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41

This is actually a very tricky question, mathematically. Physicists may think this question to be trivial. But it takes me one hour in a math summer school to explain the notion of gapped Hamiltonian. To see why it is tricky, let us consider the following statements. Any physical system have a finite number of degrees of freedom (assuming the universe is ...


40

If you're pushing a 10-ton truck and it's not moving, you are not doing any work on the truck because the distance $ds=0$ and the nonzero force $F$ isn't enough for the product $F\cdot ds$ to be nonzero. Your muscles may get tired so you feel that you're "doing something" and "spending energy" but it's not the work done on truck. You're just burning the ...


38

The main distinction you want to make is between the Green function and the kernel. (I prefer the terminology "Green function" without the 's. Imagine a different name, say, Feynman. People would definitely say the Feynman function, not the Feynman's function. But I digress...) Start with a differential operator, call it $L$. E.g., in the case of ...


34

A second-order tensor can be represented by a matrix, just as a first-order tensor can be represented by an array. But there is more to the tensor than just its arrangement of components; we also need to include how the array transforms upon a change of basis. So tensor is an n-dimensional array satisfying a particular transformation law. So, yes, a third-...


33

The definition for the cesium clock is: 9192631770 cycles per second is frequency of the radio waves which cause maximum resonance, a physically measurable condition, in the cesium atoms. This corresponds to a particular tuning of the radio. Keeping it tuned provides the reference frequency cited.


31

Which year? The sidereal year? The tropical year? The anomalistic year? The calendar year (and whose calendar)? The sidereal year is the average amount of time it takes the Earth to make one complete orbit about the Sun with respect to the fixed stars. The tropical year is the amount of average amount of time between successive spring equinoxes. The ...


28

You're correct and the video is mistaken. In fact, if cesium atoms were constantly oscillating between the two hyperfine states, cesium beam clocks wouldn't work at all! In its simplest form, a cesium beam clock uses a magnet to separate a stream of atoms into two streams based on their hyperfine state; one state is selected to continue down the tube to be ...


26

Our physics prof once put it informally that way: A state is a set of variables describing a system which does not include anything about its history. The set of variables (position, velocity vector) describes the state of a point mass in classical mechanics, while the path how the point mass got from point $A$ to point $B$ is not a state.


25

Pluto is now classified as a dwarf planet. The main difference between a planet and a dwarf planet has to do with the requirement that a planet clear out the material in and near its orbit. Planets do this, dwarf planets do not. The reclassification was triggered by the discovery of many additional object (the Edgeworth-Kuiper Belt) out beyond the orbit ...


25

Depends on what you mean by 'central force'. If your central force is of the form ${\vec F} = f(r){\hat r}$ (the force points radially inward/outward and its magnitude depends only on the distance from the center), then it is easy to show that $\phi = - \int dr f(r)$ is a potential field for the force and generates the force. This is usually what I see ...


24

Weight is the force with which gravity pulls on a mass. Maybe the simplest way to explain the difference is that on the Moon or on Mars, your weight is reduced because gravity is weaker there, but your mass is still the same.


24

The definition of a state of a system, in physics, strongly depends on the area of physics one is dealing with and it comes as one of the initial definitions once such underlying theory has to be set up. In particular one has: classical mechanics: a state of a system is a point $m\in TQ$ (or equivalently $T^*Q)$ in the tangent bundle of the configuration ...


21

Matrices are often first introduced to students to represent linear transformations taking vectors from $\mathbb{R}^n$ and mapping them to vectors in $\mathbb{R}^m$. A given linear transformation may be represented by infinitely many different matrices depending on the basis vectors chosen for $\mathbb{R}^n$ and $\mathbb{R}^m$, and a well-defined ...


21

Using the distance between the Sun and the Earth, at least for distances within the Solar system, just gives a better feel for the scales involved. You can't really imagine a distance of, say, 1000000000 kilometers -- or at least I can't. (I deliberately didn't include commas in that number, to illustrate the point.) But using a concrete physical distance ...


18

Officially, no -- but there is a weak case to be made that the Moon orbits the Sun rather than the Earth. If you trace the Moon's path in a Sun-centric frame of reference, that path is completely convex. Quoting this Wikipedia article: Unlike most other moons in the Solar System, the trajectory of the Moon is very similar to that of its planet. The ...


18

One Celsius (or Kelvin) degree as a temperature difference was defined as 1/100 of the temperature difference between the freezing point of water and boiling point of water. We call these points 0 °C and 100 °C, respectively. The number 100 arose because we're used to numbers that are powers of ten because we use the base-ten system. The Celsius degree is "...


17

You can always decompose a motion like this into two parts: (1) rolling without slipping and (2) slipping without rolling. What is slipping without rolling? It means the object moves uniformly in one direction along the surface, with no angular velocity about the object's own center of mass. For instance, a box that is pushed along the ground can easily ...


17

Charge is a fundamental conserved property of particles. It is, if you like, a measure of how much a particle interacts with electromagnetic fields. A particle with charge can produce and be affected by electromagnetic fields. This is what we mean when we say a particle has electric charge. It might help to think of it as a simple quantised way to measure ...


16

The constellations are the 88 internationally recognized stellar groupings in the sky that toghether cover the entire celestial sphere. They typcially correspond to a recognziable pattern and many are named from mythology. However, in modern usage, a specified constellation tecnically referres to the entire region of the sky, not just the recognizable star ...


16

The mass, strictly the inertial mass, relates the acceleration of a body to the applied force via Newton's law: $$ F = ma $$ So if you apply a force of 1 Newton to a mass of 1kg it will accelerate at 1m/s$^2$. This is true whether the object is floating in space or in a gravity field e.g. at the Earth's surface. The weight is the force a body exerts when ...


15

It is important to remember that words like this get used in spit-balling sessions and then stick. You have to think of a couple of guy sitting by a black board, coffee in hand saying something like OK, OK! SO that doesn't work. But what if we assume these things come in three flavors and ... It's just a word made up on the spot. That said, I think ...


13

As there wasn't a formal definition of a planet until recently, there still isn't one for a moon. But, a few guidelines: It should be in an orbit which is cleared of other objects, that is, not a bunch of objects in the same or very similar orbit. The typical minimum size for consideration is around a km. There is a smaller class known as moonlets that are ...


13

The ordering ambiguity is the statement – or the "problem" – that for a classical function $f(x,p)$, or a function of analogous phase space variables, there may exist multiple operators $\hat f(\hat x,\hat p)$ that represent it. In particular, the quantum Hamiltonian isn't uniquely determined by the classical limit. This ambiguity appears even if we require ...


13

Pressure is defined as force per unit area applied to an object in a direction perpendicular to the surface. And naturally pressure can cause stress inside an object. Whereas stress is the property of the body under load and is related to the internal forces. It is defined as a reaction produced by the molecules of the body under some action which may ...


13

Not all nonlinear systems are chaotic. However a chaotic system is necessarily nonlinear. There doesn't exists a definition for chaos but using the one given by Strogatz, ref 1: Chaos is aperiodic long-termed behavior in a deterministic system that exhibits sensitive dependence on initial conditions. Like explained in the text: aperiodic long-termed ...


13

it turns out this is exactly 273.15°C less the melting temperature of water. Actually, "Kelvin" and "degrees Celsius" are defined such that there are 273.16 degrees between absolute zero and the triple point temperature of water. Degrees Celsius are defined as $K - 273.15$. The freezing point of water is a measured quantity and is not exactly 273.15K nor ...


13

The entropy of a system is the amount of information needed to specify the exact physical state of a system given its incomplete macroscopic specification. So, if a system can be in $\Omega$ possible states with equal probability then the number of bits needed to specify in exactly which one of these $\Omega$ states the system really is in would be $\log_{2}(...


13

Physically speaking, there is no preferred origin in the spacetime of special relativity, therefore an affine space (equipped with a Lorentzian scalar product) is a better model than a vector space. The Lorentz group acts on the tangent space at each event, this space being isomorphic to the space of four-displacements. The whole invariance group is the ...


12

Yes, to some extent. Once you choose which of the electron or positron is to be considered the normal particle, then that fixes your choice for the other leptons, because of neutrino mixing. Similarly, choosing one quark to be the normal particle fixes the choice for the other flavors and colors of quarks. But I can't think of a reason within the standard ...



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