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44

The division is conventionally made at the boundary between where stars end their lives as white dwarf stars and where more massive stars will end their lives in core collapse supernovae. The boundary is set both empirically, by observations of white dwarfs in star clusters, where their initial masses can be estimated, and also using theoretical models. ...


35

There is a consistent definition, but it involves a couple of arbitrary thresholds, so I doubt you'd consider it rigorous. The construction $X \gg Y$ means that the ratio $\frac{Y}{X}$ is small enough that subleading terms in the series expansion for $f\bigl(\frac{Y}{X}\bigr) - f(0)$ can be neglected, where $f$ is some relevant function involved in the ...


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.


23

"A state of rest" is a relative term. Relative means - measured in comparison to the things around it. When you sit in a train and sip from a cup of coffee, you can do so because the cup is still relative to you even though both of you might be hurtling through the countryside at 200 km/h. For most experiments, objects can be considered "at rest" if they ...


22

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 ...


13

A c-number basically means 'classical' number, which is basically any quantity which is not a quantum operator which acts on elements of the Hilbert space of states of a quantum system. It is meant to distinguish from q-numbers, or 'quantum' numbers, which are quantum operators. See http://wikipedia.org/wiki/C-number and the reference therein.


11

Informally speaking, a complete description of a physical system is referred to as its state. Completeness of the state of a system means that it provides all the possible information about the system, i.e. everything that can be possibly known about the system has to be contained in the specification of its state. Every physical theory is ultimately based ...


11

Different people have different definitions of dynamical phase transition. At present, a widely accepted one is by Heyl et al. See their original paper Dynamical Quantum Phase Transitions in the Transverse Field Ising Model. Basically, it means some quantity (e.g., the fidelity) as a function of time is non-analytical at some critical times. See the cusps ...


9

Your question is not specific to inflation, and really applies to any case where a bosonic quantum field behaves semiclassically due to macroscopically large occupation numbers. One very simple example of this is the Stark effect in quantum mechanics, where a Hyrodgen atom is placed in a uniform electric field. The atom is treated as a quantum mechanical ...


8

In a very mathematical sense, more often than not a mode refers to an eigenvector of a linear equation. Consider the coupled springs problem $$\frac{d}{dt^2} \left[ \begin{array}{cc} x_1 \\ x_2 \end{array} \right] =\left[ \begin{array}{cc} - 2 \omega_0^2 & \omega_0^2 \\ \omega_0^2 & - \omega_0^2 \end{array} \right] \left[ \begin{array}{cc} x_1 \\ x_2 ...


8

Of course the name implies that time is involved somehow. People talk about dynamical thermal and quantum phase transitions and in one case you will rapidly change temperature, while in the other state defining parameter (say pressure or field etc.). We will consider thermal PT. Now what does it mean rapidly? Let us consider 2-d order phase transition as ...


8

Roughly, an additive quantum number is the log of a corresponding multiplicative quantum number. Mathematically, this comes from the difference between the representations of a group and a Lie algebra; in the former, the natural operation is multiplication and in the latter it is addition. Many quantum numbers we care about come from continuous symmetry ...


8

A Hilbert space $\cal H$ is complete which means that every Cauchy sequence of vectors admits a limit in the space itself. Under this hypothesis there exist Hilbert bases also known as complete orthonormal systems of vectors in $\cal H$. A set of vectors $\{\psi_i\}_{i\in I}\subset \cal H$ is called an orthonormal system if $\langle \psi_i |\psi_j \rangle =...


7

It isn't necessary to introduce the effective potential in orbital mechanics but it is really useful. Let's say we have a particle moving in a central gravitational potential. Newton's laws give you a vector equation of motion \begin{equation} m \ddot{\vec{x}} = - \nabla U \end{equation} where $U = - G M m /r$. In a general coordinate system this is a ...


7

For any operator $\hat A$ an eigenstate $|\psi\rangle$ is one for which: $$\hat A|\psi\rangle=\lambda |\psi\rangle$$ Where $\lambda$ is a constant, and is called the eigenvalue of that state. If $\hat A$ is an observable, then $\lambda$ will be real. A stationary state is an eigenstate of the Hamiltionain $\hat H$ (the energy operator). It is called ...


7

Kinetic energy of two free particles is additive: the total energy is just the sum of the individual energies: $$ K=K_1+K_2 $$ Another example is charge: the charge of a multiparticle system is the sum of the individual charges. Parity is multiplicative: the parity of a two-particle system is the product of the parities of the inidividual particles: $$ \Pi=...


6

I agree with your definition of locality (probably not surprising :)). Causality I would say is the statement that an event in the future should not affect an event in the past. We can formulate this in classical physics terms. Causality is necessary in order for there to be a well defined initial value problem: I should be able to choose an initial time ...


6

Hertz should be understood to mean "periodic events per second". I your case the events are the display of frames, so yes, you would be perfectly justified in using $\mathrm{Hz}$. That said, as several commenters have already mentioned, the unit "Hertz" does not specify what kind of periodic behavior is being counted. So the author(s) or speaker must make ...


6

Revolving around the sun is equivalent to free fall around the sun, so the revolution allows you not to 'feel' the sun's gravity. The rotation of the earth is something that can be measured: (i) a centrifugal force which is a small offset on gravity, and (ii) causes the coriolis force. Both these are small effects, so can often be ignored for laboratory ...


6

The term c-number is used informally in the way Meer Ashwinkumar describes. As far as I know, it doesn't have a widely promulgated formal definition. However, there is a formal definition for c-number that agrees with the way the term is used in many cases, including the case you're asking about. As you may know, you can think of the operator formalism for ...


6

In relativity (both special and general) one of the key quantities is the proper length given by: $$ ds^2 = g_{\alpha\beta}dx^\alpha dx^\beta \tag{1} $$ where $g_{\alpha\beta}$ is the metric tensor. The physical significance of this is that if we have a small displacement in spacetime $(dx^0, dx^1, dx^2, dx^3)$ then $ds$ is the total distance moved. You ...


6

The effective in effective action has nothing to do with the effective in effective field theory. An effective field theory is a low-energy theory (described by some action $S_{eff}$ and cut-off $\Lambda_{eff}$) of some given higher energy theory (with action $S$ and cut-off $\Lambda\gg\Lambda_{eff}$). The effective action $\Gamma$, which is sometimes ...


6

Like Wikipedia says: "Moment is a combination of a physical quantity and a distance." This 'physical quantity' could be various things. To take the examples you mention: Moment of momentum (commonly known as angular momentum) is expressed as $\vec{L}=\vec{r}\times m\vec{v}$, and is a measure for the rotational momentum of an object around some axis. Moment ...


6

The term rest mass is a poor one because it implies it's the mass measured in the rest frame. But photons have no rest frame, and indeed any particle subject to some form of confinement has a $\Delta p\gt 0$ so its rest frame is somewhat poorly defined. The modern term is invariant mass, which is simply the mass in the equation for the total energy: $$ E^2 ...


5

Both concepts are mathematical in character and they ultimately describe the same characteristics or situations. "Invariance" is a more technical word because it says "what has to be equal to what" for us to say that the symmetry exists. In particular, the "invariance under a symmetry transformation" means that an object, like the action $S$, has the same ...


5

I think the answer is no. It generally precedes some approximation method with a bounded error, but there are so many approximations methods in physics -- some rigorous, some nonrigorous -- that it's way too presumptuous to give it a rigorous definition. Generally, it means one of several things: If $a\ll b$, expanding in powers of $\frac{a}{b}$ is ...


5

I have a feeling this has been answered before, but basically it is because H and He dominate the elemental abundances in the universe. When we look at what else there is we are guided by the elements we can ascertain are present in the photospheres of stars. It just so happens that the most prominent sgnatures are those due to atomic and ionic absorption ...


5

As Qmechanic pointed out in the comments, you're mixing Einstein and abstract index notation a bit. To make things absolutely clear, we will use early Latin indices for abstract indices $(abc)$ and Greek indices for component indices $(\mu\nu\rho)$ and will always indicate Einstein summation explicitly. First and foremost, an abstract index is nothing more ...


5

Just a coincidence. There are too many quantities and not enough letters. It probably does make a difference that the fields in which these two equations exist (material science and electromagnetism) are well enough separated that you typically won't see them both in the same papers or textbooks; if that weren't the case, people would start using different ...


5

Regular functions are well defined (finite). Irregular functions tend to infinity in the limit of approaching some point. In this case, all the Bessel functions tend to zero (except j0 which goes to 1) as you approach the origin. The Neumann functions approach +infinity as you approach the origin from the positive side.



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