Directly measuring Wave-function?

[Reality Check]

I disagree entirely. The Heisenberg Uncertainty principle (as is used today, not necessarily as it was formulated) is a mathematical statement about waves. It's far more useful when describing things that aren't being measured......

I disagree completely. The Heisenberg Uncertainty principle as is used today is a statement about uncertainties in measurments. It can be derived from operators acting on the wave function (not waves).
You may be thinking of the similar uncertainty principle in Fourier analysis.

Your example is wrong: The reason that electrons do not spiral into the nucleus is because of the quantization of their energy.

 

[Yoron]

It's a very good question. What the he* do we mean by a wave function. We have two types of 'waves'. One is called 'standing waves' as in 'matter waves' and the other is 'moving waves' as light propagating. And when discussing a 'wave function' we define it as we're measuring a probability of 'something' that we can't pinpoint as a single event. The only way to pin point it is if annihilating, as a detector interacting with radiation.

Then we come to what it is before measuring, btw, all of this goes back to one simple fact. That a photon is no football coming at you. In the reality I have I can see that football coming at me, I can follow its 'path'. But that is not true with radiation, there you have only two effects observable, its recoil and subsequent annihilation. Light is defined by some principles, it's always at 'c', it's the best clock I know of, it's massless, and so has no acceleration or inertia, and we can't discuss any intrinsic 'time' for it. As I see it, related to its masslessness, that also is what allows it 'c'. The timelessness is directly following the masslessness to me.

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A alternative way of defining its 'timelessness' is the one where you define it as the 'arrow itself'. Then it's the beat/clock that 'ticks' for us, but each 'photon' will then be without 'time', as you could see it as a 'tick', forever frozen as you watch it 'propagate'. Which you can't, btw
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So, defining a probability you use the idea of waves moving, then you decide whether you expect them to exist even if unobservable. If you expect them to be 'really there' in some QM remarkable way you can do as Feynman, in where the 'wave' as such takes all paths possible simultaneously. Those paths can then be seen as waves interfering with each other, some reinforcing other quenching, in the end, at the detector leaving only one path possible. Or you can do as in the Copenhagen definition which explicitly state that if you can't measure it, you can't either define it as 'existing' meaning that the only thing you can count on is the probability of where/how it will 'exist', and that is when the detector interact with its probability, breaking it down into a 'history' of knowables.

In the Copenhagen definition the 'wave function collapse' is when the wave function, that from probability can be described as a superposition of all possible states it possibly can have before being measured reduces to a single state, in its interacting with a detector/observer. And lights 'eigenstates', before the detector finalize it into a outcome, is just the ways we expect a wave/particle to behave, like its spin, phase etc as I understands it. The formalism of it involves a lot more things though.

 

[Dilb]

I disagree completely. The Heisenberg Uncertainty principle as is used today is a statement about uncertainties in measurments. It can be derived from operators acting on the wave function (not waves).
You may be thinking of the similar uncertainty principle in Fourier analysis.

Your example is wrong: The reason that electrons do not spiral into the nucleus is because of the quantization of their energy.

It's the exact same uncertainty principle. Position and momentum are related by Fourier transforms. The thing is derived from the commutation of operators, not by considering anything related to an actual measurement (and I'm aware Wikipedia implies differently, but that's not how physicists today would derive it).

Saying 'energy is quantized' doesn't explain why electrons can't get arbitrarily close to the nucleus: I can count down 0, -1, -2, -3, ..., right down to negative infinity no problem. It's only by considering why a sharply peaked wavefunction isn't lower energy that we can say why electron orbitals are stable.

What are the units of this eigenvalue? What is knowable about a wave when one knows its eigenvalue? What kind of tool does one measure waveforms with? When one measures a waveform by measuring particles how are the particles affected?

The units would be whatever the units are of what you're measuring. Joules for energy, metres for position, m/s for speed, etc.

You don't know anything about a wavefunction just by knowing the eigenvalue. However, if you have a reasonable guess about what sort of environment the wave is in (free space, around a nucleus, in a piece of metal, etc.), you can figure out what sort of wavefunctions should exists, and match up the measured eigenvalue to whatever wave(s) have that eigenvalue in your model (the eigenfunction).

The wavefunction of a particle can initially be in a superposition of eigenfunctions. After measurement, though, it's always only in an eigenfunction with the measured eigenvalue. To figure out the initial superposition (i.e. to "measure the particle's wavefunction") you need a large number of identical particles and measure them all. You'll get a distribution of eigenvalues. You then deduce that the initial wavefunction of those particles is that combination of eigenfunctions.

 

[Vorpal]

I disagree entirely. The Heisenberg Uncertainty principle (as is used today, not necessarily as it was formulated) is a mathematical statement about waves. It's far more useful when describing things that aren't being measured.

On a purely mathematical level, HUP is a statement about any pair of self-adjoint operators on any inner product space whatsoever; there isn't even any assumption about having any waves or being a Hilbert space or being a Fourier transform pair or any such thing.

But it's part of the formalism of QM that states are in an certain kind of inner product space and observables are certain kinds of operators on it. So physically HUP becomes a statement about the product of standard deviations of measurements performed on an ensemble of systems in identical states. What's the problem?

The uncertainty principle tells us what's wrong with this: as the electron gets closer to the nucleus, it's position gets more and more exact, so it's momentum must get less exact. We started with zero-momentum (so the electron doesn't fly away from the nucleus), so we have to add larger and larger values of momentum. Larger value of momentum implies more kinetic energy. Now we know why electrons won't spiral into the nucleus: orbitals need to balance the loss of electrostatic energy with a gain of kinetic energy.

So you have a bound for ΔxΔp so that tending Δx to zero means upping Δp and hence kinetic energy. So what? Since in the simple point-like nucleus model we're discussing electrostatic potential is unbounded from below, HUP by itself is compatible with a bound electron with arbitrarily large kinetic energy, as all it means is that the potential should tend to negative infinity to compensate. So your reasoning does not establish the lack of nuclear-spiraling of the electron.

RC's reason is simple enough and almost complete: energy is quantized in such a way that there is a lowest eigenvalue. Electrons with higher energy than this do radiate, and do "spiral" (well, transition) closer to the nucleus, unless the environment inhibits this in some way. Those at the lowest eigenstate don't because there is nothing for them to transition to.

 

[Dilb]

On a purely mathematical level, HUP is a statement about any pair of self-adjoint operators on any inner product space whatsoever; there isn't even any assumption about having any waves or being a Hilbert space or being a Fourier transform pair or any such thing.

But it's part of the formalism of QM that states are in an certain kind of inner product space and observables are certain kinds of operators on it. So physically HUP becomes a statement about the product of standard deviations of measurements performed on an ensemble of systems in identical states. What's the problem?

My issue is that it's still important even if you don't ever intend to make a measurement. It's not a principle relating the practical limitations of a measurement on a single particle, it's a limitation on what states actually exist. Otherwise you might think electrons are still point particles, and it's only the way they are measured that's inherently fuzzy.

So you have a bound for ΔxΔp so that tending Δx to zero means upping Δp and hence kinetic energy. So what? Since in the simple point-like nucleus model we're discussing electrostatic potential is unbounded from below, HUP by itself is compatible with a bound electron with arbitrarily large kinetic energy, as all it means is that the potential should tend to negative infinity to compensate. So your reasoning does not establish the lack of nuclear-spiraling of the electron.

RC's reason is simple enough and almost complete: energy is quantized in such a way that there is a lowest eigenvalue. Electrons with higher energy than this do radiate, and do "spiral" (well, transition) closer to the nucleus, unless the environment inhibits this in some way. Those at the lowest eigenstate don't because there is nothing for them to transition to.

I've given a qualitative argument sure, but my version at least gives some possible explanation for why a lowest energy state exists. Saying "energy is quantized in such a way" is just begging the question.

 

[Vorpal]

My issue is that it's still important even if you don't ever intend to make a measurement. It's not a principle relating the practical limitations of a measurement on a single particle, it's a limitation on what states actually exist. Otherwise you might think electrons are still point particles, and it's only the way they are measured that's inherently fuzzy.

Since the formalism of QM refers to ideal measurements, of course it's not just a practical limitation. Yours is an important point for people to learn, but there is no contradiction in regards to what spawned this argument.

I've given a qualitative argument sure, but my version at least gives some possible explanation for why a lowest energy state exists. Saying "energy is quantized in such a way" is just begging the question.

The HUP argument suffers exactly the same flaw as you used to dismiss energy quantization as an explanation. The HUP has absolutely nothing to do with whether or not the Hamiltonian is bounded from below, so it's an equally good reason to dismiss either one.

Though "energy is quantized in such a way" is not begging the question in the sense it is directly provable from QM's formalism: there exists a lowest eigenvalue of the Hamiltonian of the proton-electron system, and it's a simple exercise show this. So the only thing we're assuming is QM itself.

 

[Dilb]

Though "energy is quantized in such a way" is not begging the question in the sense it is directly provable from QM's formalism: there exists a lowest eigenvalue of the Hamiltonian of the proton-electron system, and it's a simple exercise show this. So the only thing we're assuming is QM itself.

If we are comfortable doing that then the uncertainty principle is irrelevant. We can simply work out the actual position and momentum distributions for any and all states, and we don't need any inequalities.

Personally I wouldn't call solving the hydrogen atom a simple exercise. Without doing that, all you've done is state what happens: electrons have a ground state separated by a discrete amount of energy. That really doesn't give you much physical insight as to why electrons have such large orbitals compared to nuclei.

 

[Yoron]

Dilb, reading "To figure out the initial superposition (i.e. to "measure the particle's wavefunction") you need a large number of identical particles and measure them all. You'll get a distribution of eigenvalues. You then deduce that the initial wavefunction of those particles is that combination of eigenfunctions. "

That is what I call weak measurements, assuming that you do it on a series of entangled particles you might get away with calling them 'identical' but if you go after the definition we use classically there are no such things as 'identical' anything as long as they are separated. And weak measurements are always separated in time. On the other hand, one could argue that bosons can be superimposed, maybe And QM is after all statistical, quantized, approach to reality. So yeah, maybe?

And I agree with your definition of HUP. HUP speaks about a impossibility of a simultaneous measurement (in time) of, for example, both a photons momentum and position, whereas weak measurements speaks about a statistical significance, created from several readings treated statistically. This one describes quantum states pretty well.

As for HUP I think this, and then the 'quantum casino' do a good job describing both HUP and the 'wave function' Dave. HUP is not something you can blame on the quality of measurement, it's deeper than that, to me it's a principle of QM. And Dave, if you follow the links, (try the quantum casino too), you probably will end up knowing more than you ever wanted too, methinks, about both HUP and wave functions

 

[Vorpal]

If we are comfortable doing that then the uncertainty principle is irrelevant. We can simply work out the actual position and momentum distributions for any and all states, and we don't need any inequalities.

Personally I wouldn't call solving the hydrogen atom a simple exercise.

We don't need anything near that complicated. Getting the radial equation from separability is easy, and looking for a series solution in a simple-minded way gets you the energy eigenvalues at only the cost of a bit of algebraic gymnastics. You don't need any insight into actually solving the equations to do this. Neither radial nor spherical harmonic parts of the wavefunctions are actually needed in their closed form. Yeah, I'd call it simple.

Without doing that, all you've done is state what happens: electrons have a ground state separated by a discrete amount of energy. That really doesn't give you much physical insight as to why electrons have such large orbitals compared to nuclei.

If the HUP doesn't even establish that there is a ground state in the first place, how in the world does it establish its scale? You're ascribing to HUP jobs it just isn't capable of handling alone.

I've no doubt that intuition in regards HUP is quite important in physics, but it simply fails to do what you want it to do here. Given the standards of rigor you've applied to other attempts at explanation, you either solve for whether the energy eigenvalues are bounded below or you're just out of luck.

 

[Yoron]

Hmm, had to reread you guys to understand what you argued about The funny part is that you both seem to agree on that HUP is about the impossibility of a simultaneous reading, of momentum/position for example. So what you seem to be arguing about is how to see a electron staying in its 'probability cloud', if i got it right? My five cents comes here.

As a electron could be seen to 'rotate', if applied from a classical approach (Newton), you then could define it as constantly 'accelerating'. But if it was so it should radiate, shouldn't it? And as far as I know, it doesn't. The only time it radiates is when it changes state ('jumps' between shells). We know that the orbitals are quantized according to the Schrodinger equation into orbitals ranging from lowest (near nucleus) to highest energy (away from nucleus).

As I understands it QM defines it as if was in a 'probability cloud', unable to define it to any specific 'place' in that 'orbital' circumventing the nucleus. Instead it exist in a so called superposition defined by the 'cloud/orbital'.

Assume that you could define it to a specific point (position). If you could you would now have to allow its momentum to become indefinably large (HUP), and so also its 'energy'. As long as it's 'smeared out' around the nucleus it to me seems to be 'everywhere' or 'nowhere' and so also be in a balance, unmoving if you like.

Those 'balances' are quantized into orbitals, separated from other orbitals according to the Pauli Exclusion Principle. The properties of orbitals and their electrons are defined using quantum numbers "Principal, n - 1, 2, 3, defining energy level ; Angular momentum, 0 to n-1 defines the orbital shape ; Magnetic, ml - l to + l defines its spatial orientation and degeneracy ; Spin, Ms ± 1/2 defines the electron spin"

Knowing the orbital, you will know those values, as they defines the' orbital'. So a electron can't 'exist' other than as a probability defined by its orbital, and the orbital, or probability cloud, is a function of the electrons properties. It's a sort of symmetry to me, having nothing to do with how we define something classically. I don't think you can expect the electrons closest to the nucleus to have the most energy, even though I can see how you think there. As far as I know it's the other way around. The further away the orbital, the higher their average 'energy'. But it is a interesting point to make, as you could see it as a 'smaller box' and so increase the 'energy'. All as I understands it.

 

[Dilb]

We don't need anything near that complicated. Getting the radial equation from separability is easy, and looking for a series solution in a simple-minded way gets you the energy eigenvalues at only the cost of a bit of algebraic gymnastics. You don't need any insight into actually solving the equations to do this. Neither radial nor spherical harmonic parts of the wavefunctions are actually needed in their closed form. Yeah, I'd call it simple.

Well you're far better at math than I am then.

If the HUP doesn't even establish that there is a ground state in the first place, how in the world does it establish its scale? You're ascribing to HUP jobs it just isn't capable of handling alone.

I've no doubt that intuition in regards HUP is quite important in physics, but it simply fails to do what you want it to do here. Given the standards of rigor you've applied to other attempts at explanation, you either solve for whether the energy eigenvalues are bounded below or you're just out of luck.

Sure, the uncertainty principle does not prove that a ground state exists. Saying energy levels are quantized doesn't prove it either (energy was being quantized 20 years before anyone had the Schrodinger equation, remember). However, if we accept the Rutherford experiment, then the uncertainty principle does at least give a reason for a ground state to exist.

You get a scale from the fact that momentum is related to mass. Electrons, being much lighter than protons, are that much harder to localize, so most of the volume of atoms should be from the electrons, not the nucleons. You can even play with this if you have particles with different masses. Muons are about 200 times heavier than electrons, so replacing an electron with a muon gives you a much smaller atom (small enough to undergo fusion, even).

 

[Vorpal]

Sure, the uncertainty principle does not prove that a ground state exists. Saying energy levels are quantized doesn't prove it either (energy was being quantized 20 years before anyone had the Schrodinger equation, remember). However, if we accept the Rutherford experiment, then the uncertainty principle does at least give a reason for a ground state to exist.

I remember; that's one reason I am puzzled why you're injecting the HUP into this specific matter. The HUP is important in rejecting classical orbits, but it's neither necessary for this matter nor does a good job in justifying it. The real workhorse is de Broglie's λ = h/p, which with λ~r allows you to find the radius which minimizes the total energy, to within an order of magnitude. The HUP can get you there, but Δx~r has similar justification and you'll be making assumptions as to minimal uncertainty in addition to that just to retrace the same steps.

De Broglie is cleaner, and if you're willing to suffer a similar amount of handwaving HUP approach would require, you might as well expand the de Broglie approach with a standing wave condition, get all the energy levels at once, and explain the EM radiation during transitions as a beat formed by the frequencies of the energy levels.

 

[Dilb]

That's certainly a nicer way to actually get a value for the size of an atom. In any case, I don't think either of us will convince the other that our hand-wavy justifications for ground state energies is the better hand-wavy explanation, so I'm ready to drop this topic.

If I'm hung up on the uncertainty principle, it's because I'm irritated by the way that it's frequently confused with an observer effect. Yes it can appear in measurements if you go and measure it, but it still applies even when you aren't trying to measure position and momentum simultaneously.

 

[Reality Check]

That's certainly a nicer way to actually get a value for the size of an atom. In any case, I don't think either of us will convince the other that our hand-wavy justifications for ground state energies is the better hand-wavy explanation, so I'm ready to drop this topic.

I am not certain by what you mean by 'hand-wavy' justifications. The energy levels that are possible for an electron in an atom are straight forward QM solutions, e.g. the energy levels for the electron in a hydorgen-like atom:

.

This is the energy relative to the electron at infinity, thus the negative sign. The quantum number n is 1, 2, 3, 4, etc. Note that putting n = 0 is obvously wrong and using a negative value does not matter to the energy.

Thus you can see that the existence of a ground state is explained by the quantization of the energy from the QM solution.