Quantum Physics

Not one mill or waterwheel in sight, Mr Halfwise! "Science"? Idle self-indulgent fantasizing in my view!

Many people at the time had similar complaints. Many still do. And I'm talking about physicists mind you!

Not to rush you Halfwise but can we expect another instalment soon?

I should have it ready tomorrow night (given the time zone difference I wouldn't wait up for it in Scotshobbitland, more like thursday morning).

The difficulty is in explaining all these things without assuming a physics background. I've realized that I thought this would be easier than it is because I had taught this in high school, but that was with kids who I had been preparing for over a year. And I could use math with them, which streamlines the logic. I'm thinking maybe I should have skipped the historical development and just jumped to the clean modern understanding of quantum mechanics, but too late for that now.

I do fear that if I have this much trouble writing it, it will be an even greater chore to read it. We'll see. Assuredly the next installment will be easier both to write and understand.

I await to have my mind boggled!

Part II: History of a Revolution

This installment is a guided tour of the labyrinth the creators of QM threaded their way through enroute to a redesigned view of reality. By intention you will not have a clear idea of quantum mechanics at the end, because neither did they. The chapter will end at the point where the broad outlines were beginning to gel. This will allow appreciation of the clarity of vision presented in the third installment, drawn out of the tangled mess of observations and the nascent theories used to explain them. The achievement of the first generation of quantum scientists is equivalent to inferring the logical geometric pattern of a Moorish tapestry (quantum mechanics) based on the tangled ends of threads along the edge (the observations).

Thermal Radiation

We begin with thermal radiation inside an opaque cavity, such as in metal. Your oven will do quite nicely as an example. When we say ‘radiation’, we mean electromagnetic waves, which includes radio waves, microwaves, infrared, visible light, ultraviolet, x-rays…the whole shmear. If you want to observe what the radiation is doing inside, you can cut a small enough hole that it won’t have much effect, and study the radiation coming out. Not unlike peering through a keyhole to see what nefarious things your neighbors are up to.

It seems like a rather cerebral and unimportant case, but the idea is to take a very well defined problem and then use it as the standard for less well defined problems. For example, a fire burning in your fireplace is not enclosed in a cavity, but if you put that same fire in such a cavity so that all light is confined and accounted for, the light emitted from that fire would be somewhat different but fairly close to the open fire (assuming the oxygen feeding it etc would be the same). Therefore if you calculate the light from a fire inside a cavity with a small hole in it, you have a pretty good estimate for what an open fire would do. And a star. And a cloud. And a person. All of these measurements of the temperature of objects based on their emitted radiation is based on this ideal case which is never quite satisfied, but often does well enough for practical purposes.

It turns out a whole range of different wavelengths of radiant energy are being emitted from your oven, most of it in infrared wavelengths. What physicists wanted to understand is how the amount of energy emitted depends on the temperature of the oven and the wavelength of light you choose to look at. Before we look at the theory let’s build up some intuition with an example we are all familiar with:


If you look closely at the candle flame you see a range of colors. Nearest the wick there appears to be a dark area, which is actually pouring out ultraviolet light, we just can’t see it. Next in terms of distance is blue, then you’ll see a breath of green, yellow, and furthest away, red. The colors closest to the wick represent the light emitted from the hottest part of the gas in the flame, which means blue is the highest energy and red the lowest and coolest. (Actually there’s infrared on past the red, but we can’t see that either.) The colors are not pure, but each region described has the largest amount of light in that wavelength, and the middle white region has a nearly uniform mixture of them all. So we expect that light from a heated object will have a mixture of wavelengths with a peak that moves from red to blue (including above and below) as the temperature increases.

The physicists knew they had to come up with this same answer using the wave theory of light. They knew there were a bunch of different standing waves that could fit in a cavity, and they assumed there had to be equal amounts of each one.

“Why must there be equal amounts of each one?” you ask astutely. The answer is, (and I’m not making this up and it actually works extremely well for a wide variety of cases), “Because we can’t think of any reason why one wave should get more than any other type of wave.” But since the shorter waves have more energy than the longer ones, you get more and more energy with higher frequency (shorter wave = higher frequency). This is shown by the classical line in the graph below that shows energy emitted for each frequency: as the frequency increases the theoretical intensity shoots off to the moon.


In other words, because all wavelengths are equally possible and shorter wavelengths have greater energy, as soon as you peek in the oven your eyes would be burned out by ultraviolet radiation. In fact if by some miracle the oven didn’t melt, a single pinprick would burn your house down, followed by the entire known universe. This was known as the “Ultraviolet Catastrophe”, and it was a big problem for classical physics.

The reality is shown in the colored curves with each temperature represented by a different color: if the cavity has a temperature of about 3000 Kelvin, it will have a gentle peak in the red, but all other frequencies/wavelengths are represented. As the temperature increases the total amount of energy emitted increases, but the peak also becomes better defined and moves to the blue. Note that the blue in the candle flame looks more pure than the red. We should note that the classical prediction (made for 5000 K) starts out at very low frequencies matching the real case, but then takes off like a rocket. What is needed is some way to make the higher energy waves be less favored: the battle between these two tendencies will result in the peak we see.

Max Planck was trying to come up with a way to make this happen. His solution was to assume that the energy emitted had to come in packets proportional to the frequency of the waves. The lower frequency (longer wavelength) light would be emitted in small packets, the higher frequency (shorter wavelength) light would be emitted in larger packets. The higher frequencies would therefore be more difficult to emit. It’s sort of like if you were filling a hole with rocks of all different colors. In the classical picture the fill would consist of pebbles all of the same size but of different colors. Let’s say the energy applied to shovelling represents the temperature: higher temperature, more active shovelling. No matter how strongly you employed the shovel, all colors would be mixed equally. In the new quantum picture, each color of pebble would be a different size, with infrared ones being small sand and ultraviolet being large rocks. Higher temperatures would mean stronger shovel work that allow more of the large rocks to go flying.

Anyway, in 1900 Max Planck applied this type of idea, and did some curve fitting to find the proportionality between frequency and energy. The fit was perfect. The ratio between frequency and energy that he found was forever after known as Planck’s Constant, denoted by ‘h’, and part of the fundamental structure of the universe and Quantum Mechanics. If h goes to zero, quantum mechanics reduces to classical mechanics: it is the lynchpin of the new physics. The equation he came up with was too complex and worked in too many situations to be coincidence. There was no doubt energy was being quantized, but what did it mean? Everybody was happy to assume the quantization was due to the still unknown structure of matter, resonant vibrations or some such. It never crossed anybody’s mind that light itself was quantized and came in packets, for after all, didn’t all the experiments show conclusively that light was a wave?

The Photo-Electric Effect

Matters might have remained in this state for decades if not for another phenomena that the wave theory of light could not explain: the photo-electric effect. It’s a staple of modern technology, used anywhere from the light sensor in your camera to solar cells. The effect is simple: when light hits certain metals (all oxidation cleaned off) electrons are emitted. The number of electrons emitted is proportional to the amount of light, but the kinetic energy is proportional to the frequency of the light waves. Energy per electron and number of electrons are independent: turn high frequency light down low and you still get high energy electrons, just less of them. Moreover, if the frequency of the light becomes too low, if doesn’t matter how much light hits the metal - you can blast it with a laser; the electrons will stubbornly stay in put.

Physicists around the turn of the century could easily accept that it took a certain amount of energy to free an electron, but waves were continuous: when you shine light on an object, no matter how weak, eventually the electrons should soak up enough energy to bust free. The amount of time it took to free an electron should only depend on the light intensity. But real electrons didn’t listen to the wave theory: they popped out the instant light of high enough frequency hit the metal, regardless of the total light energy. Wave theory could not explain this.

Enter Albert Einstein. He realized that the electrons must be receiving their energy in packets (he got this idea from Planck’s paper), and any extra energy went into the kinetic energy of the ejected electron. The energy of each photon is the frequency (often written v) multiplied by Planck’s constant h. This simple idea is shown in the figure below of photons (Einstein came up with the word, which in my mind is as great as the theory itself) hitting metal and causing electrons to be ejected.

If low frequency, long wavelength red photons hit the metal, they don’t have enough energy to kick loose an electron. It doesn’t matter how bright you turn up the light because each individual photon either has what it takes or it doesn’t. If you use higher energy photons (green or blue) the excess energy is given to the electrons, so that blue light causes electrons to be popped off with higher velocities. (For most metals though, visible light doesn’t have enough energy to free electrons, you need ultraviolet light - which is why your lawn chair doesn’t zap you on a sunny day.)

This simple model of light coming in packets explained the observations perfectly, but since everyone knew light was a wave (it makes interference patterns, for chrissakes!), nobody really knew what to do with it at first. The synthesis of this wave/particle duality is the heart of quantum mechanics, and is the source of all quantum weirdness. 50 years after he invented the photon, Einstein himself admitted he didn’t understand it, and claimed anyone who said they did was fooling themselves. If it’s true of Einstein, I’m sorry to inform you it is true of everyone. I will deliberately skirt this wierdness until we have the full framework of QM in place, otherwise you will be too busy battling with your mental dragons to pick up anything new. Somehow, somehow, light is both a particle and a wave: let’s just nod our heads and agree to it, and move on.

The Bohr Atom

You should recall from the waves installment that when light of a constant wavelength passes through two slits, it produces an interference pattern that looks like ripples on the screen across from the slits. The spacing between the ripples is proportional to the wavelength of light. It turns out that if you have a whole lot of slits, the ripples sharpen into spikes. This is important because if you send two different wavelengths through the slits, broad ripples would overlap and be confusing, while sharp spikes would be easier to see. An example with red and blue light is shown below:

We can see that the red spikes are further apart than the blue spikes. If we had a continuous band of colors, the spikes would form a rainbow. You in fact can see such a rainbow on a CD: it has grooves in it close enough together to form an interference pattern out of visible light.

So what happens if we look at the light emitted from a pure sample of heated atoms? If It’s hydrogen, we see the spectrum below:


Well, that’s interesting….instead of a rainbow we see very sharp lines. This means that only specific wavelengths are being emitted. What does that tell us about the atom?

In the classical world, we have protons and electrons, and they attract each other in a way that looks very much like gravity (inverse square law). An atom should therefore look like a tiny solar system, with electrons orbiting the proton nucleus like planets around the sun. Now it turns out that each planetary orbit corresponds to a certain amount of energy: the further out, the higher the energy. If you want Mars to fall into the Earth’s orbit, you actually have to slow it down (take away energy) so that it falls out of it’s orbit, then you can redirect it into Earth’s orbit. We can take energy out of a planetary orbit by firing big rockets or something; for a an electron changing orbits in the atom, the energy would come by emitting light. The difference in energy between orbits would equal the energy of light emitted.

Here’s the problem: in our classical world, the planets can be orbiting at any distance from the sun, so we should see a continuous rainbow of wavelengths emitted from an atom. It gets worse: since anything in orbit is constantly changing it’s velocity, a charged object should be constantly emitting waves, and spiral into the nucleus in a fraction of a second (planets could do this too, but the emission of gravitational waves is an almost infinitely slower process).

Not only are atoms stable, but they don’t have a continuous spectrum: they have discrete wavelengths being emitted. Neils Bohr realized the two were related: if only discrete energies are allowed, the electron can’t make a continuous spiral into the nucleus. He was also inspired by Einstein’s construction of the photon, which meant the old wave theory of radiation may not apply on the atomic level (though he rejected the photon itself). All he had to do was come up with a reasonable mathematical model to get the right energy levels. It should be remarked here that the spectrum of hydrogen did not show energy in equal increments, it seemed to go as 1/n^2, where n was a series of integers. But how to come up with that pattern?

What Bohr settled on was quantizing the angular momentum: each orbit had a different angular momentum in equal increments of Planck’s constant h, and when you did all the orbital math (which I will NOT drag you through), it resulted in the right energy levels!


The picture is shown above, with each orbital n having an angular momentum nh, and an energy calculated from this. When an electron drops from one orbit to the lower one, it emits a photon with energy equal to the difference between the two levels. If a photon of the right energy hits the atom, an electron can be promoted to a higher level. If the energy is wrong, no absorption occurs! (Though Bohr did not at first accept the photon, the picture is much clearer if we do.)

Bohr published his model in 1913, 8 years after Einstein’s photon paper. I’d like to say it was embraced with open arms, but like the photon it was just too strange. The idea of quantized angular momentum seemed to come out of thin air. But the hydrogen spectrum was more important than the photoelectric effect, so it could not be ignored. More people began paying attention to the idea of quantized phenomena, but it’s safe to say nobody was comfortable with the idea.

Side Developments: 1913-1925

A long decade followed where the main progress was variations on the Bohr model: elliptical orbits, magnetic interactions between the electron and proton, and there seemed to be some agreement with the fine structure seen in the hydrogen spectra.

Some statistical work by the Indian physicist Bose showed that thermal radiation could be derived by using photons instead of waves, which made everyone take photons seriously.

Werner Heisenberg, a colleage of Bohr’s, decided that all the classical analogies were causing distractions that obscured the strange truths behind quantum mechanics. He worked to devise a system based only on what could be observed: the frequencies and intensities of light emitted; rather than on things that could not be observed, such as the electron’s speed and actual position inside the atom. His only other requirement was that when Planck’s constant went to zero the equations would be classical. Though he came up with what is considered to be the first complete theory of quantum mechanics, his procedure to arrive at his final result was so abstract and ‘magical’ (his colleage’s words) that to this day nearly nobody understands his thinking process. Steven Weinberg, one of the creators of the Standard Model of particle physics and someone emininently qualified to follow esoteric arguments, once tried to read the original paper and admitted he could not make heads or tails of it.

So we will skip Heisenberg’s approach, and come back to such results as the Heisenberg Uncertainty Principle in the next installment on theory and applications of quantum mechanics, all based on nice comfortable waves. Nothing is lost, for the wave theory we are following is in every way equivalent and in many ways superior to Heisenberg’s version.

Matter Waves

A French graduate student called Louis de Broglie imagined Einstein’s concept of turning light waves into particles could perhaps be reversed, so that particles of matter could be turned into waves. The only difference between the two is that matter has mass and light does not. Einstein had related energy and frequency, but the rest energy of massive particles (E=mc^2) would complicate this relationship, so de Broglie instead related wavelength to momentum:

Wavelength = h/momentum

This means the wavelength of a particle will decrease with speed and mass (momentum). You and I with our huge lumbering masses would have wavelengths too tiny to detect no matter how slow we move, which is why we don’t notice wave effects in every day life.

At the time it was not possible to do scattering experiments with atomic particles to prove they were waves, so de Broglie applied his wave idea to the hydrogen atom. The idea is that as the electron orbit gets further and further from the nucleus, the speed needed to maintain orbit decreases. So the wavelength of electrons in higher orbitals will become longer. But for a wave to exist in orbit, it’s going around biting it’s tail in a circle. To avoid canceling itself out, it must be just the right size to form a standing wave!


This means that only certain orbits are allowed, the electron can’t be just anywhere. When de Broglie worked out the math, he came up with orbits of the Bohr atom! Suddenly the strange quantized angular momentum made sense as a set of standing waves, like vibrating strings but in a circle around the nucleus.

De Broglie’s advisor was afraid to accept such an extreme idea: all matter was made of waves? So he sent the thesis draft to Einstein, who wrote back enthusiastically. The thesis was published and de Broglie’s idea became one of the foundations of quantum mechanics. His is the only PhD thesis in history to win a Nobel prize.

Quantum Wave Mechanics

Erwin Schroedinger was a classical physicist: he didn’t like photons, he didn’t like the Bohr atom. But the quantum jumps seen in atomic spectra needed to be explained, and de Broglie’s wave theory was the closest thing to a continuum approach out there. The problem was that it was too simplistic: waves did not move along one dimension, they were 3 dimensional apparitions. Physicists knew how to calculate multi-dimensional waves, as seen in the simulation of waves in a drum head shown in the animation below:

http://en.wikipedia.org/wiki/Vibrations_of_a_circular_membrane
(scroll down the webpage to see more examples)

Note that there are several different ways the drum head can vibrate, each with different energies, but there can’t be anything in between or else you won’t get a standing wave and it will cancel itself out. Sounds a lot like atomic spectra.

During a two-week vacation to a cabin in the Swiss Alps, Schroedinger figured out how to convert de Broglie’s equation into multidimensions. Then he put it into spherical coordinates and calculated the standing waves in the hydrogen atom:

the result was what you find in chemistry books, with the electron smeared into these oddly shaped clouds, each with a different energy. These are now referred to as ‘shells’. There may be multiple orientations of each standing wave, leading to subshells in each shell. The theory explained all the minutia not just of the hydrogen atom, but (with significant effort and nowadays computers) all the atoms and molecules. For example, in the helium spectrum shown below every blur is accounted for. At the atomic level, the problem was completely solved.


At least it was solved if you just looked at the spectrum of light emitted by atoms and didn’t think about the underlying theory too hard. Let’s see what we have:

• Light definitely acts like a wave, but sometimes we see particle like behavior.
• Electrons, protons, etc are definitely particles, but to fit with the observations of atomic spectra, they must be treated as waves.
• Both of these are tied together by Planck’s constant h, which seems to be a measure of the granularity built into the universe. If h was equal to zero, classical mechanics would reign supreme.

So how to put this together into a consistent, understandable picture? That’s the topic of the next installment.

Ok my head hurts- but in a good way!
Im still a little vague on why they treat particles as waves just cause it fits better- that seems like cheating! Bit like saying this elephant works better if you treat is like a small dog- which it might but at the end of the day that wont stop it leaving a dump the size of a small car on your front lawn!


ps- and at least your subject matter leaves you less likely to get pages of comments about Eru's arse making abilities!

that's the central mystery of quantum mechanics: you treat everything as a wave....until you look at it.

This problem won't go away in the next installment, it will become more clearly defined. There is no way to think about QM without your head hurting. Eventually you just accept the conflict and the headache goes away.

I picked a really bad time to read this... most of the way through my working day having had a very tiring bank holiday weekend and days since.

I need to revisit this when I am in a better frame of mind.... I did all of the wave stuff when I was doing Music Production in college... and when I was at school. Went straight over my head today.

After a lifetime of listening to Geordie La Forge and Samanta Carter on T.V.... this is definitely something I would like to have a grasp of. If for no other reason than to shout at the telly when they get it wrong!

If you've done waves before then the previous installment will be a nice review, though I have avoided all the Fourier decomposition stuff you'll be familiar with.

Its very interesting stuff, its quite a challenge for my wee brain but I will try to follow. is there any possibility we will discover new colours? probably not if its all set in the visible spectrum, I know our eyes cant see IR and UV and Radiation but maybe there is some strange new colour out there in the Universe.

Interestingly enough Schroedinger pioneered the vector theory of color which was later discovered to roughly reflect the three color receptors in our eye. Since there's so many artist types here I think a discourse on the biological basis of color is in order. Interestingly, one of our color sensors actually is sensitive to ultraviolet, but since UV gets absorbed in the eyeball fluid we don't see it. This may be your missing color!

somebody who could see the near UV: http://starklab.slu.edu/humanUV.htm

I have used UV light and IR photography and its weird stuff, Lead White comes up black if memory serves me right, another interesting phenomena are those people who see colours as smells, or hear colours, cant remember the name of the condition, but it must be odd if you suddenly develop it.

I know of one person with perfect pitch who hears the tones in music as colors - sort of like the visi-sonar in the Foundation trilogy.

wow I would really like to experience that for a day or two, I wonder if we all have that capacity or is it a genetic fluke?

Pettytyrant101 wrote:Ok my head hurts- but in a good way!
Im still a little vague on why they treat particles as waves just cause it fits better- that seems like cheating! Bit like saying this elephant works better if you treat is like a small dog

Actually that's not a bad analogy. When Native Americans first met horses many cultures saw them as giant dogs, which gave them a model for working with the strange creatures. Then they started exploring the differences, eventually producing some of the best horsemen in the world.

Likewise when Francis Drake was liberating South American Spanish of their wealth, he commented on their peculiar giant sheep....that 3 men can sit upon....with necks like camels. To have said, "treat it like a llama" would have been meaningless to a common British sailor, so they adopted the sheep model as something they already understood.

Similarly the word "wave" can be useful but deceptive. It's all goes back to an analogy to surface waves on water, but when it's applied to pressure waves such as sound, the analogy isn't perfect. Electromagnetic "waves" have even more quirks. Quantum mechanics takes the wave analogy to the breaking point and beyond.

"Wave" is an extremely useful analogy when trying to visualize things that are invisible, but you've got to be able to drop it when it stops working.

I definitely see patterns in my mind when I listen to music, but I think it's just me imagining the musical lines and textures. I don't see quite the same thing twice. If you're happy with your imagination that may be good enough!

that's a good point Dave. the term 'wavicles' has been proposed to classify what we see in QM.

I always thought 'wavilces' kinda smacked of desperation to just classify something they could not explain. And as invented scientific words go its not a patch on 'photon'.

I have expericienced sound as colour, as a physical visible mass without any sound component- I was however a lot younger and full of lsd at the time. However I have never adhered to the, it was just an hallucination explaination, as that explains nothing. For me it demonstrates that humans are capable of interpreting the universe in a variety of ways and that way we do normally is learned. But thats a conclusion coming at it from the experimental point of view lets say!

It's not desperation - you need to understand that our limited view of reality is not sufficient to cover all the concepts needed to explain the universe. So you coin a name for a new concept.

We see two ends of a phenomena, the wave end and the particle end. There's a lot going on in the middle that we are not familiar with. Instead of boxing things into our physical intuition, we have to follow wherever observations and logic lead us.

When asked to define the electron, Bertrand Russell replied "an electron is a logical construction."

It just sometimes seems like scientists like to give something a name as a way of not actually having to find out what it is!

'What is it?"
"Well its a a wave, until you look at it then its a particle."
"Well thats weird. How do you explain it."
"I dont but I've thought of a cool name for it, a wavicle."
"Right, lets hope no one asks any questions then."

Pettytyrant101 wrote:I always thought 'wavilces' kinda smacked of desperation to just classify something they could not explain.

So what would you suggest doing when it becomes clear that your classification system is too small? (Think about horses to aboriginal Americans. )

What Dave said.

Well nothing wrong in coming up with a new classification, but 'wavicle'? It seems desperate partly because its such a rubbish obviously made up word got by mashing two exisitng words together- it means nothing in of itself.