Friday Science: Where are we in time?

Review Post #2

As I look to a possible rhythm of posting for this fall, I'm thinking of looking at Fridays as a science/math type day. So today I thought I'd review the second chapter of Max Tegmark's, Our Mathematical Universe. Some thoughts on the first two chapters are here.

1. So if the previous chapter tried to locate us in space, the third chapter tries to locate us in time. I don't think there were many controversial matters in the previous chapter. But when you begin talking about the age of the universe, some do get nervous.

So before I dive in the chapter, it's worthwhile to do a little Christian processing first. My sense is that the real rub here has to do more with the question of evolution than it does the actual age of the earth. So even John Piper himself does not think the question of the earth's age is restricted by the Bible.

Here the "apparent age" theory, I think, opens the door to let science be science. It would be problematic to my faith if we had to "cook the books" when it came to science because of a supposed interpretation of the Bible, especially when 1) there are so many contradictory interpretations of the Bible and 2) as a Bible scholar, I know that those who care the most about these sorts of things are usually the most ill-equipped interpreters of all.

The apparent age theory is the idea that God created the universe to look like it was really old, even though it is actually very young. I am not endorsing this theory. I am simply saying that, if we allow for the possibility that God made the universe to look old--whether it is or isn't. Then we are free to let the chips lie where they fall as we look at the evidence. Faith isn't at stake so we can be honest with the evidence.

"Whew," so we don't have to be worried about how old the evidence seems to suggest the universe is.

2. It was an interesting thought Tegmark suggests at the beginning of the chapter that, for the Romans, the laws governing the skies appeared to be different than the laws governing the earth. For example, why doesn't the moon fall out of the sky to the ground the way things I throw up do? I didn't see a footnote here, so I'd like to hear it from the ancient horse's mouth.

But it does make a little sense. Many Jews at the time of Christ thought of the stars as angelic beings. (Which makes the star over Bethlehem potentially get very interesting) Some Greeks and Romans thought of stars as great heroes from the past. These ideas would at least fit with what Tegmark says because, in such cases, the things in the sky would be beings, not rocks. (although Anaxagoras did think the sun was a very hot rock centuries before Christ... he was kicked out of Athens)

By contrast, Newton assumed that the objects in the sky followed the same rules as things on earth. He calculated how fast something orbiting the earth would have to be moving to stay in orbit. His answer was 7.9 kilometers per second. Friction of tides and such is slowing the spinning of the earth, so things were spinning faster in the past than at present.

Apparently, if you extrapolate back, the earth can't be more than 4 or 5 billions years old, because it would have been spinning too fast then to stay together.

3. I suspect the current theory for how the solar system, the galaxies, and the universe formed fits with a host of data. It is not proven, mind you. It is just impressive in its ability to explain a great deal of evidence. If you have a cloud of hydrogen gas--the simplest of all elements, one proton and one electron--two forces will tug at each other.

The one is the force of gravity. It pulls the gas in on itself. The other is the heat of hydrogen running into itself. This heat pushes expansion.

So what happens in mathematical models of this sort of thing is that you eventually get a swirling pizza with a star at the center. The hydrogen in the middle gets so hot that it fuses to form helium, the second simplest element (two protons, two electrons, and two neutrons). The energy levels are also just right for these to fuse to form carbon and from there all the other elements as this plasma mix cools off.

It is a theory, of course. But it seems to make sense, and I can't think of anything in the Bible that would make it problematic.

4. So how old is the earth? Here we get to radioactive decay. The atomic bombs we dropped on Hiroshima and Nagasaki seemed to suggest that our notions of radioactive decay work pretty well with the evidence. What happens is that certain versions of uranium (we call them isotopes) are unstable and tend to disintegrate into lead. This is a process whose rate we have measured.

The "half life" of a radioactive element is how long it takes for half of that element to deteriorate into a simpler element, lower on the periodic table. So at present, it sure seems like we can date things by seeing, for example, how much lead and how much uranium are in a given rock sample. They have done such with certain rocks in Australia, as well as with meteorites and the result is the suggestion that the earth is about 4.4 billion years old, while the solar system is about 4.5 billion years old.

Now these conclusions are based on theories, but they seem to have consistently great explanatory power. Again, for those who think the Bible absolutely teaches a young earth, there is a release valve. If we allow for the possibility that God created rocks with some of the uranium already decomposed, we don't have to worry about this issue. We can take the science at its face value and say that the earth and solar system at least look to be billions of years old.

5. The universe is expanding. This was a hard won conclusion of twentieth century science. There were several people early on who suggested that this was a natural conclusion of Einstein's general relativity. But they were not popular voices. It is a great example of how peer pressure and popularity contests take place in science too.

So a guy named Fred Hoyle had a winning personality and he was saying what people wanted to hear--that the universe wasn't expanding but was pretty much just going along as usual. Meanwhile, Georges Lemaître, a Catholic priest, was ignored when he suggested that the universe was expanding.

And here let me pause to express my confusion as to why some Christians think the idea of a "big bang" at the beginning is problematic. What is a big bang but the indication that the universe had a beginning? It seems to fit very nicely with the cosmological argument for God's existence, that the bang needed someone to trigger it. All I can think of is that people associate a big bang with evolution or assume it means there is no one doing the banging.

On the contrary, you might paraphrase Genesis 1:1 to say, "In the beginning, God caused a big bang, and the heavens and earth were the result." Could it be that the reason so many rejected the expanding universe of Lemaître and Friedmann is precisely because of how well it fit with the idea of a creation?

But the discovery of the "blue shift" of all galaxies we can see (they are all "moving away" from us, like the Doppler effect when an ambulance passes you and the sound gets stretched out) suggests the universe is expanding. When you do the math, it suggests that the universe is about 14 billion years old.

6. So here's a question. If we can see stars that are more than 14 billion light years away (a light year is the distance light travels in a year, going 186,000 meters per second), then doesn't the universe have to be older than that for us to see them?

Here is a key insight from Einstein's general relativity. It's not really that the stars are moving away from us. It's that space--the emptiness itself--is expanding. The space between things is getting bigger everywhere. Fascinating!

Nothing can move faster than the speed of light through space. "But space itself is free to stretch however fast it wants to" (48).

7. Looking into space is like looking back in time. And our telescopes get better and better. Thank the Lord that NASA somehow got money out of Congress to send up telescopes and probes into space.

So the further you look out into space, the older the things you are seeing. We can now see galaxies forming, and what we see looks just like the theory of coagulating gases I mentioned above. The further out you go, you get to the plasma pool of hydrogen atoms. And finally you get background radiation, the afterglow of the beginning. You hear the echo of God's word, "Let there be light."

I'll leave the chapter at that. Tegmark does not believe it all goes back to a singularity, an infinite point. We'll see what he is thinking here in later chapters, I'm sure. But he thinks that the quantum mechanics breaks down about a second before "nucleosynthesis," the formation of nucleuses.

We'll see what that means. By "big bang," he means that "everything we can observe was once hotter than the core of the Sun, expanding so fast that it doubled its size in under a second" (65). Not my definition, but we'll see where he goes with it.

Feynman 3: The Bomb and then Depression

This is my third installment reviewing a biography about Richard Feynman called Quantum Man, one of the greatest physicists of all time.

So far:
Chapters 1-2: High school, MIT, and Princeton
Chapters 3-5: The Path to a Doctorate

Now chapters 6-7.

Chapter 6: Loss of Innocence
Finishing his PhD degree at Princeton was a condition for Richard and his fiancee Arline to get married. And so, even though she was deathly ill with tuberculous--and generating significant friction between Feynman and his mother--they got married. She would die two months before the bomb dropped on Hiroshima.

If you remember from last week, Feynman had helped work on the question of separating Uranium 235 from 238. He had worked with Robert Wilson's team on this, and it had been accomplished. Then next step was to build a nuclear reactor to do the separation, and this was taking place in Chicago with Enrico Fermi and John Archibald Wheeler, Feynman's doctoral mentor.

So Feynman went to Chicago. Soon after he arrived, he blew away the "theory group" by performing a calculation that had eluded them for months (78). Robert Oppenheimer picked Los Alamos as the place where the Manhattan Project would play out, and he picked Feynman to come with the first wave of scientists in 1943.

Oppenheimer was an unusual scientist because, as Feynman himself put it, he "was extremely human" (79). He not only understood the science. He had organizational skills and cared about people. He would notoriously regret the role he played in the creation of the atomic bomb. His words at the first successful testing were from the Bhagavad Gita: "Now I am become Death, the destroyer of worlds." Feynman merely grinned as he contemplated the physical causes of the mushroom cloud and sonic boom.

Feynman, once again working best in conversation, ended up almost by accident in a conversation in Los Alamos with a seasoned theoretician named Hans Bethe. Bethe would bounce ideas off of Feynman, who being very loud could be heard to cry out, "No, no! That's crazy." From Feynman's recollection, Bethe always proved to be right. From Bethe's recollection, Feynman was probably the most ingenious person in the whole division.

Bethe was the scientist who discovered that fusion fueled the sun. Bethe said of Feynman that "he could do anything, anything at all" (87). He was put in charge of the computing division. You have to wonder whether the Manhattan Project would have ended before the war if Feynman hadn't have been there.

For example, Feynman developed a mathematical method for integrating third-order differential equations that was more accurate than what they were doing with second-order differential equations. When several boxes full of the parts of an IBM computer arrived, Feynman and another person managed to put them together before the professionals from IBM arrived. It had never been done before.

Oppenheimer would say of Feynman that "He is by all odds the most brilliant young physicist here" (92).

Chapter 7: Paths to Greatness
As an academic Dean, I think I am somewhat unusual. I absolutely love knowledge for its own sake. My boss often quotes me as saying, "No one loves the irrelevant more than I do." But I am a pragmatist and a realist. In most cases, it doesn't matter how brilliant a teacher is if he or she can't teach. And in most cases, it doesn't matter how excellent your program is if no one is buying.

Of course there are top flight research institutions that are so heavily endowed that their professors can push the bounds of knowledge without a care and survive off of some small number of purely genius students. Feel free to hire me to teach there. But that is just not where the majority of academic institutions are. And, for all your pretense to greatness, most purists seem to have a penchant for self-destruction (and an overestimation of how great they are).

So I won't tell you what I wrote in the margins of this biography on reading about how the chair of Berkeley delayed making an offer to Feynman on Oppenheimer's recommendation. When he finally did, he had the gall to tell Feynman that no one had ever refused an offer from Berkeley's graduate department of physics.

Feynman did. He went to Cornell, who had the smarts to hire him two years earlier and give him a leave of absence while he was at Los Alamos. Stupid Berkeley.

But Feynman himself was pessimistic and depressed. What future was there, now that there was such a bomb. "What one fool can do, another can," he said (93). Indeed, it is amazing seventy years later that the bomb has not been used again.

It was natural that Feynman would feel like he had wasted three prime years of his intellect--the greatest discoveries of a physicist are usually made in one's twenties. His father had died of a stroke a year after his wife Arline. Teaching takes a whole lot more work than most imagine and back then there was no training in how to teach.

Feynman was being showered with praise from all corners, but he felt like his best years were behind him. Others thought he was incredible. He felt stupid. "They were absolutely crazy" (96), he thought of offers from Princeton and the Institute for Advanced Study.

Bethe, upon hearing of Feynman's depression, remarked that, "Feynman depressed is just a little more cheerful than any other person... exhuberant" (97).

There is a great story about how hard it was for Feynman to finally write up his dissertation for publication. Apparently, two of his friends forced him to write it up while he was visiting them in the summer of 1947. They practically locked him up in a room. It was easy for him to express his ideas in conversational form. But to write them down in a beginning to end argument in detail, with everything exactly write. That he found a hard time doing.

[I know a couple people I'd frankly like to lock in a room to crank some of their gems out. On the other hand, one might argue that I would have written more scholarly pieces these last ten years if I hadn't started blogging.]

However, writing it up seemed to get him over a hump. Quantum mechanics began to be more visual for him. For the first time, he began to describe quantum mechanics in the language of sums over paths, with each path having an amplitude. It was a fundamental reformulation of quantum mechanics on which all the quantum mechanics since is based. His next task was to relativize it, to incorporate Einstein's relativity into his new version of the older model.

The rest of the chapter mostly flashes back to Dirac's relativizing of the original quantum mechanical equation of Schrodinger. We hear about the spin of particles called fermions, after Enrico Fermi. We hear about boson particles that don't spin. We hear of Wolfgang Pauli's exclusion principle and Dirac's theoretical discovery of antiparticles.

Meanwhile, Feynman was trying to find a way to picture the overall paths of these particles in a way that incorporated relativity the way Dirac had for a single particle at particular time and momentum...

Greatest Physics Genius of the 1900s

Last week I finished blogging through Thirty Years That Shook Physics, a great book by George Gamow on the first thirty some years of the 1900s, when the groundwork of modern physics was laid.

If I were to pick, I would pick Einstein as the dominant figure of the first twenty years of the twentieth century. Max Planck may have suggested the quantum, but Einstein ran with it and, in the meantime, established both special (1905) and general (1915) relativity.

For the next twenty some years, 1920-40, I pick Paul Dirac as the greatest mind in physics. There are lots of candidates. I pick Niels Bohr as the most influential figure, but I don't think the greatest mind. Heisenburg, Schrodinger came up with the most central concepts of quantum mechanics, but at times it seems like they just were the ones who did what someone else would have done anyway. It was Dirac who combined Schrodinger's famous wave equation with relativity, the last time that longed for synthesis was successfully accomplished.

But I'm not sure that any of those I've mentioned were as genius as Richard Feynman. I don't feel like I know enough to have a firm opinion yet, but Feynman may have been the most brilliant physicist of the modern era. The book I'm reading for the next few weeks for my Science Fridays is Lawrence Krauss' biography of Feynman, Quantum Man.

Introduction
Krauss begins with some fun memories of his own personal engagement with Feynman. A book given to him in high school about Feynman's notion that antiparticles were particles moving back in time (it was originally John Archibald Wheeler's idea). In college he had Feynman's lectures on hand. And of course he mentioned Feynman's moment in the spotlight, when he showed everyone why the Challenger blew up.  (He dropped an O ring into ice water on television)

1. Feynman in High School
Feynman was born in 1918. He was obviously a genius. Unlike Dirac, from whom you could hardly extract a word, Feynman was flamboyant and was loud. But he could also tunnel into a problem for long periods of time. He kept notebooks with multiple solutions to the same problems. he was methodical. He taught himself subjects years before he got to them in school.

He was incredibly gifted at math--not all physicists are, Einstein for instance. Of course you have to remember that when I say, "not gifted at math," I mean that Einstein didn't invent a new branch of math. Heisenburg reinvented matrix mechanics on his own. Dirac made major contributions to linear algebra for his relativistic quantum mechanics. So Feynman would invent path integrals. Meanwhile, Einstein couldn't have worked out general relativity without the help of David Hilbert.

One principle Feynman learned in high school that he didn't like but that would be key to his later fame was Fermat's principle of least time: light takes the path through various media that gets it most quickly to its ultimate destination. It is a weird concept because it seems to imply planning on the part of light, as if it knows where it wants to go and plans accordingly.

Apparently, another way to state this principle is this: an object will take that path where the total sum of the difference between its kinetic energy at each point and its potential energy at each point is lowest. This sum is called the "Lagrangian" and the "action" of an object.

2. Feynman at MIT and Princeton
Feynman declared as a physics major at the end of his first year at MIT. It was just right for him between the non-concrete pure math and the all concrete engineering. He and a student named Ted Welton started taking advanced graduate physics courses their sophomore year. One of those professors met with the two of them their junior year to teach them quantum mechanics. This was about 1938.

Then he turned down Harvard to go study at Princeton, which at that time had Einstein and John Archibald Wheeler. Feynman was Jewish, and his parents worried both about whether he could make a go at physics and whether he would be accepted as a Jew there. He and Wheeler (the guy that coined the phrase, "black hole") worked well together.

It was during this period that Wheeler suggested some particles might move back in time. The specific problem they tackled was the self-energy of an electron. They unsuccessfully tried to eliminate that factor from the energy equation because it went to infinity according to calculations at that time. Their suggestion was to try to eliminate the idea of a electromagnetic field. The idea was that interactions would only take place by the direct interaction of charged particles, by the exchange of photons. The prevailing wisdom was that forces interact by "action at a distance," which is what fields allegedly do.

Wheeler and Feynman were apparently wrong, but it was exactly the kind of interaction that fed Feynman's growing sense of possibilities.

Krauss takes a couple pages in this chapter to give a quick two points on the nature of quantum mechanics. The first point is that, in QM, objects can be in different places doing different things simultaneously. The second is the Heisenberg uncertainty principle. There are certain pairs of qualities (position and momentum, energy and time) where the more we know one, the less we know the other. The product of our uncertainty in knowing one and the uncertainty of knowing the other will never be smaller than Planck's constant.

That's probably enough for one day...

Science Friday: The End of Gamow

Finally finished this classic of popular physics I've had since the 80s, George Gamow's classic, Thirty Years That Shook Physics.

The previous posts were:

1a. Planck's Quantum
1b. Jumping Photons (Einstein and the Photoelectric Effect)
1c. The Compton Effect (Proof of Energy Packets)

2a. Thomson and Rutherford's Atoms
2b. Bohr's Contributions (How electrons fill the atom)

3a. Pauli Exclusion Principle (no two electrons at any one energy state)
3b. The Pauli Neutrino

4a. De Broglie's Wavy Particles
4b. Schrödinger's Wave Equation

5. Heisenberg's Uncertainty Principle
6. Dirac's Anti-Particles
7. Chapters 7-8: Fermi and Yukawa

And now the final chapter, "Men and Work" and the playful stunt play, "The Blegdmasvej Faust"

1. Gamow was writing in the 1960s and, from his perspective, not much momentous had taken place in the some thirty years since the content of the book. To be sure, we have seen the perfection of the Standard Model with numerous particles not mentioned in Gamow's books (e.g., quarks). But as Lee Smolin's book, The Trouble with Physics, reflects, the breakthrough that Gamow hoped would take place before the year 2000 is yet to happen. [As an addendum, see this post of mine.]

Not only has Gamow sat on my shelves for years, there are a number of "physics classics" there waiting to be read in this series, QED, The First Three Minutes, A Brief History of Time. All in good time. But I think I'm going to go next to a biography of Richard Feynman. If Einstein typified 1900-1920, if Bohr dominated 1920-1940, Feynman is the central figure of quantum physics in the mid-twentieth century, it seems to me. So next Friday I hope to post on the first chapter of Quantum Man.

2. Gamow, I think, gives some great insights in this final chapter. He suggests that one path forward might be to explore the significance of a new set of fundamental "dimensions" in the quantum world. In the large world we inhabit, length, time, and mass are the key dimensions from which all other physical quantities and units are derived.

But what if there are really much more fundamental units when we get right down to the bottom line of existence at the quantum level. It's not hard to figure out what the first two of the three most fundamental constants might be. Planck stumbled on one--h or Planck's constant. It seems to pop up all over the place in the atomic world. The speed of light, c, is another, the dandy of Einstein.

But what might the third be? Gamow suggests that there might be a fundamental unit of distance, λ, anticipated from Pythagoras to Heisenberg. He suggests it is somewhere around 2.8 x 10^-13cm, the range of forces acting between nucleons and the point at which all the calculations seem to go to infinity. I don't know whether this specific suggestion ever went anywhere, although I do believe the notion that there is a quantum distance of this sort is quite commonly held (e.g., a Planck length).

2. He ends with an informal play performed in 1932 in Copenhagen, a riff off the German play Faust, in which a genius is bored with all normal learning and sells his soul to the Devil. Gamow was prevented from attending because the Soviet's wouldn't let him go from Russia at that time.

Faust itself is rather sacrilegious, so it might be a bit jarring. God stands for Bohr in the play, and Pauli is the Devil (Mephisto). God is pleaing for Mephisto not to take away all of his prized notions, like mass and charge. All the Devil is interested in is the neutrino.

Probably the best known part of the play is Faust's opening monologue and it's fun to see what they've done with it. "I have--alas--learned Valence Chemistry... Yet here I stand, for all my lore, No wiser than I was before..."

The play seems to jump from image to image. No doubt those at the conference found much of it hilarious (such as things falling into Dirac's holes). Mephisto (Pauli) longs for his neutrino (Gretchen) but, in the end, Chadwick's neutron is what shows up.

Finis

Science Friday: And Inside the Neutron Is...

Second to last post on George Gamow's classic, Thirty Years That Shook Physics.

The previous posts were:

1a. Planck's Quantum
1b. Jumping Photons (Einstein and the Photoelectric Effect)
1c. The Compton Effect (Proof of Energy Packets)

2a. Thomson and Rutherford's Atoms
2b. Bohr's Contributions (How electrons fill the atom)

3a. Pauli Exclusion Principle (no two electrons at any one energy state)
3b. The Pauli Neutrino

4a. De Broglie's Wavy Particles
4b. Schrödinger's Wave Equation

5. Heisenberg's Uncertainty Principle
6. Dirac's Anti-Particles

Chapters 7 and 8 of Gamow's book are fairly short. We have more or less reached the end of the thirty years that shook physics from 1900 to the early 1930s. I've pretty well decided which book to follow Gamow with, and I'll announce it next week.

With chapters 7 and 8, we are beginning to see the first hints of what will become the standard model of physics, with its panoply of particles. Little did we know when we learned about the proton, electron, and neutron, that we were just scratching the surface of what lies below.

[As an aside, my Dad remembered talking to army buddies in Europe about whether they would ever split the atom. Little did he know that Hiroshima was less than a year away!

Actually, it had already been done. Gamow mentions more than once the telegram Niels Bohr received at a 1939 conference at George Washington University relaying that two professors at Berlin University had split a uranium atom using a slow neutron bombardment, liberating vast amounts of energy.]

1. You might remember that Wolfang Pauli had suggested the existence of a very small, chargeless particle that accounted for some of the energy missing when "beta decay" takes place. Beta decay is basically when an unstable, radioactive nucleus emit an electron. In the mid-20s, they could account for where some of the energy went from the decay but not all of it.

Pauli suggested there might be a very, very small particle that carried off the missing bit of energy. In his conversations and discussions, he was calling it a "neutron." But in 1932, when James Chadwick actually discovered a neutral particle in the nucleus of almost all atoms, Chadwick called that particle a "neutron." And Chadwick called it a neutron in print.

So it was the Italian, Enrico Fermi, who called Pauli's particle a "neutrino" (a "little neutron" in Italian). Its existence wasn't finally confirmed until 1955. But long before all the details were experimentally demonstrated, Fermi was accurately predicting how such decays would play out.

It's not exactly accurate to say that a neutron has a proton and an electron inside it, but it works for a blog. In a certain state, a neutron will decay into a proton, while releasing an electron and a neutrino. Next to photons, neutrinos are the most populous particles in the universe.

This decay relates to one of what are currently considered the four basic forces of nature--the so called weak nuclear force (the others are gravity, electromagnetism, and the strong nuclear force). The details weren't worked out when Gamow wrote this book. A big piece of the puzzle fell into place in 1968, the year he died.

2. Chapter 8 is very short and is about the suggestion of Hideki Yukawa in 1935 that there was another yet undiscovered particle that created the "strong force" that holds protons together in the nucleus without them flying apart (because they have the same charge). This strong force that sticks them together only operates at a very, very short distance (10^-13cm).

They knew about what mass the "meson" (started out as "mesotron," the idea being a middle sized particle, in between the mass of an electron and a proton) should be. In 1937, such a particle was discovered among cosmic radiation from the sun. But it turned out to be a little too light. Finally, in 1947, balloons captured an earlier version of this meson in the upper atmosphere amid cosmic radiation, and it had the right mass.

Thus two more particles in the growing pantheon: the pi meson (pion, the heavier) and the mu meson (muon, the lighter).

Science Friday: Dirac's Antiparticles

Not long now. Here's chapter 5 of George Gamow's classic, Thirty Years That Shook Physics.

The previous posts were:

1a. Planck's Quantum
1b. Jumping Photons (Einstein and the Photoelectric Effect)
1c. The Compton Effect (Proof of Energy Packets)

2a. Thomson and Rutherford's Atoms
2b. Bohr's Contributions (How electrons fill the atom)

3a. Pauli Exclusion Principle (no two electrons at any one energy state)
3b. The Pauli Neutrino

4a. De Broglie's Wavy Particles
4b. Schrödinger's Wave Equation

5. Heisenberg's Uncertainty Principle

When I was at Florida State University in the summer of 1983 for Boy's State, I used some free time to find the office of P. A. M. Dirac and touch his door. He would die the next year.

One of the challenges of the burgeoning quantum mechanics of the late 1920s was that the Schrodinger equation did not account for relativistic factors that came into play with particles that might approach the speed of light. One factor that kept anyone from "relativizing" it was that the derivatives in the equation were of different orders, if I am understanding correctly. So part consisted of a first order derivative and another part of a second order derivative. It proved difficult to introduce Einstein's famous Lorentz transformation to such a mixed order equation.

So one evening in 1928 while Dirac sat in his Cambridge study at St. John's College, he thought to reduce the second order derivatives in such a way that the equation would consist entirely of first order derivatives. The hesitation to do this is that Schrodinger's equation, when of a first order level, includes imaginary numbers, that is, numbers that include the square root of -1 (i), which is difficult to conceptualize or really know what it means.

Nevertheless, it not only worked, allowing Dirac to create a relativistic version of Schrodinger's equation that would work when particles were approaching the speed light. It also accounted for another dimension of atomic description that had been discovered, namely, spin. The equation predicted that there were two values of spin that any electron could have in any orbital.

Another thing that his equation predicted were antiparticles. If one solution to his equation involved a negatively charged electron with positive mass, then another might be a positively charged electron. Dirac initially wondered if this "positive electron" might be the proton, but he did not have to wait long. In 1932, the existence of the "positron" was officially verified.

Richard Feymann would later suggest that a positron was an electron moving back in time, one of his many quirky ideas. When electron and positron meet, they annihilate and release 2mc squared of energy. Feymann suggested this was the electron changing directions in time.

The antiproton and antineutron would be discovered later. The question would arise. Are there equal portions of matter and antimatter in the universe? Might whole galaxies out there consist of antimatter, just as ours consists of matter?

Fun story about when Dirac first met Richard Feymann. Apparently he asked him, rather matter of factly, "I have an equation. Do you?"

Science Friday: Schrödinger's Wave Equation (Birth of Quantum Mechanics)

Now to finish chapter four of George Gamow's classic, Thirty Years That Shook Physics. The last part of the chapter presents Schrödinger's earth shattering wave equation.

The previous posts were:

1a. Planck's Quantum
1b. Jumping Photons (Einstein and the Photoelectric Effect)
1c. The Compton Effect (Proof of Energy Packets)

2a. Thomson and Rutherford's Atoms
2b. Bohr's Contributions (How electrons fill the atom)

3a. Pauli Exclusion Principle (no two electrons at any one energy state)
3b. The Pauli Neutrino

4a. De Broglie's Wavy Particles

4b. After the personal investment Gamow showed in the material thus far, I was a little struck by how short and impersonal his discussion of Schrödinger's wave equation is. While Heisenberg gets an entire chapter, including a fair amount of formulaic discussion, Schrödinger gets short shrift--he doesn't even get a chapter for himself when his wave equation is the foundation of quantum mechanics!

Did the two never meet? Schrödinger was in Austria, Gamow in Copenhagen and Cambridge, later America. I have a feeling (please correct me someone), that Gamow did not particularly think too much of Schrödinger.

So de Broglie was "too slow" to develop a mathematical theory for his matter waves. Although I haven't read chapter 5 yet, I get the impression that Gamow was partial to another version of Schrödinger's equation, a more complicated, less intuitive approach created by Heisenberg the year before Schrödinger. In short, Gamow was part of what would become the Copenhagen interpretation of quantum physics, and Schrödinger really wasn't.

I have two other books at my side as I work through Gamow. One is The Amazing Story of Quantum Mechanics, an attempt to present the development of quantum mechanics something like a comic book. (explanations are excellent, but the combination of comic books and science fiction seems a little awkward to me sometimes). The other is Quantum Generations.

Quantum Generations quotes Schrödinger as saying that he was "repelled" by Heisenberg's matrix approach (164). Even when it came to his own equation, Schrödinger thought Max Born's interpretation of it was nuts (Amazing Story, 69). Like Planck, Schrödinger sounds like he didn't actually like what he had discovered. He considered Bohr's theory of light emission "monstrous" and "really inconceivable" (Generations, 165).

The origins of his equation came after a lecture he was presenting on de Broglie's particle-wave duality of electrons. A senior colleague in the audience, Peter Debye, challenged him--if it is is a wave, then it should have a wave equation. The author of Amazing Stories then tells us that Schrödinger then went on Christmas break in the Swiss Alps with a woman who wasn't his wife and developed the equation (65).

Here is the equation, which I have wanted to understand since high school:

Amazing Stories actually does a pretty good job of unpacking it, although I'm not there yet. Interestingly, although the equation eventually worked to predict line spectra, Schrödinger wasn't sure what it meant physically. Since it took of the form of something per volume, he thought it might be a way to calculate charge per volume.

But Max Born--who seems to be one of the smartest cookies in the room as far as I can tell, just not the one who made the key discoveries themselves--suggested that this is a "probability distribution," something that was not to Schrödinger's taste at all. Born was suggesting that the electron cannot be pinned down to one location at any one time. It is not a particle in an orbit. There are rather probabilities that the electron will be in particular places.

____________________
Here is how a person might go from algebra to understanding this equation (something like what Collier has done with relativity):

1. Waves - basic relationships between wavelength, frequency, and velocity (define k and w)
2. Trigonometric functions
3. Radian measure
4. Imaginary numbers
5. Momentum
6. Planck's constant and equation
7. exponential functions
8. basic Ψ wave function - Ψ = e to the i(kx-wt). See DrPhysicsA here.
8. kinetic energy and potential energy
9. differentiation - first and second derivatives
10. simple time independent Schrodinger equation in one dimension in a simple form (See here)
11. partial derivatives
12. The nablus operator
13. Hamiltonians, eigenvalues, Laplace operators
13. Time dependent Schrodinger equation
14. The next steps might look at the predicted hydrogen spectra and quickly lead to Dirac's relativistic version.

There are no doubt other steps I'm missing, but am not far enough along to see.

Science Friday: De Broglie's Wavy Particles

On to chapter four of George Gamow's classic, Thirty Years That Shook Physics. Chapter 4 is on "L. de Broglie and Pilot Waves."  The previous posts were:

1a. Planck's Quantum
1b. Jumping Photons (Einstein and the Photoelectric Effect)
1c. The Compton Effect (Proof of Energy Packets)

2a. Thomson and Rutherford's Atoms
2b. Bohr's Contributions (How electrons fill the atom)

3a. Pauli Exclusion Principle (no two electrons at any one energy state)
3b. The Pauli Neutrino

4a. Now the plot thickens with de Broglie (pronounced, de-broy - he was French and of course they never pronounce words the way they look... much worse than English on that score). Again, Gamow's personal recollections of these individuals are priceless.

He recounts how he decided to spend Christmas in the late 20s in Paris and tried to meet up with de Broglie while he was there. De Broglie was from a family of some prominence, as it turns out, a family of diplomats. Gamow tells how when he arrived at the family mansion, he was corrected by the butler to know that he had not come to see "Professeur de Broglie" but "Monsieur le Duc de Broglie," the seventh Duke de Broglie.

Gamow assumed during the visit that de Broglie did not speak English, as he struggled to speak in French to him, made diagrams, etc. As it turned out, de Broglie spoke perfect English. But when you're in France...

De Broglie won the Nobel Prize in physics for an idea that he proposed in his PhD dissertation in 1925 and which was verified less than five years later. Einstein had proposed, following on Planck, that light was not just a wave (as James Clerk Maxwell had demonstrated in the 1800s) but that it also functioned as a particle. That is to say, if you break down a light stream into individual photons, you will see that the wave properties of light are a statistical phenomenon. The fewer the photons involved, the more light behaves like individual particles.

De Broglie, on the hunch that there would be a symmetry between energy and matter on this score, suggested that matter, which we think of as particles, might have wave properties as well. He used existing formula to predict how those waves would function if they were associated with particles like, say, electrons. When electrons were then later diffracted, sure enough, they showed wave light patterns.

For de Broglie, these waves were like pilot waves associated with the particles. But de Broglie had opened up a can of worms. Next week, we will see the transition from the "old quantum theory" of Einstein to the "quantum mechanics" of the Copenhagen school, which will soon become the new norm in physics.

Science Friday: The Pauli Neutrino

Today I finish chapter three of George Gamow's classic, Thirty Years That Shook Physics. Chapter 3 is on "W. Pauli and the Exclusion Principle."  The previous posts were:

1a. Planck's Quantum
1b. Jumping Photons (Einstein and the Photoelectric Effect)
1c. The Compton Effect (Proof of Energy Packets)

2a. Thomson and Rutherford's Atoms
2b. Bohr's Contributions (How electrons fill the atom)

3a. Pauli Exclusion Principle (no two electrons at any one energy state)

The remainder of chapter 3 deals with modifications to Pauli's exclusion principle and his prediction of the neutrino. Pauli's initial formulation of the exclusion principle was that only two electrons could exist at any one energy state. This proved inadequate because when the lines of the spectra of elements was put through a strong magnetic field, more lines came out than the three components Pauli had come up with could account for.

So it was suggested that an electron had a certain "spin" or "angular momentum" The two electrons might be at the same orbit, but one would have one spin and the other the opposite. This led to the modified Pauli exclusion principle that only one electron could be at each orbit, with the two electrons at roughly the same orbit "spinning" in opposite directions and moving around the nucleus in opposite directions.

The chapter ends with Wolfgang Pauli's suggestion that there might be another particle with no charge and almost no mass. It had seemed like beta particles (i.e., electrons) did not conform to the Law of the Conservation of Energy, another one of the long established principles of physics. For Bohr, what was one more sacred cow. He was willing to throw it out the window as well.

But Pauli suggested that there might be a much smaller particle that could not be detected by charge or mass but that could carry off some of the energy of a beta particle. He called it initially a neutron, but then the neutron with which we are familiar was discovered by Chadwick. Nevertheless, in 1955, the existence of the "neutrino" was confirmed.

Science Friday: Wolfgang Pauli's Exclusion Principle

I'll muster the will to make my Friday post on George Gamow's classic, Thirty Years That Shook Physics. This week I started chapter 3, "W. Pauli and the Exclusion Principle."  The previous posts were:

1a. Planck's Quantum
1b. Jumping Photons (Einstein and the Photoelectric Effect)
1c. The Compton Effect (Proof of Energy Packets)

2a. Thomson and Rutherford's Atoms
2b. Bohr's Contributions (How electrons fill the atom)

Again, it's delightful to read someone with a playful personality who knew the key players in the quantum revolution. Before he gets down to business, Gamow talks of how Pauli, though German, was an anti-Nazi. He talks of how once with his arm in a cast that propped it up, he refused to be photographed with his arm in that position.

Pauli apparently laughed quite loudly and his round figure jiggled when he laughed.  Gamow also playfully speaks of the "Pauli Effect," whereby things tended to break when he was around. In words that reminded me of Sheldon Cooper on the Big Bang Theory, Gamow writes, "It is well known that theoretical physicists cannot handle experimental equipment; it breaks whenever they touch it" (64). [Might I add that in early high school I thought of going into theoretical physics...]

Forgive me for not going into detail on the exclusion principle. The problem that Pauli first addressed was the question of why electrons did not collapse toward the lowest quantum state. His finding was basically that there is a "quota" for each electron state. Once the lower levels are filled, electrons have to fill higher levels. No more than two electrons can exist on any one level.

It's like assigned seating in an auditorium. Only two electrons can sit on the first row, right by the nucleus platform. Then there are only two seats in the second row as well. Then it gets a little more complicated. Interestingly, the average diameter of the atom remains constant no matter how big the atom is.

With these basic principles in view, Pauli and Bohr were able to model all the atoms from hydrogen to uranium.

Fin de siècle (1800s) Physics

Forgive me for picking up another book on physics this week: Quantum Generations: A History of Physics in the Twentieth Century. I had high hopes after enjoying so much The Perfect Theory on general relativity.

It's not a good sign that I am a little bored and disappointed after the introduction and the first chapter. This sentence in the introduction in and of itself may have predicted the doom of this book for me: "I have described, rather than analyzed, important parts of the development of physics between 1895 and 1995" (xiv).

The first chapter, on "end of the century" physics at the end of the 1800s, kept the author's promise. It seemed to me a cluttered and somewhat boring catalog of people and their ideas at the end of the 1800s. I feel like I could write a more interesting account... who cares that it wouldn't be nearly as accurate. :-) Give me the flavor. Give me the controversy. Give me the narrative. Give me Gamow. Give me Hawking.

The standard narrative is that physics was all but putting on the finishing touches and letting the paint dry at the end of the 1800s. Kragh says not so fast. He no doubt treats these figures much more fairly. But, come on, I want to feel smart by making these guys look silly.

Maybe someday I can write the Mickey Mouse version of 20th century physics...

Science Friday: 2b. Bohr's Contributions (5)

I've been reading through George Gamow's classic, Thirty Years That Shook Physics. This week finishes chapter 2, "N. Bohr and Quantum Orbits."  The previous posts were:

1a. Planck's Quantum
1b. Jumping Photons (Einstein and the Photoelectric Effect)
1c. The Compton Effect (Proof of Energy Packets)

2a. Thomson and Rutherford's Atoms

The rest of chapter 2 was a delight, not because of the equations but because of the stories. The rest of the chapter primary deals with Niels Bohr, who took the notion of energy packets and applied it to the atom.

1. Why doesn't the energy of electrons dissipate away? Because it has an energy "ground state," lower than which its energy cannot go. Further, electrons can only gain energy in the discrete packets that Planck suggested.

At this point, Bohr pulled together the results of several different experiments. As hydrogen is excited, say by running electricity through it, it produces a characteristic set of lines if light from it is passed through a prism. This "line spectrum" is like hydrogen's signature.

What interested Bohr about hydrogen is that he supposed that it might only have one electron. It was thus perfect for exploring the possible energy states an electron could have. The real innovation was his hypothesis that the different line spectra of hydrogen might correspond to different circular orbits hydrogen's electron might have.

Several scientists had already derived equations for different parts of hydrogen's spectrum (Balmer, Lyman, Paschen, and Brackett). Bohr connected these different parts of hydrogen's spectrum into a model in which you had a number of energy states at one orbit and then a number at another orbit, then more at another, and so forth.

Here are two diagrams to depict his theory. The first shows the series of electrons he proposed at each orbit. The second relates these to the various series of line spectra found for hydrogen.

a.

b.

Sommerfeld would then, like Kepler, suggest elliptical orbits. Any student of chemistry today will recognize these hypotheses as the beginning of what we now know as the various orbitals of electrons within various shells (1s2, 2s2, 2p6, 3s2, 3p6, 4s2, 3d10...).

2. But the most enjoyable part of the chapter were the personal reminiscences of Gamow about Bohr. I had never gotten this far in the book before. I guess the Carlsberg Brewery of Denmark fame has an scientific institute in the center of its property, Blegdamsvej 15, at that time. The most famous Danish scientist of the time is supposed to live there, and Bohr took up his residence there for many years.

I guess many of the most famous physicists of those days spent at least some time there with Bohr, who apparently was a delightful, whimsical person who "fathered many scientific children" (51). Gamow contrasts Einstein, who generally worked alone, with Bohr, who apparently loved to mentor and work with others.

Gamow describes a community of geniuses that I suspect most of us interested in ideas dream of being a part of. But alas, it doesn't happen very often, and who of us would qualify? There was Cambridge in the 20s with philosophers and writers. There was Princeton for the relativists with Jim Peebles. There's the Chicago School in economics...

What a remarkable time to be a physicist! Gamow had managed to go to Germany from Russia for a couple months in the summer of 1928. He relates that the Soviet Union had not yet closed the door to such things yet at that time. He had come up with a brilliant idea about the nucleus in those months and had sent off a paper for consideration to be published.

On his way back to Russia, he had just enough money left to go by Copenhagen for a day and try to meet Bohr, unannounced, just a hope to meet the man. As an indication of the kind of man Bohr was, Gamow ended up staying for a year there on a fellowship. The chapter continues with reminiscences of Bohr's whimsy.

Gamow ends the chapter remembering the time in 1939 at a conference in DC, where he had gone to teach in 1933. Bohr was at the conference, as was Enrico Fermi (coming up in chapter 7). In the middle of the conference, they received word that an experiment had been conducted in Berlin in which uranium basically split in half when bombarded with neutrons.

The experiment was partially reproduced that evening in DC and would lead to the atomic bomb, six years later.

Next Week: Wolfgang Pauli

Science Friday: The Bohr Atom (4)

I've been reading through George Gamow's Thirty Years That Shook Physics. This week starts chapter 2, "N. Bohr and Quantum Orbits."  The previous posts were:

1a. Planck's Quantum
1b. Jumping Photons (Einstein and the Photoelectric Effect)
1c. The Compton Effect (Proof of Energy Packets)

Thomson
Hindsight is 20/20. You can see the stubbornness of scientists, what Kuhn called "normal science," as you look back at history. This chapter starts by recounting J. J. Thomson's stubbornness as the Dane Niels Bohr suggested that his model of the atom wouldn't work given the recent developments of Planck and Einstein.

Thomson's Plum Pudding Model

Thomson had proved the existence of electrons, negatively charged particles that could be dislodged from an atom. But Thomson had suggested a model something like a pudding with raisins stuck in it. When Bohr arrived at Cambridge, Thomson wasn't at all excited about these new ideas. Bohr finally switched to Manchester, where Ernst Rutherford was bombarding atoms with alpha particles (basically, helium atoms without the electrons).

Rutherford
What Rutherford found was that while most alpha particles passed right through foil, some actually were deflected at significant angles. His conclusion was that most of an atom must be empty, with the positive charge in an atom concentrated in a very small space in the middle (the nucleus).

Rutherford proposed a model of the atom that was much like the solar system, a heavy nucleus in the center with orbiting electrons.

The Rutherford Model of the Atom

But Rutherford's model couldn't explain why the energy of the electrons didn't gradually dissipate. That would take Bohr...

Thomson on left, Rutherford on right

Next Week: Bohr's contribution

Science Friday: Proof of Energy Packets (3)

I'm reading through George Gamow's Thirty Years That Shook Physics. This post finishes chapter 1, "M. Planck and Light Quanta."  The previous posts were:

1a. Planck's Quantum
1b. Jumping Photons (Einstein and the Photoelectric Effect)

Today ends chapter 1 with "The Compton Effect."

The last part of chapter 1 is barely more than a page. It simply recounts the experiment of Arthur Compton in 1923. If on the smallest level light is made up of packets of energy, then these "photons" should bounce off of electrons.

But how to set up an experiment. Light at low frequencies in the best stream of electrons they could muster would be like dropping a baseball from a plane and trying to randomly hit a bird (sorry, my illustration). So Compton used X-rays, which because of their high frequency have very large quanta of energy. And he used carbon, which is relatively speaking a smaller element in which the electrons aren't as tightly bound.

The results of the experiment were consistent with Planck's hypothesis. Photons from the X-ray were absorbed by the electrons of the carbon and then expelled at a lower frequency in a deflected direction. Like it or not, Planck had been right.

Next Friday: Theories of the Atom

Science Friday: Jumping Photons (2)

Last week I (foolishly) started blogging through George Gamow's Thirty Years That Shook Physics. (Foolish because I wonder whether my puny brain will even make it through the first chapter).

When last we met our hero (old quantum theory), Max Planck had just suggested that the energy of light radiation does not exist at every continuous quantity but that the higher the frequency of light radiation, the bigger the "packets" in which that energy existed. Planck didn't like it. The suggestion was an "act of desperation." He did it to make the theory match the experiment. But he hoped it wasn't part of the basic nature of energy but some quirk having to do with atoms.

But Einstein dashed his hopes to pieces, five years later in 1905. Einstein didn't win the Nobel Prize for relativity, special or general. Einstein didn't get the Nobel Prize for E = mc². Einstein won the Nobel Prize for demonstrating that Planck's "quantum" was in fact very much real. Even though light behaved like a wave on a large scale, on the smallest scale, light was little packets or "quanta" of energy, which Einstein called, "photons." The size of these quanta increased with frequency.

What Einstein did was show that Planck's quantum explained one of the puzzles of late 1800s physics: the photoelectric effect. The photoelectric effect is the fact that, below a certain threshold frequency, the electrons in the surface of a metal plate jump free with the same amount of energy no matter how intense of light radiation you shine on it. As you increase the intensity of the light on the surface at a lower frequency, the number of electrons jumping increases but the amount of energy each of them has does not.

However, once you reach the threshold frequency, which is different for different metals, the amount of energy the electrons have as they jump begins to increase. In fact, after not increasing at all, the amount of energy begins to increase evenly with the increase in frequency.

Planck's E_photon = hf explained this phenomenon. The energy of a photon increased evenly with the increase of frequency.

Next week: The Compton Effect

Science Friday: Planck's Quantum

Since I was in high school, I've wanted to understand quantum physics. In my first year at college, I felt called to ministry and so never continued that path. But in mid-life I've tried to get back into some of those old aspirations on the side.

One book I've had for years is George Gamow's, Thirty Years that Shook Physics (it has a Joseph Beth's bookmark in it from Asbury days). This past week I reread the section on Max Planck. Planck's reluctant proposal in 1900 launched modern physics. He didn't want to. He called his proposal "an act of desperation." He was not at all an "outside the box" type.

According to the story, Planck was working on what had come to be known as the "ultraviolet catastrophe" (I don't actually think that's quite right, but this is the best way to tell the story, it seems). If you have a bunch of gas molecules in a chamber, you expect over time that the energy of the molecules will eventually even out as the molecules bump into each other and share energy (the equipartition theorem). It's the same idea as what happens when you open the door between a hot room and a cold room. Eventually the temperature will even out.

(In reality, there is still somewhat of a variation, but most molecules statistically will have a certain average amount of energy at a particular temperature and those with more or less energy will be in decreasing numbers.)

Two scientists from the late 1800s (Rayleigh and Jeans) supposed that the distribution of energy in a "blackbody" (which absorbs all wavelengths of light)--or in a cavity made to mirror the effects of a blackbody--would also spread out similarly into waves of all the frequencies and wavelengths. Following the idea of equipartition, the total energy would begin to distribute itself into to each wavelength of light until every possible wavelength had roughly the same amount of the total energy.

Gamow has a helpful illustration. It would be like playing a note on the piano without a damper between the strings. In such a situation, the energy from playing one note would seep into all the other strings until each string in the piano corresponding to each key had 1/88th of the initial energy.

This isn't what happens, and it's a good thing too. What would happen if the energy of a fireplace soon spread into other wavelengths so that before long, the fireplace was spewing out ultraviolet and x-rays. Gamma rays would follow and you would soon die.

Planck's suggestion was that energy couldn't seep into higher frequencies of radiation as easily as into the lower frequencies. The higher the frequency, the higher the threshold of energy it took to distribute into that part of the spectrum. Light was not behaving like a continuous wave that spread out evenly to all frequencies but more like discrete packages of energy of differing sizes. The size of these packages was different at different frequencies.

I've tried to think of a good illustration, even to help myself understand exactly what Planck is suggesting. Consider this one a work in progress. Let's say there is an auditorium where the seats are arranged somewhat strangely. The front row has lots of cheap seats. The farther back in the auditorium you go, the seats are fewer in each row and they cost more.

So let's say you start off with a certain amount of energy in a blackbody, based on the temperature. It is like having a certain amount of money to spend on seats in the auditorium. The energy is distributed. Most of it goes to the lower frequencies because those seats are cheap. Yes, some of it goes toward the more expensive seats, but not nearly as much.

How much does a seat cost? Planck found he could make the graph in theory look like the real graph by introducing a number that would eventually be called Planck's constant (6.626 x 10^-34 Joule-seconds, given the symbol h). A packet of energy at a particular frequency, E, was h times the frequency:

E = hf

That's how much a seat cost. A seat at a low frequency didn't cost that much. But the higher the frequency, the more expensive the seat.

Einstein would seal the deal on Planck's desperate theory five years later. Light was not a continuous wave. When you break things way down, you get to something like atoms of energy smaller than which you cannot go--photons.

Next Friday: Einstein and the photoelectric effect

Science Friday: Atomic Numbers

It continues.
________________
... Meanwhile, Spruce continued to talk to himself.  "It's the protons that tell you what element it is.  That's because it's the only particle whose number is always the same no matter what form the element comes in."

Stefanie carefully removed a piece of parsley from a delicious strip of fettuccine.

"You see, we normally think an atom has the same amount of protons and electrons.  When it does, it has no charge or a neutral charge.  The negatives of the electrons outside the nucleus balance out the positives of the protons in the nucleus."

Stefanie next eyed a strip of fettuccine with some extra Alfredo sauce on it.  It looked absolutely delicious.

"But some atoms can give away some of their electrons or borrow some from another atom.  When that happens, the atom can become an ion with either a positive or negative charge.  So you can't count the electrons to decide what kind of an atom it is.  The number can change."

"That's interesting," Stef said, eyeing a lovely bit of broccoli.

"You can't count the neutrons," Spruce continued, "because the same element can come with different numbers of neutrons.  They're called different isotopes of the element."

"What's an element again?" she asked, not noticing that Spruce had used the word repeatedly.

"An element is a kind of atom.  So gold is an element, a particular kind of atom.  Silver is an element, a particular kind of atom."

"And diamonds are a kind of atom," she said.

"No, I think diamonds are made up of carbon atoms or something."

Oooo, Stef thought, a mushroom.  It had been hiding under several fettuccine noodles.

"So that's why they use the number of protons to identify an atom, because if you change the number of protons, you've changed the kind of element."

With no more fettuccine, broccoli or 'shrooms on her plate, Stef finally was forced to look at him.

"And I need to know this why?"

"The number of protons in an atom tells you the atomic number," Spruce answered. "It tells you which element it is.  You know the periodic table, that chart on the inside of every chemistry book?"

"Yeah?" and indeed she did.  There had been a huge one in her high school chemistry room, a picture full of boxes stacked on top of each other, like you were looking through the walls of some huge condominium with a tower on each end.

"The periodic table starts at the top left with the number 1 and moves across each row with the atomic number counting up one by one.  The first row only has two boxes on each end, numbers 1 and 2."

"So you're telling me number 1 has one proton in its nucleus and number 2 has two protons in its nucleus?"

Spruce was actually surprised she had paid that much attention.

"Yeah, hydrogen has one proton and helium has two.  Then the next row keeps counting up: 3, 4, 5, 6, 7, 8, 9, 10."

"Why does it skip the space in the middle at the top," Stef surprisingly asked, having stared at the chart a time or two in high school.

Spruce thought like he was about to say something profound and finally came forth with his answer.  "No idea"...