[An immediate continuation of Part 1, which you should definitely read first; today’s post is not stand-alone.]
The Asymmetry Between Location and Motion
We are in the middle of trying to figure out if the electron (or other similar object) could possibly be of infinitesimal size, to match the naive meaning of the words “elementary particle.” In the last post, I described how 1920’s quantum physics would envision an electron (or other object) in a state |P0> of definite momentum or a state |X0> of definite position (shown in Figs. 1 and 2 from last time.)
If it is meaningful to say that “an electron is really is an object whose diameter is zero”, we would naturally expect to be able to put it into a state in which its position is clearly defined and located at some specific point X0 — namely, we should be able to put it into the state |X0>. But do such states actually exist?
Symmetry and Asymmetry
In Part 1 we saw all sorts of symmetry between momentum and position:
- the symmetry between x and p in the Heisenberg uncertainty principle,
- the symmetry between the states |X0> and |P0>,
- the symmetry seen in their wave functions as functions of x and p shown in Figs. 1 and 2 (and see also 1a and 2a, in the side discussion, for more symmetry.)
This symmetry would seem to imply that if we could put any object, including an elementary particle, in the state |P0>, we ought to be able to put it into a state |X0>, too.
But this logic won’t follow, because in fact there’s an even more important asymmetry. The states |X0> and |P0> differ crucially. The difference lies in their energy.
Who cares about energy?
There are a couple of reasons we should care, closely related. First, just as there is a relationship between position and momentum, there is a relationship between time and energy: energy is deeply related to how wave functions evolve over time. Second, energy has its limits, and we’re going to see them violated.
Energy and How Wave Functions Change Over Time
In 1920s quantum physics, the evolution of our particle’s wave function depends on how much energy it has… or, if its energy is not definite, on the various possible energies that it may have.
Definite Momentum and Energy: Simplicity
This change with time is simple for the state |P0>, because this state, with definite momentum, also has definite energy. It therefore evolves in a very simple way: it keeps its shape, but moves with a constant speed.
How much energy does it have? Well, in 1920s quantum physics, just as in pre-1900 physics, the motion-energy E of an isolated particle of definite momentum p is
- E = p2/2m
where m is the particle’s mass. Where does this formula come from? In first-year university physics, we learn that a particle’s momentum is mv and that its motion-energy is mv2/2 = (mv)2/2m = p2/2m; so in fact this is a familiar formula from centuries ago.
Less Definite Momentum and Energy: Greater Complexity
What about the compromise states mentioned in Part 1, the ones that lie somewhere between the extreme states |X0> and |P0>, in which the particle has neither definite position nor definite momentum? These “Gaussian wave packets” appeared in Fig. 3 and 4 of Part 1. The state of Fig. 3 has less definite momentum than the |P0> state, but unlike the latter, it has a rough location, albeit broadly spread out. How does it evolve?
As seen in Fig. 6, the wave still moves to the left, like the |P0> state. But this motion is now seen not only in the red and blue waves which represent the wave function itself but also in the probability for where to find the particle’s position, shown in the black curve. Our knowledge of the position is poor, but we can clearly see that the particle’s most likely position moves steadily to the left.
What happens if the particle’s position is better known and the momentum is becoming quite uncertain? We saw what a wave function for such a particle looks like in Fig. 4 of Part 1, where the position is becoming quite well known, but nowhere as precisely as in the |X0> state. How does this wave function evolve over time? This is shown in Fig. 7.
We see the wave function still indicates the particle is moving to the left. But the wave function spreads out rapidly, meaning that our knowledge of its position is quickly decreasing over time. In fact, if you look at the right edge of the wave function, it has barely moved at all, so the particle might be moving slowly. But the left edge has disappeared out of view, indicating that the particle might be moving very rapidly. Thus the particle’s momentum is indeed very uncertain, and we see this in the evolution of the state.
This uncertainty in the momentum means that we have increased uncertainty in the particle’s motion-energy. If it is moving slowly, its motion-energy is low, while if it is moving rapidly, its motion-energy is much higher. If we measure its motion-energy, we might find it anywhere in between. This is why its evolution is so much more complex than that seen in Fig. 5 and even Fig. 6.
Near-Definite Position: Breakdown
What happens as we make the particle’s position better and better known, approaching the state |X0> that we want to put our electron in to see if it can really be thought of as a true particle within the methods of 1920s quantum physics?
Well, look at Fig. 8, which shows the time-evolution of a state almost as narrow as |X0> .
Now we can’t even say if the particle is going to the left or to the right! It may be moving extremely rapidly, disappearing off the edges of the image, or it may remain where it was initially, hardly moving at all. Our knowledge of its momentum is basically nil, as the uncertainty principle would lead us to expect. But there’s more. Even though our knowledge of the particle’s position is initially excellent, it rapidly degrades, and we quickly know nothing about it.
We are seeing the profound asymmetry between position and momentum:
- a particle of definite momentum can retain that momentum for a long time,
- a particle of definite position immediately becomes one whose position is completely unknown.
Worse, the particle’s speed is completely unknown, which means it can be extremely high! How high can it go? Well, the closer we make the initial wave function to that of the state |X0>, the faster the particle can potentially move away from its initial position — until it potentially does so in excess of the cosmic speed limit c (often referred to as the “speed of light”)!
That’s definitely bad. Once our particle has the possibility of reaching light speed, we need Einstein’s relativity. But the original quantum methods of Heisenberg-Born-Jordan and Schrödinger do not account for the cosmic speed limit. And so we learn: in the 1920s quantum physics taught in undergraduate university physics classes, a state of definite position simply does not exist.
Isn’t it Relatively Easy to Resolve the Problem?
But can’t we just add relativity to 1920s quantum physics, and then this problem will take care of itself?
You might think so. In 1928, Dirac found a way to combine Einstein’s relativity with Schrödinger’s wave equation for electrons. In this case, instead of the motion-energy of a particle being E = p2/2m, Dirac’s equation focuses on the total energy of the particle. Written in terms of the particle’s rest mass m [which is the type of mass that doesn’t change with speed], that total energy satisfies the equation
For stationary particles, which have p=0, this equation reduces to E=mc2, as it must.
This does indeed take care of the cosmic speed limit; our particle no longer breaks it. But there’s no cosmic momentum limit; even though v has a maximum, p does not. In Einstein’s relativity, the relation between momentum and speed isn’t p=mv anymore. Instead it is
which gives the old formula when v is much less than c, but becomes infinite as v approaches c.
Not that there’s anything wrong with that; momentum can be as large as one wants. The problem is that, as you can see for the formula for energy above, when p goes to infinity, so does E. And while that, too, is allowed, it causes a severe crisis, which I’ll get to in a moment.
Actually, we could have guessed from the start that the energy of a particle in a state of definite position |X0> would be arbitrarily large. The smaller is the position uncertainty Δx, the larger is the momentum uncertainty Δp; and once we have no idea what the particle’s momentum is, we may find that it is huge — which in turn means its energy can be huge too.
Notice the asymmetry. A particle with very small Δp must have very large Δx, but having an unknown location does not affect an isolated particle’s energy. But a particle with very small Δx must have very large Δp, which inevitably means very large energy.
The Particle(s) Crisis
So let’s try to put an isolated electron into a state |X0>, knowing that the total energy of the electron has some probability of being exceedingly high. In particular, it may be much, much larger — tens, or thousands, or trillions of times larger — than mc2 [where again m means the “rest mass” or “invariant mass” of the particle — the version of mass that does not change with speed.]
The problem that cannot be avoided first arises once the energy reaches 3mc2 . We’re trying to make a single electron at a definite location. But how can we be sure that 3mc2 worth of energy won’t be used by nature in another way? Why can’t nature use it to make not only an electron but also a second electron and a positron? [Positrons are the anti-particles of electrons.] If stationary, each of the three particles would require mc2 for its existence.
If electrons (not just the electron we’re working with, but electrons in general) didn’t ever interact with anything, and were just incredibly boring, inert objects, then we could keep this from happening. But not only would this be dull, it simply isn’t true in nature. Electrons do interact with electromagnetic fields, and with other things too. As a result, we can’t stop nature from using those interactions and Einstein’s relativity to turn 3mc2 of energy into three slow particles — two electrons and a positron — instead of one fast particle!
For the state |X0> with Δx = 0 and Δp = infinity, there’s no limit to the energy; it could be 3mc2, 11mc2, 13253mc2, 9336572361mc2. As many electron/positron pairs as we like can join our electron. The |X0> state we have ended up with isn’t at all like the one we were aiming for; it’s certainly not going to be a single particle with a definite location.
Our relativistic version of 1920s quantum physics simply cannot handle this proliferation. As I’ve emphasized, an isolated physical system has only one wave function, no matter how many particles it has, and that wave function exists in the space of possibilities. How big is that space of possibilities here?
Normally, if we have N particles moving in d dimensions of physical space, then the space of possibilities has N-times-d dimensions. (In examples that I’ve given in this post and this one, I had two particles moving in one dimension, so the space of possibilities was 2×1=2 dimensional.) But here, N isn’t fixed. Our state |X0> might have one particle, three, seventy one, nine thousand and thirteen, and so on. And if these particles are moving in our familiar three dimensions of physical space, then the space of possibilities is 3 dimensional if there is one particle, 9 dimensional if there are three particles, 213 dimensional if there are seventy-one particles — or said better, since all of these values of N are possible, our wave function has to simultaneously exist in all of these dimensional spaces at the same time, and tell us the probability of being in one of these spaces compared to the others.
Still worse, we have neglected the fact that electrons can emit photons — particles of light. Many of them are easily emitted. So on top of everything else, we need to include arbitrary numbers of photons in our |X0> state as well.
Good Heavens. Everything is completely out of control.
How Small Can An Electron Be (In 1920s Quantum Physics?)
How small are we actually able to make an electron’s wave function before the language of the 1920s completely falls apart? Well, for the wave function describing the electron to make sense,
- its motion-energy must be below mc2, which means that
- p has to be small compared to mc , which means that
- Δp has to be small compared to mc , which means that
- by Heisenberg’s uncertainty principle, Δx has to be large compared to h/(4πmc)
This distance (up to the factor of 1/4π) is known as a particle’s Compton wavelength, and it is about 10-13 meters for an electron. That’s about 1/1000 of the distance across an atom, but 100 times the diameter of a small atomic nucleus. Therefore, 1920s quantum physics can describe electrons whose wave functions allow them to range across atoms, but cannot describe an electron restricted to a region the size of an atomic nucleus, or of a proton or neutron, whose size is 10-15 meters. It certainly can’t handle an electron restricted to a point!
Let me reiterate: an electron cannot be restricted to a region the size of a proton and still be described by “quantum mechanics”.
As for neutrinos, it’s much worse; since their masses are much smaller, they can’t even be described in regions whose diameter is that of a thousand atoms!
The Solution: Relativistic Quantum Field Theory
It took scientists two decades (arguably more) to figure out how to get around this problem. But with benefit of hindsight, we can say that it’s time for us to give up on the quantum physics of the 1920s, and its image of an electron as a dot — as an infinitesimally small object. It just doesn’t work.
Instead, we now turn to relativistic quantum field theory, which can indeed handle all this complexity. It does so by no longer viewing particles as fundamental objects with infinitesimal size, position x and momentum p, and instead seeing them as ripples in fields. A quantum field can have any number of ripples — i.e. as many particles and anti-particles as you want, and more generally an indefinite number. Along the way, quantum field theory explains why every electron is exactly the same as every other. There is no longer symmetry between x and p, no reason to worry about why states of definite momentum exist and those of definite position do not, and no reason to imagine that “particles” [which I personally think are more reasonably called “wavicles“, since they behave much more like waves than particles] have a definite, unchanging shape.
The space of possibilities is now the space of possible shapes for the field, which always has an infinite number of dimensions — and indeed the wave function of a field (or of multiple fields) is a function of an infinite number of variables (really a function of a function [or of multiple functions], called a “functional”).
Don’t get me wrong; quantum field theory doesn’t do all this in a simple way. As physicists tried to cope with the difficult math of quantum field theory, they faced many serious challenges, including apparent infinities everywhere and lots of consistency requirements that needed to be understood. Nevertheless, over the past seven decades, they solved the vast majority of these problems. As they did so, field theory turned out to agree so well with data that it has become the universal modern language for describing the bricks and mortar of the universe.
Yet this is not the end of the story. Even within quantum field theory, we can still find ways to define what we mean by the “size” of a particle, though doing so requires a new approach. Armed with this definition, we do now have clear evidence that electrons are much smaller than protons. And so we can ask again: can an elementary “particle” [wavicle] have zero size?
We’ll return to this question in later posts.
One Response
Have you every heard of “aether bubbles”?
I found an unserious write-up: https://liquidgravity.nz/DoubleSlitExperiment.html, about these aether bubbles and how they can explain the double-slit experiment. Yes, this is way out there in left field, but it could still be within the ball park.
What if, :-), these aether bubbles are actual fundamental black holes, composed of spheres (bubbles) of empty space, really empty (0 Kelvin) “trapped” by a symmetrical high density energy flux. We are talking at or below Planck’s scale.
a) Atoms are composed of these BH bubbles up to a limit to give the fundamental atom structure we observe, beyond this limit the decay into electromagnetic waves propagating between these BH bubbles. (similar to the attached write-up).
b) EM spectrum is determined by the source “package” of these trapped BH bubbles that make up the atoms (nucleus), to more compact (higher densities) the smaller the wavelengths emitted, gamma rays. So, as the energy flow at higher radii the wavelengths increase, to radio waves.
This “aether bubbles”, I like to call them Blackhole bubbles could also explain both quantum gravity and GR, depending on the densities of these structures.
– a retired engineer throwing monkey wrenches, 🙂