by Brian Cox
1. The fact that the gravitational potential exactly maps the terrain is because, in the vicinity of the Earth’s surface, the gravitational potential is proportional to the height above the ground.
2. They are in fact described by Bessel functions.
3. This is obtained using the fact that the energy is equal to ½ mv2 and p = mv. These equations do get modified by Special Relativity but the effect is small for an electron inside a hydrogen atom.
4. This is a big ball and we don’t need to worry about any quantum jiggling. But, if the thought crossed your mind, it is a good sign: your intuition is becoming quantized.
5. Actually, musicians probably don’t say this either, and drummers definitely don’t, because ‘frequency’ is a word with more than two syllables.
6. i.e. n = 1 in the case of the square well potential.
7. Incidentally, if you know that E = cp for massless particles, which is a consequence of Einstein’s Theory of Special Relativity, then E = hc/λ follows immediately by making using of the de Broglie equation.
1. Technically, as we mentioned in the previous chapter, because the potential well around the proton is spherically symmetric rather than a square box, the solution to the Schrödinger equation must be proportional to a spherical harmonic. The associated angular dependence gives rise to the l and m quantum numbers. The radial dependence of the solution gives rise to the principal quantum number n.
2. We will learn in Chapter 10 that accounting for the possibility that the two electrons interact with each other means we need to calculate the probability to find electron 1 at A and electron 2 at B ‘all at once’ because it does not reduce to a multiplication of two independent probabilities. But that is a detail as far as this chapter is concerned.
3. In units of Planck’s constant divided by 2π.
1. This is an excerpt from his 1956 Nobel Prize-winner’s speech.
2. For the sake of this discussion we are ignoring the electron’s spin. What we have said still applies if we imagine that it refers to two electrons of the same spin.
3. Recall we have in mind two identical electrons, i.e. they have equal spin.
4. Providing the protons are not moving too rapidly relative to each other.
5. This is true for standing waves, where the clock size and the projection onto the 12 o’clock direction are proportional to each other.
6. You might think there are four more wavefunctions, corresponding to the ones we have drawn turned upside down, but, as we have said, these are equivalent to the ones drawn.
7. The electron volt is a very convenient unit of energy for discussing electrons in atoms and is widely used in nuclear and particle physics. It is the energy an electron would acquire if it were accelerated through a potential difference of 1 volt. That definition is not important, all that matters is that it is a way of quantifying energy. To get a feel for the size, the energy required to completely liberate an electron from the ground state of a hydrogen atom is 13.6 electron volts.
1. This definition is purely a matter of convention and a historical curiosity. We could just as well define the current to flow in the direction that the conduction band electrons move.
1. The propagator shrinks the clock as well, in order to make sure that the particle will be found with a probability of 1 somewhere in the Universe at time T.
2. We met this idea before, when we tackled the Pauli Exclusion Principle in Chapter 7.
3. This is a technical point because the clock-winding and -shrinking rule we have used throughout the book to this point does not include the effects of Special Relativity. Including these, as we always must if we are to describe photons, means that the clock-winding rules are different for the electron and photon.
4. g is related to the fine structure constant:
5. This is a technical point, to ensure that the electron feels roughly the same sized magnetic force as it moves around.
6. The one first anticipated by Bohr back in 1913.
1. An ‘event’ is a single proton–proton collision. Because fundamental physics is a counting game (it works with probabilities) it is necessary to keep colliding protons in order to accumulate a sufficient number of those very rare events in which a Higgs particle is produced. What constitutes a sufficient number depends on how skilful the experimenters are at confidently eliminating fake signals.
2. Our ability to think of a massive particle as a massless particle supplemented with a ‘kink’ rule comes from the fact that P(A,B) = L(A,B)+L(A,1)L(1,B)S+L(A,1)L(1,2)L(2,B)S2+L(A,1)L(1,2)L(2,3)L(3,B)S3 +…, where S is the shrinkage factor associated with a kink and it is understood that we should sum over all possible intermediate points 1, 2, 3 etc.
3. This is a subtle point and derives from the ‘gauge symmetry’, which underwrites the hopping and branching rules of the elementary particles.
4. He was far too modest to call them by that name.
1. Recall from Chapter 5 that particles of definite momentum are in fact described by infinitely long waves and that as we allow for some spread in the momentum so we can start to localize the particle. But this can only go so far and it makes no sense to talk about a particle of a certain wavelength if it is localized to a distance smaller than that wavelength.
2. We can generalize to the entire star because we are not being specific about where the cube actually is. If we can show that a cube located anywhere in the star does not move then that means all such cubes don’t move and the star is stable.
3. It is of course possible to compute more precisely how the electrons move around but at the price of introducing more mathematics.
4. Newton’s second law can be written as F = dp/dt. For constant mass this can be written in the more familiar form: F = mdv/dt = ma.
5. Here we have combined the exponents according to the general rule xaxb = xa+b.
6. For those of a mathematical bent, show that , i.e. that the function g (a) is actually determined once we know the function f (a).