"D"erring-"D"o with Physics Vocabulary Quiz

Answer: Derivative

Taking the "derivative", or instantaneous rate of change, of a function is a crucial part of any physicist's toolbox and is one of the most fundamental operations of calculus. Let's look at a simple example to show how useful it can be.

Suppose that you throw a ball straight up into the air and you want to understand its motion. With a ruler and a fast camera, you realize that you can describe its height x as a function of the time t since you threw the ball: x(t) = h0 + v0*t - 0.5*g*t^2. Here, h0 is the height from which you released the ball, v0 is the initial velocity you gave it, and g is the gravitational constant where you performed the experiment.

The first derivative of x(t) is the rate at which the position changes -- the velocity. This works out to v(t) = v0 - g*t. At t=0, the moment of release, the ball has exactly the same velocity you gave it, but with each passing moment its upward speed is more and more reduced by the force of gravity. At its greatest height, v0 = g*t and the ball has no velocity at all -- and then it starts moving downward.

The second derivative of x(t) is also the first derivative of v(t): the rate at which the velocity changes, otherwise known as the acceleration. This works out to a(t) = -g, which makes perfect sense: since we're neglecting drag, gravity is the only force acting on the ball during its motion, and it produces a constant downward (negative) acceleration. See how handy derivatives can be?

Answer: It reduces the size of an oscillation -- its amplitude

An ideal oscillation goes on forever, each time the same as the time before: pull down a mass on a spring, and it will bounce back and forth between the same two spots. In the real world, however, oscillations are damped. There's friction. There's drag. There's metal fatigue. All these things make the system lose energy, so the size of the oscillation gets smaller and smaller.

But damping can also be useful if you want to stop an oscillation quickly once it's started. Shock absorbers on cars, for example, are designed to damp the oscillations that start with every bump on the road. There's no need for damping to dampen your spirits!

Answer: It is infinity at one point and zero everywhere else.

A one-dimensional Dirac delta function, represented by the lower-case Greek letter delta, depends on a single variable x. It's useful for describing sudden changes in distributions -- like the sudden force applied by a baseball bat to a ball, or like the change between empty space and a point charge (such as an electron). It has the interesting attribute that -- despite the fact that its value is infinite where its argument is equal to zero -- it does NOT have an infinite integral. An integral is a way of measuring the area under a curve, and the area under the delta function is 1, which makes it tremendously for simplifying integrals: it "picks out" certain values of the variable and drives the contributions of all other values to zero. It's often used to set constraints on a problem; for example, particle physicists use it to enforce conservation of momentum and energy.

All this talk of a Dirac delta "function" tends to annoy mathematicians, who have a much purer conception of "functions" than physicists do. Strictly speaking, the Dirac delta function does not satisfy many important requirements for mathematical functions -- for example, it does not have the same integral as the function y = 0, which has the same value everywhere but a single point. It would be more accurate to call this useful construction a Dirac delta "distribution", but that would require extra syllables, so physicists continue to speak of it a bit less rigorously.

Answer: Two or more distinct states have the same energy.

Studying quantum-mechanical systems is all about identifying their states: which states they're in, and which are available. (This doesn't translate to deterministic knowledge: in quantum mechanics, a system could easily be in several states at once.) Most of the time, one thinks about eigenstates with well-defined energies, which are solutions to the Schrödinger equation.

Sometimes, though, there's information in a quantum state that doesn't affect its energy. Take the electron in a hydrogen atom, for example. It could be in its ground state, close in to the proton; it could be in an excited (higher-energy) state further away. But that ground-state electron has a spin, which could be up or down. Both have the same energy, so the electron's spin states are two-fold degenerate. Turning on an external magnetic field "breaks" the degeneracy: the spin states now have different energies in the new hydrogen-atom-plus-magnetic-field system, depending on their alignment relative to the field.

Answer: Waves

The word "diffraction" refers to a whole suite of phenomena that arise when waves interact with obstacles. Picture a water wave striking a boulder, for example: the wave breaks around the rock and the water comes in from a different angle. You can think of each point on a wave front as being like the source of a whole new wave. When the original wave is unobstructed, these new secondary waves are no different - but when there IS an obstacle, then the secondary waves produced along that obstacle give rise to complex interference patterns, with areas of high intensity and of low intensity. You can see these characteristic fringes in the borders of a shadow - that's light diffracting around the edges of the object that's otherwise blocking it.

In the early 20th century, diffraction experiments with two tiny slits were instrumental in developing quantum mechanics. If you shine light through these two tiny gaps, you end up with a characteristic fringed diffraction pattern on the other side; if you pass electrons through the same slit arrangement, then you get the same fringed pattern on your detector, even if you're only providing one electron at a time! Electrons and other quantum-mechanical particles are thus both particles and waves, a duality which is still difficult to understand.

Answer: Momentum

Louis de Broglie's equation, which he proposed in the PhD thesis he wrote in 1924, says that a particle's wavelength - the distance between one peak and the next - is equal to Planck's constant, h, divided by the particle's momentum. A fast-moving particle thus has a shorter wavelength than a slow-moving particle with the same mass. A thermal neutron at room temperature has a de Broglie wavelength of 1.4 Angstroms (one Angstrom is one ten-billionth of a meter); a dust particle, moving more slowly but with a much higher mass, has a de Broglie wavelength of about seven millionths of an Angstrom.

What does it mean for a particle to have a wavelength? If you send the particle into some structure about the same size as its de Broglie wavelength, you'll see that the particle interacts with the structure in a wavelike way: you'll see diffraction patterns. If you're using, say, electrons to try to get an image of something very small, you're won't get a clear picture of anything smaller than their de Broglie wavelength. This is one reason why particle accelerators strive for higher and higher energies: to get smaller and smaller wavelengths.

Answer: Diode

A "diode" is a passive, solid-state device that is represented in circuit diagrams by a triangle pointing along the wire, with a little perpendicular line at its tip. It's a nifty tool: in its typical operating range, it maintains a constant, small voltage drop (usually around half a volt) across itself, allowing current to flow only one way. (If you apply a large enough voltage in the other direction, you can send a diode into "reverse breakdown," which is not desirable behavior for diodes in most applications.)

Diodes have a variety of uses, such as "rectifying" an alternating-current signal into a precursor to a direct-current voltage. And then there are the specialty diodes, which can regulate voltages (Zener diodes), tune TV receivers (varactor diodes), and even glow brightly (light-emitting diodes, or LEDs).

Answer: The sound waves seem distorted when the source is moving relative to the observer.

Christian Doppler noticed and named this effect in 1842. Sound, like other waves, is described by its wavelength (the distance between two consecutive peaks): a shorter wavelength means a higher frequency (the number of cycles per second) and a higher pitch, and a longer wavelength means a lower frequency and a lower pitch.

The Doppler effect arises because, when the source of a sound is moving toward you, the wavelength appears compressed, so you hear a higher-pitched sound; the opposite occurs when the source of the sound is moving away from you.

This effect applies to all waves, not just to sound: astronomers see large Doppler shifts in the light from galaxies that are moving very fast relative to us.

Answer: It does not interact with electromagnetic radiation (light).

Light -- electromagnetic radiation -- is our only real way of observing the universe. We discover stars by the light they emit. We discovered the other planets of our solar system by the light they reflect. Even black holes have distinctive radiation signatures. Dark matter doesn't shine, so we can't see it -- but it interacts gravitationally, so we can tell something is there. Galaxies, for example, rotate in a different way from how they would if their visible components were all they had. Galaxy clusters have a larger gravitational pull than their visible mass can account for. And, on a more fundamental level, we need dark matter to explain why the universe is structured the way it is.

So what is dark matter? We just don't know. We know that a very small part of it is neutrinos, and another small part of it may be invisible but ordinary objects like never-ignited stars, but most of it probably consists of things we can only speculate about. Astrophysicists and particle physicists alike are engaged in a search for dark matter, both in space and in the laboratory -- but even after this problem is solved, there's still dark energy.

Answer: The universe's expansion rate is speeding up.

We know that the universe is expanding because all the distant galaxies we see are moving away from ours. There's nothing special about our patch of the universe; it isn't plausible that the Milky Way has body odor that bad. If the universe is expanding, though, then any observer will see other galaxies racing away from her.

But the universe isn't just expanding at a constant rate: the expansion is speeding up. This was first observed in studies of faraway supernovae. You can tell the distance of these exploding stars from their brightness and the universe's expansion rate from the Doppler effect on their light. A more distant supernova gives you a snapshot further back in time, so you can see how the expansion rate has changed over billions of years.

It's not clear yet what dark energy is, but it's the best explanation we've got, for this very precise measurement of the cosmos and for others.