The Quantum Universe_ Everything That Can Happen Does Happen - Brian Cox [113]
5. This way of arriving at the Uncertainty Principle did, however, rely on the de Broglie equation in order to link the wavelength of a clock wave to its momentum.
6. In the jargon, the momentum space wavefunctions that correspond to particles with definite momentum are known as momentum eigenstates, after the German word eigen, meaning ‘characteristic’.
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