a chance or if MWI is true there are an infinite number of universes where an army of 100 story tall pie eating sumo robots spontaneously materializes in NYC?

If the wavefunction of the 100 story tall pie eating sumo robots allows the state of NYC, then yes.

Interesting that in Quantum Philosophy by Roland Omnes, says:
If we consider from this perspective an ordinary object, an empty bottle, say, the quantum principles will only take into account the particles forming the botttle, and will therefore treat on an equal footing a multitude of different objects. This is due to the fact that the atoms that make up the bottle could, without changing their interactions, adopt thousands of shapes to form a thousand different objects: two smaller bottles, six wine glasses, or a chunk of melted glass. One could also seperate the atoms according to their kind and end up with a pile of sand and another pile of salt. A rearrangement of the protons and electrons to transmute the atomic nuclei without modifying the nature of their interactions could also produce a rose in a gold cup. All these variants belong to the realm of the possible, of the multitude of forms that the wave functions of a given system of paricles may take.

Yes, I think for for all reasonable physical systems every possible configuration will have a nonzero probability associated with it. For instance if you have a hydrogen atom the electron has a probability to be found literally anywhere in relation to the proton. However the probability is only non-negligible inside a very small volume of size about 10^-10 meters, and decays exponentially outside this volume. You can appreciate that while the probability of finding the electron on the other side of the room from the proton is nonzero, it is vanishingly small.

Similarly for a more complicated system like a chair the wave function should assign a nonzero probability to all possible configurations of the particles that make up the chair. However it is vanishingly unlikely that you will actually observe the particles of the chair adopt some configuration that is radically different from their current one, i.e. your chair is not going to spontaneously fall apart (absent some outside force like a sledgehammer).

I'm sort of confused what you're saying implies that the wavefunction has an infinite number of possibilities to collapse/decohere/do something else into but from reading the internet I get

Q11 How many worlds are there?
--------------------------
The thermodynamic Planck-Boltzmann relationship, S = k*log(W), counts
the branches of the wavefunction at each splitting, at the lowest,
maximally refined level of Gell-Mann's many-histories tree. (See "What
is many-histories?") The bottom or maximally divided level consists of
microstates which can be counted by the formula W = exp (S/k), where S
= entropy, k = Boltzmann's constant (approx 10^-23 Joules/Kelvin) and
W = number of worlds or macrostates. The number of coarser grained
worlds is lower, but still increasing with entropy by the same ratio,
ie the number of worlds a single world splits into at the site of an
irreversible event, entropy dS, is exp(dS/k). Because k is very small
a great many worlds split off at each macroscopic event.

Which seems to me to imply that there are a finite number of possibilities for the wavefunction to collapse/decohere/do something else into. Since MW claims that each possibility leads to another world.

ZapperZ
Staff Emeritus
Q11 How many worlds are there?***
--------------------------
The thermodynamic Planck-Boltzmann relationship, S = k*log(W), counts
the branches of the wavefunction at each splitting, at the lowest,
maximally refined level of Gell-Mann's many-histories tree. (See "What
is many-histories?") The bottom or maximally divided level consists of
microstates which can be counted by the formula W = exp (S/k), where S
= entropy, k = Boltzmann's constant (approx 10^-23 Joules/Kelvin) and
W = number of worlds or macrostates. The number of coarser grained
worlds is lower, but still increasing with entropy by the same ratio,
ie the number of worlds a single world splits into at the site of an
irreversible event, entropy dS, is exp(dS/k). Because k is very small
a great many worlds split off at each macroscopic event.

Moderator Edit: *** From "The Everette FAQ" by M.C. Price.

Do wavefunctions have to have every conceivable possibility? Say for instance you have a chair. Does the wavefunction of the chair necessarily have a possibility where the chair breaks apart spontaneously? Or a set of worlds where the chair breaks apart if MWI is true? Or can the wavefunction simply consist of possibilities where the chair does not splinter apart?

Does the wavefunction of a being have to have a possibility where the being changes their mind about something or can all possibilities of the wavefunction simply be ones where the being's mind stays the same?

Yes. the wave function takes into account every conceivable possibility. Parts of that wave function will project items into deep space. It's not that it is there physically... only there as a possibility, which is to add, very small indeed.

Perhaps a clairification is needed:

Does a quantum system (macroscopic in this case) have an infinite or finite amount of physical possiblities that can actualise upon measurement?

Perhaps a clairification is needed:

Does a quantum system (macroscopic in this case) have an infinite or finite amount of physical possiblities that can actualise upon measurement?

Yes, but the wavelength of matter is exceedingly small on large enough scales. Even your homework jotter will be statistically sitting as a possibility on the surface of venus as an extreme example. The only reason why it is not sitting there, is again, highly unlikely.

I guess what I'm really asking is whether there are infinite or finite possibilites.

Sorry I wasn't clear enough earlier

I guess what I'm really asking is whether there are infinite or finite possibilites.

Sorry I wasn't clear enough earlier

Same question. Everyone here seems to be leaning toward infinite but the FAQ author seems to think it is finite.

I wonder if there will be an answer anytime soon?

I wonder if there will be an answer anytime soon?

Most agree it to be infinite. It's probabilities are spread infinitely throughout spacetime.

Most agree it to be infinite. It's probabilities are spread infinitely throughout spacetime.

Okay so the FAQ is wrong

Okay so the FAQ is wrong

Fields where designed in the sense they ''needed to touch'' vast areas. Most of the fields we deal with in quantum mechanics are infinite by nature.

The wave function is also a field, and can also be infinite by nature. It is a field of infinite possibilities, or a field representive of the probabilities of events. It must be infinite in many cases. A particles possible location is not situated to a small area, but has a range which scopes from $$-\infty$$ to $$\infty$$. That means the wave function appreciates even the most unlikely of scenarios.

A. Neumaier
Okay so the FAQ is wrong

The quality of the Everett FAQ is very poor. See the section ''On the Many-Worlds-Interpretation'' of Chapter A4 of my theoretical physics FAQ at http://www.mat.univie.ac.at/~neum/physfaq/physics-faq.html#manyworlds

In particular, the entropy argument used in the answer to Q11 is funny since it implies a fractional number of worlds unless the ensemble of worlds is microcanonical. But then each world is equally probable, and we must be puzzled why we are in a world where the unlikely happens rarely...

The usual entropy formula from statistical mechanics employed only counts the number of energetically accessible energy eigenstates (not the number of all possible states at a given energy, which is infinite), and is applicable only to a bounded volume of matter in equilibrium.

But the many worlds interpretation must consider the whole universe as the physical system, and the latter is neither in equilibrium nor (most likely) bounded.

So the wavefunctions physical states are infinite - by physical states I mean states like a chair, or a computer?

So the wavefunctions physical states are infinite - by physical states I mean states like a chair, or a computer?

Well that is difficult to say. The wave length of matter at our levels of macroscopic objects are so small, that we don't even see a wave function which is physically projected through space, however an ethereal wave function exists for all objects, even your own body. Physical projections of possibilities have been observed though but they are very small objects which are not free from quantum effects.

So theoretically it would be possible to teleport from one side of the universe to the other instantaneously.

however an ethereal wave function exists for all objects, even your own body. Physical projections of possibilities have been observed though but they are very small objects which are not free from quantum effects.

An ethereal wave function? Meaning?

Again my favourite paragraph from Quantum Philosophy by Roland Omnes:
If we consider from this perspective an ordinary object, an empty bottle, say, the quantum principles will only take into account the particles forming the botttle, and will therefore treat on an equal footing a multitude of different objects. This is due to the fact that the atoms that make up the bottle could, without changing their interactions, adopt thousands of shapes to form a thousand different objects: two smaller bottles, six wine glasses, or a chunk of melted glass. One could also seperate the atoms according to their kind and end up with a pile of sand and another pile of salt. A rearrangement of the protons and electrons to transmute the atomic nuclei without modifying the nature of their interactions could also produce a rose in a gold cup. All these variants belong to the realm of the possible, of the multitude of forms that the wave functions of a given system of paricles may take.

Always wondered what was meant by 'forms that the wave functions of a given system of particles may take' until I realised that a wave function can have one solution to it, i.e. two smaller bottles, and another can have the solution, i.e. in this case six wine glasses, which can add together to form another wave function of a superposition of two smaller bottles + six wine glasses.

An ethereal wave function? Meaning?

Again my favourite paragraph from Quantum Philosophy by Roland Omnes:

Always wondered what was meant by 'forms that the wave functions of a given system of particles may take' until I realised that a wave function can have one solution to it, i.e. two smaller bottles, and another can have the solution, i.e. in this case six wine glasses, which can add together to form another wave function of a superposition of two smaller bottles + six wine glasses.

I need to be careful now because I realize the word I choose could be misconstruded as to mean something else. It was just me trying to be over-elegant.

When we talk about probabilities, we tend to wonder what we mean. Probabilities are things which happen inside our heads. Probabilities are the world of mind-stuff. This is not to mean that somehow the world is created mentally, but in many ways this part of quantum mechanics mirrors this fascinating fact rather well. The brain is physical, thoughts seem a lot less physical, almost ethereal. Thoughts or probabilities inside our heads don't objectively exist in the outside world. Physical probabilities may exist in the objective world.

This is why when the wave function was formulated, many scientists in the beginning thought that wave function was merely a statistical way for the scientists mind to make sense of an otherwise, evading reality of possibilities.

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So theoretically it would be possible to teleport from one side of the universe to the other instantaneously.

I'm not entirely sure how the subject of teleportation has arisen, as that would require to some theoreticians, the use of entangled particles.

However, with that said, there are many philosophical arguements rooted from the mathematics of such theories which cast doubt on whether teleportation is possible. Is a newly created, (new matter) but otherwise completely identical twin of an object be actually teleported? If you teleport information about a system and reconfigure those atoms into a complete duplicate, who is to say that it is the same object in question? All you have done is read from a recipe book and replicated your mothers apple pie. And the consciousness is not fully understood either... Personally I do not believe you can entangle particles over large distances and teleport something as complex as a human being. My consciousness inhabits the atoms in my body. Not the entangled states of particles over large distances.

Maybe you would like to make a post on the subject to see what others think.

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wouldn't the question of infinite/finite have something to do with whether space is actually discrete or continuous or bounded/unbounded?