Endnotes
Note 1: About a quantum’s amplitude
- Quote: For any type of vibration or wave, there is a smallest possible amplitude that it can have. If you try to make the amplitude smaller, the vibration or wave simply won’t occur.
- Endnote: This is correct for a vibration but a mild oversimplification in the case of a wave, because the actual minimum amplitude of a wave depends on the wave’s shape—for instance, on how many troughs and crests it has. Though I will stick with this manner of speaking for now, we’ll turn our focus in the next chapter from a wave’s amplitude to a wave’s minimum energy, at which point its shape will cease to matter.
- Discussion:
Later in the book, in Chapter 17, we will see that what’s actually minimal about a quantum is not its amplitude but its energy (or really, its ratio of energy to frequency.) This is encoded in Planck’s quantum formula, E = f [h].
For a simple vibration like a pendulum, the amplitude is also minimized, because energy and amplitude are directly related. But for a wave, the situation is more complicated, because the amplitude of the wave depends on its shape as well as on its energy. For instance, even for a simple wave of a definite frequency, a short wave with just a few crests and troughs needs a larger amplitude to have the same amount of energy as a long wave with many crests and troughs.
The formula for the amplitude for a wave of a definite energy shows explicitly that it depends on shape. This is discussed in my series on Fields and Particles (with math), in particular just before the figure in the fifth article in the series.
Note 2: Zero-point motion
- Quote: You would be unable to reduce the [pendulum’s] amplitude by half at each step. Instead, the pendulum would vibrate with amplitudes slightly different from the ones you wanted. Finally, it would reach its minimum amplitude, and your only options after that would be to leave it as it is, or to shut off the vibration altogether.
- Endnote: Actually, even after the vibration is shut off, there is still some random motion left over; physicists call this zero-point motion. It arises from quantum uncertainty, which we’ll get to soon.
- Discussion (click here)
Note 5: “Electricity”, “electron” and “electric field”
- Quote: Do be careful not to conflate the electron field with the more familiar electric field, as they are completely different. For one thing, the electron field is fermionic, while the electric field, a part of the electromagnetic field, is bosonic; see Chapter 15 and Table 5.
- Endnote: The similarity in the fields’ names is a historical association of both fields with aspectsof electricity, but the two fields are profoundly different in character.
- Discussion: (coming soon)
Note 6: Detecting gravitons
- Quote: Among the bosonic fields, a W boson is a wavicle of the W field, a Z boson is a wavicle of the Z field, and a gluon is a wavicle of the gluon field.
- Endnote: Gravitons, wavicles of the gravitational field, may well exist, and I usually assume that
they do. But it may be a long time before this can be confirmed experimentally. - Discussion: (coming soon)
We certainly don’t know that quantum physics and Einstein’s theory of gravity merge in the simplest possible way. But if they do, then gravitons must exist. However, the experimental difficulty of detecting gravitons is extreme.
One way to see this is that a photon with a frequency of visible light can be easily deflected by a tiny mirror, whereas a graviton with the same energy could pass through the largest star without being absorbed or significantly deflected. That’s a problem! If an experiment can’t absorb or deflect a particle, it has little hope of detecting it.
Gravitons aren’t the only particles that are difficult to detect. Neutrinos, too, can pass through huge amounts of ordinary matter without being affected by it, and for this reason, they were very difficult to observe. For both neutrinos and gravitons, the higher their energy (up to a point, but a very high one), the more likely they are to scatter off matter.
However, it’s much, much easier to detect neutrinos than gravitons. The probability for a neutrino of a given energy to scatter off a chunk of material is 1032 times larger than the corresponding probability for a graviton of the same energy, and so one needs tremendously more gravitons to have any hope of detecting even a single one. Moreover, there are strategies for making large numbers of neutrinos that don’t work for gravitons. One can rely on powerful sources of neutrinos such as the Sun or human-made nuclear reactors, or one can create powerful beams of neutrinos. (How to create such a beam is explained in this article.) These beams contain so many neutrinos of moderate energy that the probability that one of them will scatter off an atom in a large detector, over a period of hours or days, is not so small. But no natural reactor or artificial process makes large numbers of moderate-energy gravitons.
Natural sources of gravitons are the same objects that create gravitational waves — indeed these waves would be made of gravitons in much the same way that bright light is made of photons. But the energies of these gravitons are very low, making the scattering probability unbelievably small. The amplitude of realistic gravitational waves produced by astronomical effects in the distant universe is nowhere near large enough to give us any hope that its individual gravitons could be detected.
Making higher-energy gravitons either artificially or naturally is almost impossible outside of an evaporating black hole, which can only exist in the present universe in very special situations. (Generally, the cosmic microwave background radiation heats black holes faster than they can evaporate, unless they were born very small; but we don’t yet know any processes in nature that can make small black holes.) So to our current understanding, actually detecting individual gravitons lies wildly beyond our technological capabilities. The only hope is that there is some major modification of Einstein’s theory of gravity when it marries quantum physics that breaks some of the implicit assumptions that I’ve made here.
Note 7: Fields with one or two wavicles
- Quote: the easiest way to think about it is this: some fields have two types of wavicles, while others have only one. In the first case, the two types of wavicles are each other’s antiparticles, while in the second, the lone wavicle is its own antiparticle.
- Endnote: In math, the difference is simple. Some fields are described using complex numbers, and they have two types of wavicles related by complex conjugation. For other fields, real numbers suffice; they have one type of wavicle.
- Discussion: (coming soon; what is below is just a rough draft)
If a fieldis real — if its value at each point in space
and time
is a real number — then its quanta, its particle-like ripples with definite speed
, are given by sines and cosines with real coefficients
, in which the field looks like
$$ \phi(x,y,z,t) = A_1 \cos(x -v t) + A_2 \sin(x – v t) $$ .
If a fieldis complex, then it has real and imaginary parts, and each of these can have sines and cosines, combined in ways that are related by complex conjugation. We can write it as before except now with complex coefficients
$$ \Phi(x,y,z,t) = B_1 \cos(x -v t) + B_2 \sin(x – v t) $$
but in fact it is best to write it in this way
$$ \Phi(x,y,z,t) = C_1 e^{i (x-vt)} + C_1^* e^{i (x-vt)} +C_2 e^{i (x+vt)} + C_2^* e^{-i (x+vt)} $$
which makes it clear that it has two sets of particles related by complex conjugation.
Note 8: Wave functions versus wavicles
- Quote: It is easy to become confused about wavicles because a second wavy concept often arises in discussions of quantum physics. Wavicles such as electrons and photons are physical objects that carry energy; they exist within and move across the same empty space that you and I live in. An unrelated wave is called the Schrödinger wave function; it does not exist within the space that you and I live in, does not carry energy, and is not a physical object at all. Instead, it is a wave that travels in the abstract space of all possibilities for the wavicles and fields that it describes. This wave function is a mathematical tool used to calculate the probabilities for what physical objects (including wavicles and fields) may do. Wavicles, as physical objects, can have observable consequences—a wavicle can enter your body and cause damage to one of your cells, for instance—while a wave function can do no such thing. We will not encounter wave functions again in this book.
- Endnote: Mistaking wave functions for wavelike particles, and thinking them physical objects, is a common error for students learning atomic physics for the first time. Since a single electron can move around in three-dimensional space, a wave function of a single electron exists in three-dimensional space, too, and so it seems like a real object moving in the space we live in. But a wave function of two electrons already exists in six-dimensional space, because the two electrons have six dimensions’ worth of possible positions. For four electrons, the wave function is a wave in twelve-dimensional space. Once we get to quantum field theory, the wave function exists in an infinite dimensional space because a field can take on an infinite variety of wavy shapes.
- Discussion (coming soon)
