To better understand the Higgs field and how it works its magic, we need to make a detour into the history of physics and look at the similarities and differences between particles and waves. In ordinary life (what we call the ‘classical’ world) a particle is a localized object that is usually of small size, has a fairly well defined boundary, and a mass. A grain of rice and a speck of dust are particles. A wave, on the other hand, is the name we give to the pattern of vibrations traveling through some medium (think of the waves in water or sound waves traveling through air) that is extended, has no sharp edges, and does not have mass.
This distinction between particles and waves seemed pretty clear-cut for centuries. Any entity had to be either a particle or a wave because the properties of the two categories were quite distinct. However, there was one troublesome phenomenon that kept escaping from the category to which it was assigned, and that was light.
Isaac Newton (1642-1727) did a systematic investigation of light and declared that its properties were consistent with thinking of it as being composed of a stream of particles and this view persisted for over a century, dominating over the alternative wave theory proposed by Christiaan Huygens (1629-1695).
But during the late eighteenth and early nineteenth centuries, further investigations seemed to indicate that light also had wave-like properties and Thomas Young (1773-1829) and Augustin-Jean Fresnel (1788-1827) showed that light exhibited the phenomenon of interference (where light from two sources overlapped to sometimes create a large effect and sometimes no effect at all) and also that of diffraction (the ability to bend around edges). Both these properties had previously never been observed with particles but only with phenomena (like sound and water) that had been established to be waves. Thus light was also deemed to be a wave.
This immediately raised a new problem. Waves had always been thought to be disturbances in an underlying medium. What medium was being disturbed by light waves? This was especially problematic since light was reaching us from distant stars so this medium had to extend everywhere, even over what we had thought was the vacuum of deep space. This led to the suggestion that there was an elastic medium (which was given the name of ‘luminiferous ether’ or simply ether) that permeated all space even though we had not been aware of it up until that point. The sole function of the ether seemed to be to enable the propagation of light waves.
That state of affairs lasted about a century. Beginning with the twentieth century, all manner of problems with the wave-particle distinction started to emerge, not the least of which was that, despite determined efforts by some of the best experimentalists (including the famous Michelson-Morley experiment done at my own institution of Case Western Reserve University), no corroborating evidence could be found for the existence of this ether. It was Albert Einstein (1879-1955) who resolved this issue when he proposed his theory of relativity that made the ether redundant and soon it was dispensed with. But this resurrected the old problem of how light waves propagated and it was suggested that light waves, unlike all other waves, did not need a medium at all but were a disturbance of space itself.
Then further investigations and analyses by Max Planck (1858-1947) on black body radiation and Einstein on the photoelectric effect started revealing that light had particle-like properties, seemingly bring us back to the ideas of Newton. This was clearly a problem since light continued to show interference and diffraction effects, the defining characteristics of waves. It was suggested that light waves consisted of a stream of particles that on a large scale appeared like a wave but that it was when you looked at it on a small scale that its particle properties became apparent. The particle associated with light was called the photon.
So thanks to light, the wall of separation between particles and waves was being chipped away from the wave side.
That was not all. Louis de Broglie (1892-1987) suggested that those things that had been thought to be incontrovertibly particles (like electrons) also had wave-like properties, a prediction that was confirmed in 1927. But since electrons still had a mass, a defining characteristic of a particle, this meant that the wall of separation was also being undermined from the particle side.
It was a puzzle and the search was on to explain this strange behavior where things were neither purely particles nor waves but perhaps both or even neither.
Major progress in the search for a coherent unifying theory was achieved by Erwin Schrodinger (1887-1961) who in 1926 wrote down an equation called the Schrodinger equation that was a wave equation for particles with mass. This hybrid equation turned out to be extremely successful for calculating properties of some things like the size and energy levels of the hydrogen atom.
The Schrodinger equation was clearly something that straddled the wave-particle divide, thus eliminating the wall of separation. But although it was successful as a computational device, for some time it was not clear what the wave function that emerged as the solution to the equation meant. There was considerable debate over this, with a rough consensus eventually emerging that while particles remained particles (i.e., objects that had definite boundaries that were confined to a small region of space), the wave function represented information about the probability (technically the probability amplitude) that the particle would be found in a region of space. So a region of space with a large wave function for a particle meant that the particle was more likely to be found there than in a region of space with a smaller wave function. This wave function could extend over a large area (like waves do) and was the cause of the interference and diffraction effects that particles exhibited.
This interpretation of the Schrodinger equation became the standard one for about half a century since it seemed to be able to reconcile the troubling idea that entities seemed to exhibit both wave and particle properties, what we refer to as wave-particle duality.
But like all scientific theories, it solved some problems while creating new ones and it was the effort to resolve those that led to further advances.
Next: Fields as a unifying concept.
I wish I had had this explanation for my high school Physics class (1956) Our teacher then made a quip that light behaved as a wave on Monday Wednesday and Friday and as a particle the other days.
I’m reading “The Last Man who Knew Everything” a biography of Thomas Young by Andrew Robinson. A fascinating account of the life of a true genius and polymath who’s name now is almost forgotten; relegated to Young’s Modulus and the two slit experiment.
The wave function of particles is a dreadfully difficult concept for me to fathom. I had quantum tunnelling explained to me as particles that move so close to a thin barrier that their wave function overlaps the barrier, and then because the wave function is, as you said, a probability function, if you look at the distribution of each particle in its wave function, statistically some particles must be on the other side already, and so there’s a steady stream of those ones speeding off as if though there was no barrier at all! Is that even close to how reality works?
It may clarify things to think in terms of a single particle. The generalization to many particles that are all in identical states is fairly straightforward.
One needs to also specify the initial conditions. Suppose that you had a wave function that describes a single particle moving to the right that is approaching a barrier. What quantum mechanics (i.e., the solution of the Schrodinger equation for this situation) says is that the wave function would be non-zero on both sides of the barrier. This means that there is a non-zero probability of finding that particle in three possible states:
(a) On the left of the barrier and moving towards it. This is the initial state.
(b) On the left of the barrier and moving to the left (i.e., away from the barrier). We say that this particle has been reflected by the barrier.
(c) On the right of the barrier and moving to the right. We say that the particle has penetrated the barrier and been transmitted.
Although one can find the particle on the right of the barrier, it is not the same as there being no barrier at all. If there were no barrier, one would have 100% probability of finding the particle on the right at a later time. With the barrier, the probability is reduced.
If one has many particles initially in the identical state, then one can calculate the numbers in each of those final states by multiplying the total number by the probabilities for each of the three situations above.
Hope that helps.
So a thin piece of glass and a thin piece of wood just affect the wave function of particles differently? 🙂 I realize that’s probably an overly simplistic way of looking at things, but is that more or less what transparent or translucent means in terms of what we see compared to what a particle ‘sees’?
What the Schrodinger equation needs is an expression for the magnitude of the potential due to the barrier and the thickness of the barrier. A larger potential corresponds to a less penetrating barrier and would result in a lower probability past the barrier.
A thicker barrier would also reduce the probability past the barrier.
So basically one would need be able to translate the physical properties of the barrier into an expression for a potential that can be inserted into the equation.
No love for hometown heroes Michelson and Morley in failing to detect the ether? I know that didn’t perhaps seal the deal completely, but it’s one of the biggest things to happen in Case’s history and fits naturally here. (Though for reasons of length, I can see skipping them.)
Einstein probably got less phallic memorials than theirs, though.
Thank you Mano, after some thought it actually does, although now I can’t avoid being astonished how solid reality appears on the large scale when it seems so capricious at the small scale.
Good point. I have added the recognition!
Of course we Dutch always have backed Huygens ….
You stir up sweet memories of my training as a teacher maths and physics during the 80’s. The director of the physics department of my institute was the younger brother of Gerard ‘t Hooft. He predicted a Nobel Price back then, something I didn’t take seriously.
Will Heisenberg and his famous principle come along? It made me decide calling myself an atheist iso an agnost, which I had been for 10 years or so. No coming out needed; it made very little difference on the way I viewed the world already. But even today I fail to see how it can be combined with any causal god.
Only later I learned that I was offered the Copenhagen Interpretation.
I have never met ‘t Hooft but from all that I’ve read and heard about him, he is brilliant and confident but also not arrogant, a nice combination.
So I was thinking about this. We like to think of wave functions using incredibly simple examples, for instance a particle moving toward a barrier. But we have already set arbitrary limits--namely, on the time of the experiment, and the objects in play.
We could just as easily have set up two barriers. Now we have the question--what’s the probability that the particle will penetrate both barriers? My guess is it goes something like the square of the probability of only penetrating one barrier--that is, astronomically small. We were already probably talking something like 10^-30 (I’m totally making this up), but then with a second barrier, we’d get 10^-60.
Now let’s look at reality. How many “barriers” do particles have to go through? The air is chock full of gasses, there are walls everywhere, and humans operate on INCREDIBLY slow scales compared to these experiments that we’re looking at. I’d bet we have quantum tunneling happening around us up the wazoo. The only thing is--a particle might quantum tunnel a very small distance through some incredibly small barrier (e.g. through another atom), but the probability of it achieving a large chain of quantum tunneling events, a chain large enough that we’d call it “macroscopic,” is so small that it just never happens. Ditto for the probability of many objects together all quantum tunneling at the same instant, in the same direction. Take a baseball with, say, one trillion atoms (no idea if this is the right scale)--we’d have to take our probability of tunneling through something and divide it by one trillion.
So--for us to notice these quantum events on the macroscopic scale, we would need an unbelievably unlikely series of unbelievably unlikely events. The result is that these events simply do not occur!
Mano, is this anywhere near the correct explanation?
This wave/particle duality has to do with the eigenfunctions corresponding to the observables in question, right? If it is a Dirac delta “function”, that is particle like.
That is an excellent description of why such quantum effects are highly unlikely to be observed in our classical world.
Professor Singham,
I’m really enjoying this physics series, and really learning something and clarifying some confused thoughts.
Thanks so much for doing this. I’m looking forward to learning something about field theory.
I hope also you might be able eventually to write about entanglement, the Einstein-Rosen-Podalski objection, the Bell inequalities, and how violation of the inequalities proves that there are no hidden variables. My intuitiion wants there to be hidden variables. The many-worlds interpretation is interesting but also disturbing and hard to accept. Maybe you can shed some light here?
Those topics are fascinating and although I have some knowledge of them, they are somewhat removed from the focus of the present series and require a different series of posts. If I can at some future time, I will do so.
Not quite. The eigenstates depend of the operator of which it is an eigenstate. Do you mean that the eigenstate of the position operator is a particle and that of (say) a momentum operator is a wave?
This sequence at Less Wrong covers those topics very well.
Thanks for the tip Paul.
ARGH, you are right; I botched it. I meant to say that the eigenfunction of the position operator is a Dirac which yields a specific “point value” which corresponds to “particle like” behavior and the eigenfunction of, say, the momentum operator has “wave like” properties.
That is why you are qualified to teach QM and I am barely qualified to read elementary QM texts. 🙂