Atoms, electrons, nuclei, and blackbodies
Einstein published what we now call his special theory of relativity in 1905. By 1907 he had established that all of physics was consistent with his new theory, with just one exception: gravity. Newton's universal law of gravitation relied on instantaneous action at a distance, which is forbidden in special relativity (no influence can travel faster than light speed); furthermore, the force between two masses depends on their separation, which will be different for different observers moving at different speeds.
Also in 1907 Einstein hit upon a clue as to the true nature of gravity: the equivalence principle. Upon hearing a story of someone who fell off a ladder, he had what he called "the happiest thought of my life" -- namely, that someone who is in freefall feels as if there is no gravitational field. (Freefall is motion under the influence of gravity alone, without air resistance or any other forces.) This is the case because all matter in freefall accelerates downward at exactly the same rate: the inertial mass (which determines how strongly an object resists accelerating) is always exactly the same as the gravitational mass (which determines how much gravitational force an object feels). So if a painter falls off a ladder, it seems to him (until he lands) that he is floating and that his brushes and his paint can are floating alongside him, just as if he were in deep space in zero gravity.
You should understand Einstein's thought experiment of the rocket traveler out in deep space: What we see as acceleration in our inertial (i.e., unaccelerating) reference frame is experienced as gravitation in the traveler's accelerating reference frame. And Einstein now allows accelerating frames to be just as valid as inertial frames: this is the principle of general relativity, that all experiments carried out by all observers will obey the same laws of physics. Since the traveler experiences the rocket's acceleration as a gravitational field, it is a gravitational field in his reference frame, with no hedges such as "as if" or "apparent" or so on.
(You should also understand why a similar thought experiment shows that a gravitational field will bend the path of light.)
We looked at another thought experiment, this one involving measurements made by someone on the rim of a spinning turntable (or merry-go-round). By observing the turntable from a uniformly moving vantage point (i.e., hovering overhead), Einstein was able to use special relativity to analyze what he would see. We saw that the person on the rim would measure the turntable's circumference to be more than π times the diameter, contrary to Euclid. Similarly, imagine placing a clock on the rim of the turntable; in the uniformly moving reference frame, the clock would be moving and hence would run slow. The person on the turntable would perceive a gravitational field rather than acceleration, so Einstein realized that clocks would run slow in gravitational fields and that gravitation had something to do with non-Euclidean geometry.
It took him years to figure out just what it had to do with geometry; in fact he was despairing of finding an answer until he contacted his old college pal (by now a math professor), who proceeded to tutor Einstein in the complex mathematics of non-Euclidean geometry, worked out by Riemann several decades earlier. The bottom line is that gravity isn't a force exerted by one object on another, as Newton thought; rather, gravity is geometry, gravity is spacetime curvature. Think about the trampoline analogy. The bowling ball at the center doesn't reach out and pull on the marble that you roll past it; no, the bowling ball curves the trampoline's surface, and the marble then travels in a straight line on that curved surface. Matter and energy curve spacetime, and spacetime then tells matter (and light) how to move. Also think of the analogy with an airplane flying along a great-circle route from, say, Boston to Paris: the path looks curved on a map, but that's because the flat map misrepresents Earth's curved surface. And our common-sense notion that we inhabit Euclidean 3-D space misrepresents curved 4-D spacetime.
Chapter 15 is all about observational and experimental evidence for general relativity. You'll be responsible for the various items discussed in this chapter, some of which were also discussed in your oral presentations.
In Chapter 16 you are not responsible for the speculative sections about quantum gravity: theories of everything, extra dimensions, etc. You should understand that general relativity can be applied to the universe as a whole. Possible solutions for the expansion of the universe were (see chapter) negatively curved, where the expansion continues forever; positively curved, where the expansion eventually stops and then reverses to produce a "Big Crunch"; or flat (Euclidean), where the expansion never quite stops but never reverses either. Einstein at first wanted none of the above -- a static universe -- and so introduced a "cosmological constant" -- usually denoted by a Greek upper-case lambda (Λ) -- that would permit a universe that never expands or contracts. Then in 1929 Edwin Hubble discovered that the universe is expanding after all, so Einstein renounced the cosmological constant. But in 1998 astronomers made the surprising discovery that the universe is not just expanding but, contrary to all expectations, is expanding faster and faster as time goes on rather than slower and slower: we live in an accelerating universe. Thus the cosmological constant, which would be one way of explaining this acceleration, may be relevant after all.
Our current best estimate is that the universe is flat at the largest scales, with about 3/4 of the required energy supplied as mysterious "dark energy" (Einstein's Λ?) and 1/4 as matter. Most of the matter is exotic "dark matter"; only about 4% of the universe's energy is in the form of the protons, neutrons, and electrons that make up you and me and trees and planets and stars and everything else we hold dear. Stuff like us, in other words, is the exception to the rule.
I discussed the common fallacy that the universe was empty and dull and then the Big Bang was an explosion that occurred at a particular spot in that empty space at a particular time. That would be too easy to grasp. You should understand why it is that there was no preexisting space into which the universe expands, and no time before the Big Bang (and hence no "before the Big Bang" at all). You also should understand that we're not the center of the universe's expansion: everyone is at the center, or, equivalently, there is no center. After all, at the Big Bang all of space was crunched down to a zero-diameter point (singularity), so everywhere was the center.
Atoms were postulated by ancient Greeks and Romans such as Democritus and Lucretius but the idea was discarded until Dalton revived it in 1800. By 1900 this concept had proved extraordinarily powerful in chemistry, and also in physics (e.g., the idea that temperature has to do with the mean kinetic energy of randomly moving atoms). But some philosophers strongly objected to this concept, since they felt that scientists have no business talking about objects that they can't see even in principle. A major step in combating this "positivist" viewpoint was taken by Einstein in 1905, when he explained Brownian motion for his doctoral dissertation. You should know what Brownian motion is and what atoms have to do with it.
In 1897 J. J. Thomson had already shown that atoms are not indivisible as originally thought, but in fact have parts: electrons. You should be able to describe how he determined that all atoms have negatively charged parts with the same charge-to-mass ratio e/m.
No one knew just how many electrons any atom had, where they were located, or how they were moving. Thomson devised the "plum pudding" model (think "raisin pudding" if you're American rather than British) in which the atom is a sphere of positively charged fluid (the "pudding") with a number of electrons (the "raisins") randomly placed within. Remember, this was Thomson's theoretical concept ("model") of the atom, entirely separate from Thomson's 1897 experiment that demonstrated the existence of electrons. In 1911 Ernest Rutherford showed that the plum pudding model was incorrect. You should understand just how Rutherford's "gold foil" experiment was set up and run. The expectation was that since an atom's positive charge is spread throughout the atom, there would be no concentrations of positive charge that could strongly deflect any alpha particles. Yet it turned out that a small but significant number were indeed deflected (scattered) sideways or even backwards. This could only mean that the positive charge was concentrated in tiny clumps (nuclei), leaving the rest of the atom empty (save for the occasional electron).
Step back eleven years to 1900. Max Planck wanted to understand the properties of incandescent (opaque) bodies, or "blackbodies"; an example that one might study in the lab would be an oven with solid walls except for a tiny pinhole through which light can escape and be measured. Blackbodies were observed to put out more and more light as their temperatures rise, and also to put out their peak emission at shorter and shorter wavelengths. These data were a mystery, because Maxwell's electromagnetic theory predicted that since there are many more ways to fit short-wavelength waves into such an oven than long-wavelength waves, there should be far brighter short-wavelength radiation than long-wavelength radiation, regardless of the temperature: the "ultraviolet catastrophe."
First Planck searched for a mathematical formula that would fit the data, and he was successful. Next, in order to understand why blackbody radiation obeys this formula, he tried to work out how the atoms in the walls of the oven absorb and emit light. And he was successful here too, but only by making a very strange assumption, one he only resorted to "in desperation." His assumption was that atoms can't emit and absorb just any amount of light energy of a given color -- that is, of a given frequency f -- but can only emit or absorb integer multiples of a basic unit. Even stranger, the amount of energy in this unit depended on the frequency: E = hf, where the very small proportionality constant h is now known as Planck's constant.
How does this assumption explain the data? A very hot opaque body -- say, a star whose surface gases are at 50,000 K -- contains atoms that randomly race around with tons of kinetic energy. These atoms have plenty of energy to use up in producing light; they can "afford" to emit both low-energy infrared light units and high-energy ultraviolet light units in large quantities. So this star will be very bright and very blue -- although someone with UV vision would see it as ultraviolet-colored.
On the other hand, a cool object like Earth (average surface temperature = 288 K = 15°C) mostly contains "sluggish" atoms and molecules. Such atoms don't have enough kinetic energy to convert into the energy of even one UV or visible light unit. Since these atoms can't emit a fraction of a unit (according to Planck), they have to settle for emitting zero units of UV and visible light: They emit a small number of low-energy IR units instead.
The first paper Einstein published in his "miracle year" of 1905 was not about atoms or about relativity but about light. He went beyond what Planck had said about atoms emitting and absorbing light only as multiples of basic units by proposing that light only exists as such units. In other words, he proposed that light is made up of particles that he called quanta (Latin for "amounts"). Radical though Planck's hypothesis had been, Einstein's was far more so. To use Einstein's own analogy, Planck's idea was like saying that a pub will only sell you beer by the pint, whereas Einstein's idea was like saying that beer can only exist in one-pint units. Planck had claimed that atoms emit and absorb light energy in integer multiples of a base unit, but Einstein was going further and suggesting that light energy can only exist in such units. This suggestion, that light only exists as integer numbers of particles, flew in the face of 100 years of data and successful theory showing that light is not a particle but is instead an electromagnetic wave.
There's an important relationship between the wave model and the particle model:
Light that can be described as a high-frequency electromagnetic wave can also be described as a stream of high-energy particles (quanta).
Mathematically, this is just Planck's formula E = hf; conceptually, though, it's much more radical than what Planck (and virtually everyone else) was willing to accept. Instead of describing faint violet light as a high-frequency, low-amplitude pattern of oscillating electric and magnetic fields, Einstein suggested describing it as a sparse stream of high-energy quanta. Of course, violet quanta are only energetic by comparison to other visible light quanta; X-ray and gamma ray quanta put them to shame.
Einstein was motivated not only by lab data on blackbody radiation but also by recent (1902) data on the photoelectric effect. You should have some idea of the experimental setup, and should especially understand that there are two important independent variables and two dependent variables:
INDEPENDENT: brightness and wavelength of light source
DEPENDENT: photocurrent and stopping voltage
(Actually, there was a third independent variable: you could vary what kind of metal [zinc, iron, tungsten, etc.] you shined the light onto. It turns out that this just changes the threshold frequency, so let's not worry about it.)
You should understand why the photocurrent is a measure of how many electrons per second are ejected from the metal plate, and why the stopping voltage is a measure of how much kinetic energy the fastest-moving ejected electrons have.
You should understand what the experimental results were, which ones were not expected, and why they weren't expected.
Finally, you should clearly understand why the details of the data do make sense if light is a stream of quanta, and if the UV quanta have more energy than the IR quanta.
Each absorbed quantum gives up its energy all at once: no time delay.
Each UV quantum has a lot of energy to give to an electron: the photoelectron leaving the metal has a lot of kinetic energy.
Turning up the brightness of an ultraviolet lamp means more UV quanta hit the metal each second, but doesn't mean that each individual UV quantum has any more energy than before: more photoelectrons per second, but no more kinetic energy per electron.
Each IR quantum has too little energy to remove an electron from the metal: no photocurrent, no matter how bright the infrared lamp.
Einstein's quantum hypothesis was so odd, so contrary to so much that physicists felt entirely confident of, that it was almost universally rejected, chalked up to the eccentricity of a genius. Meanwhile people made some headway understanding atoms. In 1911 Rutherford discovered the nucleus (see above), and just two years later, Niels Bohr came up with a model of what the electrons are doing within the atom. Bohr's goal was to understand emission lines and absorption lines.
Solids, liquids, and very dense gases (stars) tend to be opaque, but tenuous gases are transparent. Such gases exhibit emission lines when heated up. We can understand this by supposing that the electrons within atoms and molecules are "normal" negatively charged objects that travel on "normal" circular trajectories, much as Earth orbits the Sun (except that it's electrical rather than gravitational attraction involved). For an orbit of a given size, "normal" calculations (not covered here) determine the electron's energy. But these electrons also do some strange things: they do not radiate energy while circling, contrary to Maxwell; and they are only "allowed" to have certain special energies, corresponding to orbits of certain special sizes. In a hot gas these electrons can be boosted up to high energy levels through collisions with other atoms; once there, they can spontaneously, instantaneously drop to lower allowed energy levels (i.e., to smaller orbits closer to the nucleus), so long as the energy lost by the electron is given to something else. In particular, the something else might be light. Since only certain special light energies are possible -- corresponding to the differences between the electrons' allowed energy levels -- only certain special wavelengths (colors) will be present in the spectrum.
I repeat, the energy of the emitted light is equal to the downward change in the atom's energy. Furthermore, since different types of atom (helium vs. sodium vs. neon) have different allowed energy levels, different atoms produce different types of light, yielding different emission-line spectra.
(You might have thought that Bohr would assume that the light is emitted all at once as a chunk, a particle: one of Einstein's quanta. Wrong. Bohr was one of those who rejected the quantum hypothesis and who worked very hard to avoid using it.)
It's now a simple matter (I hope) to consider a cool transparent vapor, one whose (sluggish) atoms seldom have energetic collisions. These atoms, then, will typically be in low allowed energy states, orbiting close to the nucleus. What if we now pass light through this gas? For simplicity, imagine that this light is produced by a hot incandescent lamp, so that we have white light with a continuous rainbow spectrum (i.e., light of all possible energies). Most of this light will pass through the gas unhindered, and can be viewed by us on the other side. But light that has just the right energies to boost electrons from inner orbits to outer orbits (low allowed energy levels to high allowed energy levels) can be absorbed. This light will be absorbed by the atoms, and hence will not reach us on the other side. Since we see the absence of light as black, we'll see a rainbow spectrum with black lines -- absorption lines -- at certain special wavelengths.
If you've followed the physics, it should be clear to you that the energies of the absorbed light just discussed are exactly the same as the energies of the emitted light several paragraphs back. The two processes are essentially inverses of each other. Conservation of energy dictates that whatever energy an electron loses, the light must gain, and vice versa.
(Atoms in solids and liquids behave similarly, but because the atoms are packed tightly together and influence each other, the resulting energy level diagrams are extremely complicated. In particular, they tend to have large numbers of closely spaced allowed energy levels, which overlap to form energy bands. This yields emission spectra that have large numbers of closely spaced, overlapping emission lines: emission bands. LEDs and chemical light sticks behave this way, with certain special ranges of color emitted rather than sharply defined individual colors. What would you get if these same materials absorbed light? I mentioned some everyday examples in a prerecorded lecture.)
Bohr's model, with its mix of classical (19th-century) physics and new physics, did a good job of qualitatively explaining emission lines and absorption lines -- at the expense, of course, of introducing weird assumptions about the behavior of electrons in atoms. In fact for the hydrogen atom it gave excellent quantitative (numerical) agreement with the data. But for other atoms, such as helium, the quantitative predictions didn't agree with the actual colors of the emission and absorption lines seen in the lab, so over the next decade various people tweaked the model, for example allowing electron orbits to be elliptical rather than circular. (Planetary orbits in our solar system are actually elliptical.) Bohr eventually was able to work out a lot of chemistry using his model; for example, the alkali metals such as sodium are highly reactive because they have one lonely electron in an outer ("valence") orbit and are happy to give that electron away to other atoms. But this practical success didn't change the fact that the model was based on weird, ad hoc assumptions.