A Brief History of Quantum Mechanics: From Blackbody to DeBroglie
Embedded Files
For the Reader
For the Reader
This short overview is supposed to be a chapter in the review article that i was writing on Quantum Materials. It was review article for mechanicians, introducing them to Quantum Materials. For the sake of clarity, we decided to start from the band theory. This means that even though i had worked on this part, we ended up scrapping it. I was really proud of the question that i had asked "Why Experiments & Invariance Matter in Physics", I dont want to let it go to waste. I will leave it here on my website for readers to go through, if they were interested. I have shared this as a series of linkedin posts. If you are interested, then please follow the day no. form this one onward.
Why Experiments & Invariance Matter in Physics
Why Experiments & Invariance Matter in Physics
The origin of Quantum mechanics can be traced back to Max Plank and his explanation for the ultraviolet catastrophe, which in turn is based off of Kirchoff's identification of the black body spectrum being a universal phenomena. We first take a step back and discuss the importance of experiments in physics. Experiments are not only performed to verify theoretical predictions, but also to measure material properties. When performing the latter, it is often important to design experiments such that it minimizes measurement errors. If there is some symmetry or invariance relationship that the system has, then designing experiments focused on this would give rise to reduced errors. We give a few examples, two from mechanics, one from electrodynamics and one from thermal physics. The first three serve as setup to showcase the importance of invaraince, while the latter serves as a segue to the ultraviolet catastrophe:
Invariance of isotropic elastic materials:
One of the biggest reason to put elasticity on a rigorous footing was to find restrictions on the energy density. One such case was that for isotropic elastic materials. It can be shown that the energy density depends on three principal invariants of strain (see Jog chapter 5 for discussion on strain invariants and isotropic elastic solids). Rivlin showed that for homogeneous, incompressible isotropic highly elastic solids, experiments can be done to determine the shape of the free energy density with respect to the strain invariants (see paper).
Highly elastic refers to materials for whom energy is conserved in a cycle of deformation carried out under isothermal conditions.
Dead loading device:
A deal loading device applies traction on the surface of a body that maintains its direction and amplitude and is unchanged by the deformation experienced the body. It can be shown that, provided the material is homogeneous, the body will be in a state of uniform stress (see Abeyratne, James). A similar behavior is seen for pressure loading device. A key feature is geometry independence of the undeformed configuration and the uniform stress response in the material. This allows one to focus their attention on other aspects of the experiments. For example, for dead loading device one has to maintain the direction and magnitude of the applied surface traction, while for the pressure loading device, one has to ensure that the pressure transmitting container, ie, the contained in which the body is suspended while it is subject to uniform hydrostatic pressure, must be large enough.
Van der Pauw method:
The Van der Pauw method is a way to measure the resistivity and Hall coefficient of flat samples of isotropic materials of arbitrary shape (see Philips Report). The requirement is to make a singly connected sample that is homogeneous in composition and thickness. The current contacts must be far away from the point of measurement to ensure that the current pattern near the measurement point is roughly parallel and not changed when a magnetic field is applied. The independence on the shape of the sample to the measured quantities (resistivity and hall conductivity) mean that slight imperfections in the shape won't change the measured response.
Black body Radiation:
Whenever a body absorbs radiation from a source, it emits some of it back in order to maintain a constant temperature. For energy incident on a body, one may conclude that the energies are either, transmitted, absorbed or reflected. Kirchhoff defined a black body as something that would absorb all incident electromagnetic radiation, regardless of frequency or angle of incidence. The transmission and reflection coefficients for this material are zero at all frequencies. Because it must radiate off the energy, the radiation can only depend on the temperature of the object and not on the material composition or geometry. The reasoning is purely thermodynamic. Consider two black bodies of different geometry and composition held at the same temperature. The two bodies are separated by a filter that only lets radiation of a certain frequency through. If the radiation spectrum were dependent on these parameters, then they would radiate different amounts of energy. There would be a net flow of energy from one body to the other. But this violates the zeroth and second laws of thermodynamics. Since this holds true for each frequency, we have the exact same spectrum.
Kirchoff was the one to first notice the importance of the black body radiation spectrum as being a reflection of the fundamental law of physics; the spectrum is independent of geometry and material. Otto Lummer was one of the first to do experiments on a black body by constructing a cavity with a hole. Provided the hole is small, any incident radiation on the hole will almost surely never leave. The hole acts like a black body owing to the independence of the material and geometry, this qualifies as a black body. The measured black body radiation graph is reproduced below. The beauty and power of Kirchoff's argument should slowly become clearer.
When the hole with a cavity is maintained at a fixed temperature, the hole produces radiation. Inside the cavity, electromagnetic standing waves are setup of all wavelengths λ. To compute the energy associated with each frequency (ν). The typical calculation makes use of the geometry independence and works with standing waves for a cube, for which we know the standing waves (nλ/2=L). Given the frequency wavelength relationship c=λν implies (2L/c)ν=√(nₓ²+nᵧ²+nz²), we can compute the number of modes, or allowed tuple of integers (nₓ,nᵧ,nz), in the frequency interval (ν,ν+dν) by noting that they correspond to the surface area of the sphere of radius ν (see Figure to the left).
Thus the number of modes at a given frequency is is roughly proportional to the square of the frequency. From the equipartition theorem, the amount of energy associated with each mode is assumed to be same and each given the energy kBT/2. Thus the total energy at a given frequency is the product of the no. of modes and the energy per mode is proportional to ν²T. However notice that for a fixed temperature, and at high frequencies (low wavelengths), the class law predicts infinite energy, while the experimental measurements report infinitesimal energy (see black body radiation curve above). This came to be known as the ultraviolet catastrophe, the prediction was wrong for the high frequency regime.
It was Max Plank who finally solved this problem by assuming that the energy is quantized. It turned out that the application of the equi-partition theorem was what lead to the issue. While the no. of modes increase with the energy, the energy associated with the higher modes must be less to compensate. To get around this Plank quantized the allowed energies for each mode in units which varied with the frequency. The simplest model gives us E=hν, where h is the Planks constant.
Page updated
Google Sites
Report abuse