In this note, I've demonstrated how the mathematically rigorous treatment of observables with continuous spectra works. At the end, I've introduced a little question of mine arosed while writing this note. It was represented in the 11'th session of Quantum Theory study circle at Quanta.

This paper discusses the mathematical differences between electric and magnetic dipoles. I starts with analyzing the necessity of their difference and then discuss the possibility of considering dipoles as fundamental sources of EM fields rather than electric monopoles and currents.

P.S: In the "Final Instance" part, there is a mistake negleting the effect of bound surface-charges. applying this effect, the total magnetic field is zero everywhere which leads to the fact that if you attempt to construct this magnetization using dipoles arrangement, they all cancel eachother and produce no effective field.

This paper was my attempt to formulate the probabilistic approach to classical mechanics before I get familiar with Liouville's theory; Therefore, I used a frustrating method similar to what is used for Fokker-Planck theory. Then, I derived Liouville's equation and analyzed its solution in different situations.

This paper is my attempt to build the Schrodinger equation starting from Liouville-Von Neumann equation. This ended up with the possibility of considering the Quantum Mechanics based on density matrix rather than state vectors.

This is a review paper written by S. A. Jafari in 2020 which investigates the generalized Lorentz symmetry of condensed matter systems with tilted dirac cone and the effective metric provided by them.

This is a paper written by Jörn Dunkel & Stefan Hilbert in 2013 which criticizes the usage of the so-called "negative temperature" states and suggest that the Gibbs entropy is the only reasonable (consistent) candidate for the statistical counterpart of the thermodynamical entropy. They show that this entropy, while equivalent to Boltzmann entropy at thermodynamic limit, forbids the presence of negative temperature states.

I've read this paper because i've always thought (and currently thinking!) that statistical mechanics is deeper than it appears (in standard courses) and is not investigated thoroughly as deserved (in contrast to, perhaps, QM, QFT, etc.).

This is a paper written by S. Eckel, A. Kumar, T. Jacobson, I. B. Spielman, and G. K. Campbell in 2017 which investigates the theoretical and experimental aspects of a particular analog gravity. They've prepared a ring-shaped Bose-Einstein condensate (BEC) with an excited azimuthal phonon mode. Then, expanded it rapidly and they could observe (in both theory and experiment) the redshift (as expected) and Hubble friction. They've also shown that the external radial power leaks into asimuthal energy and hence an "analog" of reheating in cosmology.

This was a part of my studies about Analog Gravity due to my project.

This is a paper written by Carlos Barcelo´, S Liberati and Matt Visser in 2001 which investigates an analog gravity model based on Bose-Einstein condensation (BEC). They consider a quantum superfluid described by Gross-Pitaevskii equation (in hydrodynamic limit; so no quantum fields appear) with some wierd (in my opinion) generalizations and show the effective metric in different approximation regimes.

This was a part of my studies about Analog Gravity due to my project.

This is a paper written by W. G. Unruh in 1981 which is the birth of Analog Gravity. We know that perturbations in a static fluid is described by a wave equation with speed of sound. Unruh asks that if the fluid is not static (but still non-rotational), where does this wave equation go? he shows that although the wave equation form is lost, it can be revived in some metric! Just analogous to the curve path taken by a light ray in the presence of gravitational field. He then shows that the dumb hole (analogous to black hole) provide a Hawking radiation with thermal spectrum just as the actual black holes.

This was a part of my studies about Analog Gravity due to my project.

This is a paper written by Asher Peres in 1995 which investigates the necessary and sufficient conditions on generalization of Schmidt decomposition to a composite system with more than two subsystems. It turns out that it's simpler that it appears!

This is a paper written by E. T. Jaynes in 1957 which argue that statistical mechanics is not actually a "physical theory" dependent on physical phenomena (such as ergodicity, metric transitivity, etc.). In this paper, statistical mechanics is considered as "the best possible estimate we can make based on available information" as a statistical inference. If ergodicity is seen to be violated, we are given some "new information" which alters our "best possible estimate" about the system so doesn't contradict other perspectives. I personally prefer This perspective to the subject and enjoyed reading this paper.

This is a paper written by E. T. Jaynes in 1971 which demonstrates the possibility of rising the Boltzmann's H function in course of time (contrary to the H-theorem). It discusses an "Empirically Rializable" way to produce H-theorem-violating systems Using a simple beautiful method. (I have presented this paper in QSC. More info can be found here in "Historical Papers" presentations).

This is a paper written by U. Klein which tries to formulate the classical probabilistic mechanics Similar to the Quantum Theory and find the "Quantization Rule".

This paper is the original paper written by W. Heisenberg in July 1925 which is considered as the birth of Quantum Mechanics translated in English (ref: Sources of Quantum Mechanics, Van Der Wearden). There is also a paper written by M. Khorrami explaining this paper with modern notation which I've found interesting.