The Hamiltonian is a concept in (unsurprisingly) in Hamiltonian mechanics which represents an operator in a system. In many cases it corresponds to the total energy of a system and so I’ll continue to use it as analogous for the total energy although this not always true. Just picture it as an equation with each term representing one of the kinds of energy a system contains. Now often you want to find the ground state, the lowest possible energy, for the system as this will provide useful information to solve it. But what if this Hamiltonian’s ground state is very complicated, if not impossible, to find algebraically? Well, this is the basis of an adapted form of quantum computing called adiabatic quantum computing.
The precise measurement of Earth’s gravitational field is very important. By timing the period of a simple pendulum it might be possible to get a measurement for the field with accuracies to three significant figures but not much more than that. A key part of geophysics is to use variations in gravitational field strength at different points around the surface of the Earth to estimate subterranean densities. These variations can be found to such resolution that they can be used to study tectonic plates, volcanoes and even the decrease in mass of melting glaciers.
The human eye is quite incredible. When it comes to resolving two images it is practically at the physical limit imposed by diffraction, it can identify over one million separate colour hues which is a whole lot more than anyone could ever need and it has intensity detection of about 5nW so maybe 150 visible photons. It’s these properties we use when we observe images as our eyes are able to detect the gradients of colour and of intensity. If we are dealing with greyscale images, in just black and white, then there is no colour and each pixel responds just to the intensity of light falling upon it. To simplify these are the kind of images that we’re discussing today.
It will be a trivial fact for most people that photons can provide energy, i.e. scatter, electrons. Photons providing energy to electrons occurs in the photoelectric effect which most people with a rudimentary knowledge of physics will know. This energy transfer, photons to electrons (or in fact any charged particle), is called Compton scattering which was covered in only slightly more detail here. This can be pictured as a stationary electron, impacted by a high frequency (therefore high energy) photon, and flying off due to receiving energy.
The simple fact is that sometimes there is no solution to a problem. This is not a response that many people would like to hear but it is an unfortunate truth. A particularly famous example would be the three body problem, where the equations of motion for three masses with arbitrary starting positions and velocities are simply unsolvable. In many ways it is the balance that scientific models strive for to both be accurate to physical reality but also solvable. If it impossible to do both then making a model accurate but solvable by numerical methods is acceptable. The worst situation is that to get an accurate model will require creating the model so that even a computer will struggle with providing solutions as complexity grows.
The Hall effect is not too difficult to understand. Electrons experience a force when moving in a magnetic field, so if a sheet of metal carrying a current was in a magnetic field we would expect the electrons flowing through it to experience a force towards one side. The electrons will then try to move straight as part of a flowing current, but will be forced to curve due to the perpendicular force they feel. Having accumulated at one side of the sheet they will have set up a noticeable potential difference across the sheet, perpendicular to both the magnetic field and the net current flow. This whole set of behaviour is the Hall effect.
The photothermal effect is an example of giving a technical name to an effect that everybody already knows. It is the idea the electromagnetic radiation will interact with the charged particles present in any material resulting in those particles being excited and the material gaining thermal energy. In other words, light can heat things up. It is also possible for the light to actually transfer some of this energy into usable mechanical work although the number of materials which possess the ability to do this is small.