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Mathematics is a tool for physics, to model pre-existing data and predict future observations and measurements. Physics is one of the areas where mathematics is used in this fashion. If one considers measuring the fields in flat land, as the Nile valley in Egypt, as data, then the origin of western axiomatic mathematics was also from fitting observations, ...


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So, can an area of space a few feet across (externally) be a light-year wide (internally) (TARDIS sort of thing)? Yes. It can and that's normal. If you made a region with less space inside than the surface area indicated that would require exotic matter. And with exotic matter you can make time machines. So ... Ironically you need something smaller on ...


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The kilogram is the base unit of mass because electrical engineers in the late 19th century chose a particular set of practical electrical units. These practical units were a success, and we are still using them today: ohm, volt, ampere, and the joule. In 1874 the mechanical units cm, g, s ('CGS') were adopted as the coherent system of units for science. ...


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See this wikipedia article. It is probable the "S" stands for Spanier as in Edwin Spanier.


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In a hydrogen atom the kinetic energy is on the order of $8$ eV. From $T = \frac{p^2}{2m}$ we get that the typical momentum is about $3$ keV/c ($m = 511$ keV/$c^2$). On the other hand $\hbar/(2a_0)\approx 1.2$ keV/c where $a_0$ is the Bohr radius, which is about the size of a hydrogen atom. Since these quantities are of similar magnitude the Heisenberg ...


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Consider this example: If I throw a ball vertically upwards then (neglecting air resistance) it moves in a straight line with the constant acceleration $-g$. At some time $T$ later the ball falls back down and I catch it. The displacement in the time $T$ is zero because the ball started in my hand and finished in my hand, so the average velocity is zero ...


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There is a reason physics starts with kinematics, description of motion and evolution is what it is about and once you throw in the causes you get dynamics, which is everything. It is not obvious that energy is secondary at all. And I dispute it. How can you tell you have something? Because it interacts in certain ways (dynamics) or it has the capacity to ...


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IMHO, the notation $\int_a^b\mathrm{d}x\,f(x)$ is much cleaner than $\int_a^b f(x)\,\mathrm{d}x$, because the integration variable ($x$) and its associated integral range $(\int_a^b$) are kept together. This is particularly important in lengthy and multi-dimensional integrals. Consider $$ \Upsilon_{pq}(k)= \int_0^\infty\mathrm{d}x ...


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Besides the reasons listed in Lubos Motl's answer, here is another reason for the $\int \!dx ~f(x)$ notation: By writing the integral sign $\int_a^b$ and $dx$ next to each other in multiple nested integrations, it becomes more easy to trace which limits belong to which integration. This becomes particularly handy when changing the orders of integration. ...


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It's not just QFT literature. Physicists, especially adult research physicists, find this notation sensible and popular – even though it may be more popular among particle physicists than elsewhere. Formally, $dx\,f(x)$ is a product of two factors and $\int$ is a form of a sum. Because product is commutative, it doesn't hurt when the order is interchanged. ...


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Self-similarity and chaos have been very influential in physics. They help us understand a lot of very interesting physical phenomena on the macroscopic scale, especially fluid flow and many-body problems. So, you can study these things and not be labelled a crank by physicists. But (you heard the "but" coming, right?) if you are talking about ...


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First off, quantum mechanics isn't "random". In QM, everything evolves perfectly deterministically, just in a somewhat strange way. Some people have the picture that a quantum mechanical universe looks just like a classical one, except that every once in a while crazy stuff randomly happens. Nope! It's not classical + randomness, it's a totally different ...


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Above is a picture of a section of diamond crystal, the hardest naturally occuring material we know and to me that least, the triangle shape is not as evident in this section as it is in any man made structure. The other point that may be worth mentioning is that, as you probably know, a diamond is one giant molecule, whereas I don't think graphene ...


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Triangles are the "strongest shape" only other things being equal, such as rigidity of sides. To the extent that the analogy works rigidity corresponds to the strength of forces between atoms in a lattice, and that depends on their type. So no, the strength of the lattice is not determined by geometry alone. For example, a triangular pattern would require ...


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In condensed matter "bulk" does not refer to the dimensionality of the problem but the location in the material. It refers to the volume of the crystal, as opposed to, e.g., surface effects. Many organic conductors behave as 1D systems, yet you can talk about bulk properties. Copper oxide superconductors have a 2D physics. However, often you will find ...


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This is a question about philosophy of science and epistemology, so you should expect varying answers with different prespectives. This is my personal approach to the question. First let's examine what does it mean to say that a scientific theory is "absolutely proven". Just as John Rennie pointed out in his answer, a scientific theory is a mathematical ...


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Photons undergo angular acceleration in very strong gravitational fields, gravitational lensing.. An acceleration can be defined in its change of direction, angular acceleration in radians/second^2, so the answer is positive, yes, light can be accelerated, but its speed will still be c, only the direction relative to the gravitational source changes.


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Classically, spinors are insignificant, and should not be expected to fulfill any physical role, since they are not proper representations of the rotation group $\mathrm{SO}(3)$ or the Lorentz group $\mathrm{SO}(1,3)$. The importance of spinors does not arise from any classically intuitive thinking about orientations. Instead, the reason why spinors appear ...


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No-one knows what dark matter and dark energy are, so any comments on your question are necessarily speculative. Having said this, dark matter is generally considered to be just matter and the adjective dark is not meant to signify anything mysterious but merely that it doesn't interact with electromagnetic radiation or charge. The most popular suggestion ...


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Spin-resolved current in the context of scanning probe methods means that due to a finite magnetisation of probe and sample the current consists of electrons of one spin in a larger quantity than of the other. Spin current usually refers to current that consists exclusively of electrons of one spin direction.


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Sure, there are a lot of them ! In my opinion, a basic conceptual reason for this is that most of the time, models used to describe a given class of physical phenomena have at their boundaries "uncontrollable" phenomena (otherwise we could extend the model straightforwardly and include them). So it is often the case that when you try to take some limits, the ...


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Sciences use mathematics only as a tool. In almost all such applications, mathematical problems (such as pointwise vs uniform convergence) are not inherent to the scientific problem at hand, but arise from the mathematical model and are indicative of its limitations. For example, when modelling a large collection of particles (be it in a solid (crystal) or ...


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If you want to hard code a specific set of qubits you could measure the spin (pass it through an inhomogeneous magnetic field) of a spin 1/2 system and then if you get the wrong spin, flip it (tune a photon to a wavelength based on a given uniform magnetic field strength). If you want to read an output you could do the same thing, measure the spin of each ...


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The truth of the Lorentz transformation as an accurate description of the co-ordinate transformation between relatively uniformly moving observers needfully implies relativity of simultaneity. Contrapositively, the Lorentz transformation cannot be sound if simulteneity is not relative. So, in the sense that the soundness of the Lorentz transformation has a ...


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As an alternative to Anna's nice historical discourse a heuristic that covers modern uses of the phrase would be that energies are "high" when the QCD can be treated as perturbative. That regime sets in considerably above the nucleon mass scale, say 10s of GeV. So LHC physics is in, JLAB physics is out (even with the 12 GeV upgrade).


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Elementary particle physics is an outgrowth of what was high energy physics, historically at the time. X-rays were high energy physics when first discovered, they are part of the tools of solid state physics now. Alpha particles and gamma rays were high energy physics at their time, they are nuclear physics now. Mesons discovered in cosmic rays started ...



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