Topic 12: Quantum and Nuclear Physics

12.1 Interaction of matter with radiation

Blackbody radiation

  • What is a blackbody – A blackbody is, as the name suggest, ‘completely black’. That is, any EM waves, no matter the wavelength, gets absorbed when it hits such a body. Hence, the colour of the such an object depends only on its own radiation spectra. This spectra depends on the body’s temperature.
  • ultraviolet catastrophy (equipartition of energy, more energy at higher freq modes in a cavity -> Rayleigh-Jeans law: radiated power proportional to freq squared) (Wien’s law on the other end). Actual history of Planck’s discovery.
  • Planck’s formulation

Photoelectric effect

  • Classically, we treat light as a wave. What happens when we shine light on a metal? Well, if it’s a wave, it imparts energy to the metal. After some time, we would expect enough energy to be transferred, freeing an electron. So we would see electrons. We can do this by measuring their current. Higher amplitude, higher frequency, these would lead to higher power. So more electrons more quickly. Except that’s not what happens.
  • Above a certain wavelength, no electrons are emitted. None. And when electrons are emitted, they are emitted immediately, without a delay. The greater the intensity, the more electrons are emitted. But again, if the wavelength is too large, then intensity does not help. This minimum wavelength depends on what type of metal it is. Why? You’ll see shortly.
  • If we treat light as particles, then this becomes easily understandable. The ‘amplitude’ of the wave is simply related to the number of particles incident per unit time. There are few incident photons compared to the number of electrons. Since time over which the excitation happens is short, the photon-electron interaction is one to one. Only one photon to one electron at a time. The energy of the photon is given by E = hf. So if the photon doesn’t have enough energy to free the electron, the electron isn’t freed, and no electrons are observed to be emitted. Hence, below a certain minimum frequency, no electrons are emitted. Above this threshold frequency, a larger intensity leads to a larger photocurrent, since more electrons are liberated.
  • The key formula for this chapter is a consequence of the conservation of energy. Ek_max = hf – workfunction. The maximum kinetic energy of the photoelectron is the energy of the incoming photon minus the work function, which is the amount of energy that is associated with the binding of the electron to the nucleus. This is similar to being in debt. In order to be liberated, you must be hit by a banknote which is worth more than your debt. Then you can be free. The amount of money you have when you are free is at most the difference between the bank note and your debt. Why maximum? Why not exactly? Well, you’ll have to take a bus to get to wherever to get the debt settled, and you might have eaten a sandwich on the way, and all these cost money. Similarly, for the electron, it may hit into various particles on the way out, and so the kinetic energy given by this formula is the maximum kinetic energy.
  • What exactly is the workfunction? An electron in a metal can be thought of being in a potential well (due to the negative potential associated with the Coloumb attraction between the electron and the positive ion lattice of the metal). In order to come out, a certain amount of energy has to be provided. This is similar to the ionisation so commonly encountered in various subjects.

Earlier, we treated light (which we normally think of as waves) as a particle in order to explain the photoelectric effect. What was traditionally considered to be waves can be treated as particles. Can we treat particles as waves?

It turns out we can. This is exactly what de Broglie did. A particle with a momentum p has an associated wavelength lambda = h / p, where h is Planck’s constant. That is, the momentum is inversely proportional to the wavelength. Everyday objects like cars and people have a fairly large momentum, hence the wavelength is too small to notice, which is why we don’t notice their (our) wave properties. However, for small particles like electrons, the typical momentums are much lower, and can thus have wavelengths comparable to that of say, inter-atomic spacing.

So now particles can have a wavelength. What about their other wave properties? Can we find a good way to describe them? How about properties other than momentum? All these are encapsulated in Schrodinger’s wave equation, which describes how the wavefunction. This wavefunction itself has no physical meeting, but if you take the mod square of the wavefunction, you get the probability distribution function of where the associated particle can be found. That is, the probability it can be found within a region deltaV of a point r is deltaV(r)*|psi(r)|^2. There are other nice things you can get from the wavefunction, but I shall not go into detail here.

Heisenberg uncertainty. Quantum tunneling.

Pair production and annihilation (feynman diagrams)

12.2 Nuclear physics

The volume of the nucleus is proportional to the number of nucleons in it. Consequently, the radius is proportional to the cube root of the number of nucleons in it. The constant of proportionality is R_0, the radius of a nucleus with only 1 nucleon (hydrogen). Radius of nucleus  R = R_0 A^1/3

Nuclear radius acting as a single slit. Sin theta = lambda / D (first minimum)

Radioactive decay and the decay constant (·A = lambda N ) (N = N_0 e^{- lambda t})