Question about Probability

[fredrik]

How is our being unable to measure all of the electron in the electron shell not a limitation of measurement? Didn't we just agree that our method of measurement collapses the electron?

 

I understand that you are out for some kind of philosophical understanding, and I'd say there there are probably *several* equivalent ways to view this.

 

a) Loosely speaking, it is on one hand clear that our measurement is a key, but it the uncertainty is not due to our ignorance to device a better more accurate measurement. It has do with with that there is a deeper connection between, measurement/interaction, and reality. An interaction/measurement by definition changes reality. Each measurement "pokes" the reality a bit.

 

I'm going to come up with a really silly analogy, so don't taken it litteraly but maybe it explains part of my point:

 

Suppose there is (as per some speculations) at some very instant a very tasty cookie somewhere in the universe. You (choose) ask yourself two questions

 

1) *Is* there really here a cookie here Now? or is the cookie an illusion?

 

2) Is the cookie really *tasty*?

 

How to find out? To be absolutely confident about (2) you may need to eat the cookie, but once you eat the cookie it doesn't exist anymore, or more specifically you loose track of when it existed, because of the time it takes to eat it. So maybe you think you could eat part of the cookie? But then, perhaps the other half of the cookie doesn't taste as well?

 

b) Quantum mechanics suggest instrinsic uncertainties between various variables, and they are called conjugates variables. Exaples are

 

momentum and position

energy and time

 

It means you can not with 100% certainty know both momentum and position at the same time.

 

Same goes with energy and time.

 

I the axiomatic approach to QM, this can be viewed simply as a consequence of how the time and momemtum measurements are *defined*

 

c) Another way to try to "understand", is that if we consider a particle to have a wave-particle duality, wether a standing wave or a wave packet (doesn't matter), we know from fourier analysis, that a wave propagation in can be written as a superposition of harmonics.

 

An implications it that it is impossible to for example have a localised pulse, that consists of only once frequency. A pulse requires a distribution of various frequencies to get it's shape.

 

In quantum mechanics and in particular the wave particle duality, the frequency or wavelenght of the wave is associate with momentum. Meaning that in order for a particle to be well localized in space (be a finely contained wave packet), this wave packet is made up from many many different momenta.

 

The fourier transform decomposes an arbitrary wave packet into it's momemum components.

 

So the conjugate variables are

 

By the same token, a system with a well defined energy has an uncertainty in time, because the energy components build up the time position just like harmonics build up a wave packet.

 

I'm not sure if it was the answer you were looking for, but it's an attempt to explain how come the uncertainty is not because of plain ignorance, but rather the definition of variables. Or put another way, some variable just aren't independent, and that's how determining one, does affect that other one, or at least our knowledge of the other one, and there is btw, no difference in this case. Because there is no other way to know that to interact.

 

/Fredrik

[gib65]

b) Quantum mechanics suggest instrinsic uncertainties between various variables, and they are called conjugates variables.

 

I always knew there was an inevitable uncertainty between position and momentum, but I didn't know there were many such pairings called "conjugate variables". Interesting.

 

What other conjugate variable pairs are there? Membrain was asking about mass as a property of a particle that one can measure, and swansont answered that there is no uncertainty about mass measurements (or charge). Does this mean we can rule out mass and charge as conjugate variable with anything else?

[swansont]

I always knew there was an inevitable uncertainty between position and momentum, but I didn't know there were many such pairings called "conjugate variables". Interesting.

 

What other conjugate variable pairs are there? Membrain was asking about mass as a property of a particle that one can measure, and swansont answered that there is no uncertainty about mass measurements (or charge). Does this mean we can rule out mass and charge as conjugate variable with anything else?

 

Energy and time are another pair. If a decay is quick, for example, there will be a larger linewidth of the emission/absorption spectrum for that transition.

 

It's not that there is no uncertainty in mass or charge measurements. There is experiemental error in any measurement. What I meant is that you cannot put the electron in a superposition of two mass or charge states that would allow the collapse of a wave function upon measurement. That is not true of energy states, where the electron in an atom could be in e.g. either of the hyperfine ground states of an alkali atom (to pull an example randomly from stuff related to what my colleagues and I do in the lab). When you measure the atom, the electron will be in one state or the other, but before that measurement, you can't say for sure. But no matter what state preparation you do, the electron will have the same mass and charge.

[Membrain]

I'm not sure if it was the answer you were looking for, but it's an attempt to explain how come the uncertainty is not because of plain ignorance, but rather the definition of variables. Or put another way, some variable just aren't independent, and that's how determining one, does affect that other one, or at least our knowledge of the other one, and there is btw, no difference in this case. Because there is no other way to know that to interact.

It sounds like you're talking about measurement. I'm OK with an electron being not measurable with any accuracy. I just wanted to make sure that we consider there to be 47 electrons in a silver atom. This means that there is no probability to the electrons' existence within the atom, right? It's certain isn't it?

[fredrik]

It sounds like you're talking about measurement. I'm OK with an electron being not measurable with any accuracy. I just wanted to make sure that we consider there to be 47 electrons in a silver atom. This means that there is no probability to the electrons' existence within the atom, right? It's certain isn't it?

 

I guess with the notion that the electrons exists within the atom you mean that the electron is bound to the atom, which is energy dependent. Given sufficient energy, the atom will ionize, and one would not longer say that the electrons are bound to the atom. So if you know the energy with certainty to forbid any electron beeing ionized it will stay bound. But if unsure there may be a probabiltiy seeing an excitation.

 

Also what is the size of the atom? Unlike the classical orbit model, the "electron clouds" of the atom while defined, really smear out to infinity. So there is a finite, but incredibly small probability to find those electrons pretty much anywhere at an arbitrary (but finite) distance, outside of the classical atom. It's just that the probability quickly gets so small that we are unlikely to ever observe such unlikely event, like detecting the electron from an atom in the lab while beeing on the loo. And if we did we are unlikely to be able to reproduce it to reach any significant experimental confidence level in a human lifetime. So if you think that we are sure that the electron is never found without som finite radius r, I think this is not quite true because of the above.

 

Another problem is that elementary particles are by their concept industinguishable. Since electrons can come from all over the place, we can not with absolute certainty trace the origin of the electron and distinguish it from other nearby electrons, only consider what is the most likely source of it. And in a controlled experiment the likely source by order of magnitudes in probability outcompetes other options.

 

For practical purposes ignoring overley utterly unlikely evens works great, but from the philosophical point of view, I wouldn't ignore it.

 

/Fredrik

[fredrik]

I always knew there was an inevitable uncertainty between position and momentum, but I didn't know there were many such pairings called "conjugate variables". Interesting.

 

What other conjugate variable pairs are there? Membrain was asking about mass as a property of a particle that one can measure, and swansont answered that there is no uncertainty about mass measurements (or charge). Does this mean we can rule out mass and charge as conjugate variable with anything else?

 

Another conjugate variables are also angular momentum, or spin vs the "rotational angle" if you use polar or spherical coordinates.

 

The odd part becomes half integer spin of fermions, because there is no way of understanding half integer sping in terms of classical rigid body systems. This is one of the reasons, the rigid body concept is inconsistent with the electron. When you try to derive the dirac equaton the half integer spin can be created from rotations in complex spacetime of an initially bosonic field. So there seems to be different interpretations, depending on what peculiarities ones finds least annoying.

 

/Fredrik

[fredrik]

When you try to derive the dirac equaton the half integer spin can be created from rotations in complex spacetime of an initially bosonic field. So there seems to be different interpretations, depending on what peculiarities ones finds least annoying.

 

Btw, this is sort of touches the concept called "supersymmetry". What I personally don't like is that often these things are treated in a matter that really abstract out the link to the previous step.

 

To speak for myself who is going the philosophical route, the concept of "supersymmetry" is also intimately related to the time evolution and time reversal. You unavoidably step into this when working you are trying to work out implications of starting points. What bugged me beyond belief as as student was that the attitude was to let's step "over it", rather than "into it", and resolve it. This is one of the things I'm goint to revise in detail as soon as I get some spare time.

 

/Fredrik

[gib65]

I just wanted to make sure that we consider there to be 47 electrons in a silver atom. This means that there is no probability to the electrons' existence within the atom, right? It's certain isn't it?

 

I think what membrain is asking here is whether the precise number of electrons is a function of probability. I don't think so. If it's a silver atom, and it's not ionized, then there are exactly 47 electrons (AFAIK). Fredrik's point that we can't know with absolute certainty whether they are in the vicinity of the atom is still true, but it is certain that 47 of them exist. It is an odd concept, isn't it - that an electron can have no definite position yet still definitely exist.

 

The odd part becomes half integer spin of fermions, because there is no way of understanding half integer sping in terms of classical rigid body systems.

 

I don't even know what "spin" means, let alone half or whole integer spins. Can you enlighten me?

[fredrik]

I don't even know what "spin" means, let alone half or whole integer spins. Can you enlighten me?

 

Classically, spin is pretty much what it sounds like. Spin or rotation is the same thing. For example the earth has as an instrinsic rotation around it's own axis, yielding day and night. The earth orbiting the sun OTOH generates a year.

 

You could say that the earth has as "spin".

 

Spinning a massive body, means we have mass motion and thus energy. Just like a mass in constant motion has a momentum and kinetic energy, a spinning massive body has a so called angular momentum and also a kinetic energy of rotation.

 

In quantum mechanics, when you quantize spin, it takes on a discrete set of values (called "eigenvalues" of the spinoperator), just like a "particle in a box" takes on discrete momemtum values.

 

As is known to the word at least from chemistry, an electron has two spin states, up or down. And it's kind of the flip of the other, corresponding to the direction of rotation.

 

In the wave picture duality a standing "wave" clearly has to match in wavelenght to it's resonance. Otherwise it's not a standing wave. At first, it seems spin ½ particles can exists in odd resonances. If you picture yourself the electron as a classical rigid body and try to solve the quantum equations, you cannot wrap your head around these half integer spins. This suggest that the electron is really kind of a weird thing.

 

So the angular momentum, or spin is quantized, but some particles (like the electron) appears to be able to possess strange values.

 

That's a simple explanation without going into all the math.

 

need to go for now...

 

/Fredrik

[fredrik]

Like I tried to say in simple, spin is intrinsic rotation, as opposed to orbital motion.

 

But other than that I forgot to add (which is a key) in experiments and atom talk is that the point is that in classical electromagnetic theory a "spinning charge" generates a magnetic field who interacts with the rest of the EM field, and thus the spin up or down have different energies as relative to the positive charge of the atom nucleus.

 

So the fact that the electron is charged, and spins, it behaves as a little dipole magnet. So does the nucles - relative to the electrons orbit.

 

/Fredrik

[fredrik]

Like I tried to say in simple, spin is intrinsic rotation, as opposed to orbital motion.

 

But then the second point was that this does not make sense for an electron to explain it's magnetic moment. Which really give new meaning to your question what is spin, or what is magnetism or what is an electron?

 

/Fredrik

[gib65]

So I guess a 1/2 spin is only problematic in the quantum paradigm, not the classical. Would it still increment/decrement by full integers (i.e. add an extra quantum of energy to give it 3/2 spin)?

 

Anyway, at some point here, I wanted to derail the course of this thread so that I could ask a question that's becoming more and more evident the more I think about it. It's not much of a derailment since it's still well within the scope of the general topic at hand. I want to know if I'm understanding this whole "collapse of the wave function" thing properly. Given the HUP, it only seems logical that there isn't just one wave function - there's a distinct wave function for each variable we can measure. So far, I understand that when we measure some property (like position), this causes the "collapse of the wave function" as they say. But if the HUP is correct, it also "resurrects" the wave function for a conjugate variable (like momentum). Therefore, there can't just be one all encompassing wave function. Am I on the right track?

[Membrain]

I think what membrain is asking here is whether the precise number of electrons is a function of probability. I don't think so. If it's a silver atom, and it's not ionized, then there are exactly 47 electrons (AFAIK). Fredrik's point that we can't know with absolute certainty whether they are in the vicinity of the atom is still true, but it is certain that 47 of them exist. It is an odd concept, isn't it - that an electron can have no definite position yet still definitely exist.

Yes! I think we are getting close.

 

So where I was headed was: when it comes to probabilities, is it true that we are not talking about probabilities of an electron's existence, but probabilities that it is located "over there" as opposed to "over here"?

[fredrik]

So I guess a 1/2 spin is only problematic in the quantum paradigm, not the classical. Would it still increment/decrement by full integers (i.e. add an extra quantum of energy to give it 3/2 spin)

 

The electron as an elementary partle has a fixed intrinsic spin magnitude, it's only the direction of the spin that varies, since the spin is really a vector quantity.

 

In the hydrogen atom there are several quantum numbers. One quantum numbers for the radial "shell", and two angular orientation quantum numbers, and one for the spin orientation. These first three quantum numbers give rise to the various orbital shapes/clouds.

 

The impact of the electron spin is called the fine structure as it's notice on close up that the spectral lines are splitted. The split is exaplained by the electron spin and the difference in energy.

 

I want to know if I'm understanding this whole "collapse of the wave function" thing properly. Given the HUP, it only seems logical that there isn't just one wave function - there's a distinct wave function for each variable we can measure. So far, I understand that when we measure some property (like position), this causes the "collapse of the wave function" as they say. But if the HUP is correct, it also "resurrects" the wave function for a conjugate variable (like momentum). Therefore, there can't just be one all encompassing wave function. Am I on the right track?

 

There is only one wave function. One observer and one wavefunction for his observations. The ideas is that the wave function represents the total information the observer has of the system. But of course the system may have different, not very interacting parts, so one part of the system may "collapse" while another part does not. So in a sense you are still right in your thinking except the custom is to talk about a single wavefunction.

 

So it depends on what kind of measurement you do. A measurement is usually targeted to a particular property. In QM a measurement is represented by operators which basically perform a mathematical operation on the wavefunction.

 

Operators correspond to different observables, and compatible or so called commuting observables can be determined with certainty at the same time.

 

Non-commuting means that if you first measure for A, then for B, you get a different result than if you to B first and then A.

 

It's when two observables (and observable variable) do not "commute" that you get into the HUP between those two observables.

 

However two compatible variables can be known at the same time. For example, there is no uncertainty between the momenta in the different space dimensions. So it's possible to know the "x position" exactly at the same time as "y momentum".

 

/Fredrik

[fredrik]

Yes! I think we are getting close.

 

So where I was headed was: when it comes to probabilities, is it true that we are not talking about probabilities of an electron's existence, but probabilities that it is located "over there" as opposed to "over here"?

 

Yes that's true. The conservation of probability that is considered important requires that the electron must not dissappear.

 

But this is in basic QM. In the generalizaiton with particle-antiparticle interactions there may be fluctuations to this.

 

But yes, "summing over all possibilities" you are always 100% right so to speak. That is sort of by construction.

 

/Fredrik

[Membrain]

Yes that's true. The conservation of probability that is considered important requires that the electron must not dissappear.

 

But this is in basic QM. In the generalizaiton with particle-antiparticle interactions there may be fluctuations to this.

 

But yes, "summing over all possibilities" you are always 100% right so to speak. That is sort of by construction.

 

/Fredrik

OK thanks very much (all). I will now start a thread called "Question about Uncertainty".

 

Thanks again.