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Coherent state

In physics, specifically in quantum mechanics, a coherent state is the specific quantum state of the quantum harmonic oscillator, often described as a state that has dynamics most closely resembling the oscillatory behavior of a classical harmonic oscillator. It was the first example of quantum dynamics when Erwin Schrödinger derived it in 1926, while searching for solutions of the Schrödinger equation that satisfy the correspondence principle. The quantum harmonic oscillator (and hence the coherent states) arise in the quantum theory of a wide range of physical systems. For instance, a coherent state describes the oscillating motion of a particle confined in a quadratic potential well (for an early reference, see e.g. Schiff's textbook ). The coherent state describes a state in a system for which the ground-state wavepacket is displaced from the origin of the system. This state can be related to classical solutions by a particle oscillating with an amplitude equivalent to the displacement.

These states, expressed as eigenvectors of the lowering operator and forming an overcomplete family, were introduced in the early papers of John R. Klauder, e.g. In the quantum theory of light (quantum electrodynamics) and other bosonic quantum field theories, coherent states were introduced by the work of Roy J. Glauber in 1963 and are also known as Glauber states.

The concept of coherent states has been considerably abstracted; it has become a major topic in mathematical physics and in applied mathematics, with applications ranging from quantization to signal processing and image processing (see Coherent states in mathematical physics). For this reason, the coherent states associated to the quantum harmonic oscillator are sometimes referred to as canonical coherent states (CCS), standard coherent states, Gaussian states, or oscillator states.

Coherent states in quantum optics

In quantum optics the coherent state refers to a state of the quantized electromagnetic field, etc. that describes a maximal kind of coherence and a classical kind of behavior. Erwin Schrödinger derived it as a "minimum uncertainty" Gaussian wavepacket in 1926, searching for solutions of the Schrödinger equation that satisfy the correspondence principle. It is a minimum uncertainty state, with the single free parameter chosen to make the relative dispersion (standard deviation in natural dimensionless units) equal for position and momentum, each being equally small at high energy.

Further, in contrast to the energy eigenstates of the system, the time evolution of a coherent state is concentrated along the classical trajectories. The quantum linear harmonic oscillator, and hence coherent states, arise in the quantum theory of a wide range of physical systems. They occur in the quantum theory of light (quantum electrodynamics) and other bosonic quantum field theories.

While minimum uncertainty Gaussian wave-packets had been well-known, they did not attract full attention until Roy J. Glauber, in 1963, provided a complete quantum-theoretic description of coherence in the electromagnetic field. In this respect, the concurrent contribution of E.C.G. Sudarshan should not be omitted, (there is, however, a note in Glauber's paper that reads: "Uses of these states as generating functions for the $$n$$-quantum states have, however, been made by J. Schwinger ). Glauber was prompted to do this to provide a description of the Hanbury-Brown & Twiss experiment, which generated very wide baseline (hundreds or thousands of miles) interference patterns that could be used to determine stellar diameters. This opened the door to a much more comprehensive understanding of coherence.   (For more, see Quantum mechanical description.)

In classical optics, light is thought of as electromagnetic waves radiating from a source. Often, coherent laser light is thought of as light that is emitted by many such sources that are in phase. Actually, the picture of one photon being in-phase with another is not valid in quantum theory. Laser radiation is produced in a resonant cavity where the resonant frequency of the cavity is the same as the frequency associated with the atomic electron transitions providing energy flow into the field. As energy in the resonant mode builds up, the probability for stimulated emission, in that mode only, increases. That is a positive feedback loop in which the amplitude in the resonant mode increases exponentially until some nonlinear effects limit it. As a counter-example, a light bulb radiates light into a continuum of modes, and there is nothing that selects any one mode over the other. The emission process is highly random in space and time (see thermal light). In a laser, however, light is emitted into a resonant mode, and that mode is highly coherent. Thus, laser light is idealized as a coherent state. (Classically we describe such a state by an electric field oscillating as a stable wave. See Fig.1)

Besides describing lasers, coherent states also behave in a convenient manner when describing the quantum action of beam splitters: two coherent-state input beams will simply convert to two coherent-state beams at the output with new amplitudes given by classical electromagnetic wave formulas; such a simple behaviour does not occur for other input states, including number states. Likewise if a coherent-state light beam is partially absorbed, then the remainder is a pure coherent state with a smaller amplitude, whereas partial absorption of non-coherent-state light produces a more complicated statistical mixed state. Thermal light can be described as a statistical mixture of coherent states, and the typical way of defining nonclassical light is that it cannot be described as a simple statistical mixture of coherent states.

The energy eigenstates of the linear harmonic oscillator (e.g., masses on springs, lattice vibrations in a solid, vibrational motions of nuclei in molecules, or oscillations in the electromagnetic field) are fixed-number quantum states. The Fock state (e.g. a single photon) is the most particle-like state; it has a fixed number of particles, and phase is indeterminate. A coherent state distributes its quantum-mechanical uncertainty equally between the canonically conjugate coordinates, position and momentum, and the relative uncertainty in phase [defined heuristically] and amplitude are roughly equal—and small at high amplitude.

Quantum mechanical definition
Mathematically, a coherent state $$|\alpha\rangle$$ is defined to be the (unique) eigenstate of the annihilation operator $â$ with corresponding eigenvalue $⟨n⟩$. Formally, this reads,
 * $$\hat{a}|\alpha\rangle=\alpha|\alpha\rangle ~.$$

Since $â$ is not hermitian, $α$ is, in general, a complex number. Writing $$\alpha = |\alpha|e^{i\theta},$$ |$α$| and $α$ are called the amplitude and phase of the state $$|\alpha\rangle$$.

The state $$|\alpha\rangle$$ is called a canonical coherent state in the literature, since there are many other types of coherent states, as can be seen in the companion article Coherent states in mathematical physics.

Physically, this formula means that a coherent state remains unchanged by the annihilation of field excitation or, say, a particle. An eigenstate of the annihilation operator has a Poissonian number distribution when expressed in a basis of energy eigenstates, as shown below. A Poisson distribution is a necessary and sufficient condition that all detections are statistically independent. Contrast this to a single-particle state ($$|1\rangle$$ Fock state): once one particle is detected, there is zero probability of detecting another.

The derivation of this will make use of (unconventionally normalized) dimensionless operators, $θ$ and $X$, normally called field quadratures in quantum optics. (See Nondimensionalization.) These operators are related to the position and momentum operators of a mass $P$ on a spring with constant $m$,
 * $$ {P}=\sqrt{\frac{1}{2\hbar m\omega }}\ \hat{p}\text{,}\quad {X}=\sqrt{\frac{m\omega }{2\hbar }}\ \hat{x}\text{,}\quad \quad \text{where }\omega \equiv \sqrt{k/m}~.$$

For an optical field,
 * $$~E_{\rm R} =

\left(\frac{2\hbar\omega}{\epsilon_0 V} \right)^{1/2} \!\!\!\cos(\theta) X \qquad \text{and} \qquad ~E_{\rm I} = \left(\frac{2\hbar\omega}{\epsilon_0 V}\right)^{1/2} \!\!\!\sin(\theta) X~$$ are the real and imaginary components of the mode of the electric field inside a cavity of volume $$V$$.

With these (dimensionless) operators, the Hamiltonian of either system becomes
 * $${H}=\hbar \omega \left({P}^{2}+{X}^{2} \right)\text{,}

\qquad\text{with}\qquad \left[ {X},{P} \right]\equiv {XP}-{PX}=\frac{i}{2}\,{I}.$$

Erwin Schrödinger was searching for the most classical-like states when he first introduced minimum uncertainty Gaussian wave-packets. The quantum state of the harmonic oscillator that minimizes the uncertainty relation with uncertainty equally distributed between $k$ and $X$ satisfies the equation
 * $$\left( {X}-\langle {X}\rangle \right)\,|\alpha \rangle = -i\left( {P}-\langle{P}\rangle \right)\, |\alpha\rangle \text{,}$$

or, equivalently,
 * $$ \left( {X}+i{P} \right)\, \left|\alpha\right\rangle = \left\langle {X}+i{P} \right\rangle \, \left|\alpha\right\rangle ~,$$

and hence
 * $$ \langle \alpha \! \mid  \left( {X}-\langle X\rangle \right)^2+  \left( {P}-\langle P\rangle \right)^2   \mid \!\alpha\rangle  = 1/2 ~.$$

Thus, given $(∆X−∆P)^{2} ≥ 0$, Schrödinger found that the minimum uncertainty states for the linear harmonic oscillator are the eigenstates of $(X + iP)$. Since â is $(X + iP)$, this is recognizable as a coherent state in the sense of the above definition.

Using the notation for multi-photon states, Glauber characterized the state of complete coherence to all orders in the electromagnetic field to be the eigenstate of the annihilation operator—formally, in a mathematical sense, the same state as found by Schrödinger. The name coherent state took hold after Glauber's work.

If the uncertainty is minimized, but not necessarily equally balanced between $P$ and $X$, the state is called a squeezed coherent state.

The coherent state's location in the complex plane (phase space) is centered at the position and momentum of a classical oscillator of the phase $P$ and amplitude |α| given by the eigenvalue α (or the same complex electric field value for an electromagnetic wave). As shown in Figure 5, the uncertainty, equally spread in all directions, is represented by a disk with diameter $θ$. As the phase varies, the coherent state circles around the origin and the disk neither distorts nor spreads. This is the most similar a quantum state can be to a single point in phase space.

Since the uncertainty (and hence measurement noise) stays constant at $1⁄2$  as the amplitude of the oscillation increases, the state behaves  increasingly like a sinusoidal wave, as shown in Figure 1. Moreover, since the vacuum state $$|0\rangle$$ is just the coherent state with $1⁄2$=0, all coherent states have the same uncertainty as the vacuum. Therefore, one may interpret the quantum noise of a coherent state as being due to vacuum fluctuations.

The notation $$|\alpha\rangle$$ does not refer to a Fock state. For example, when $α = 1$, one should not mistake $$|1\rangle$$ for the single-photon Fock state, which is also denoted $$|1\rangle$$ in its own notation. The expression $$|\alpha\rangle$$ with $α = 1$ represents a Poisson distribution of number states $$|n\rangle$$ with a mean photon number of unity.

The formal solution of the eigenvalue equation is the vacuum state displaced to a location $α$ in phase space, i.e., it is obtained by letting the unitary displacement operator $D(α)$ operate on the vacuum,
 * $$|\alpha\rangle=e^{\alpha \hat a^\dagger - \alpha^*\hat a}|0\rangle = D(\alpha)|0\rangle$$,

where $â = X + iP$ and $â^{†} = X - iP$.

This can be easily seen, as can virtually all results involving coherent states, using the representation of the coherent state in the basis of Fock states,
 * $$|\alpha\rangle =e^{-{|\alpha|^2\over2}}\sum_{n=0}^{\infty}{\alpha^n\over\sqrt{n!}}|n\rangle =e^{-{|\alpha|^2\over2}}e^{\alpha\hat a^\dagger}e^{-{\alpha^* \hat a}}|0\rangle =e^{\alpha \hat a^\dagger - \alpha^*\hat a}|0\rangle = D(\alpha)|0\rangle ~,$$

where $$ |n\rangle $$ are energy (number) eigenvectors of the Hamiltonian
 * $$H =\hbar \omega \left( \hat a^\dagger \hat a + \frac 12\right)~,$$

and the final equality derives from the Baker-Campbell-Hausdorff formula. For the corresponding Poissonian distribution, the probability of detecting $α$ photons is
 * $$P(n)= |\langle n|\alpha \rangle |^2 =e^{-\langle n \rangle}\frac{\langle n \rangle^n}{n!}   ~.$$

Similarly, the average photon number in a coherent state is
 * $$~\langle n \rangle =\langle \hat a^\dagger \hat a \rangle =|\alpha|^2~$$

and the variance is
 * $$~(\Delta n)^2={\rm Var}\left(\hat a^\dagger \hat a\right)=

 * \alpha|^2~$$.

That is, the standard deviation of the number detected goes like the square root of the number detected. So in the limit of large $n$, these detection statistics are equivalent to that of a classical stable wave.

These results apply to detection results at a single detector and thus relate to first order coherence (see degree of coherence). However, for measurements correlating detections at multiple detectors, higher-order coherence is involved (e.g., intensity correlations, second order coherence, at two detectors). Glauber's definition of quantum coherence involves nth-order correlation functions (n-th order coherence) for all $α$. The perfect coherent state has all n-orders of correlation equal to 1 (coherent). It is perfectly coherent to all orders.

The second-order correlation coefficient $$ g^2(0)$$ gives a direct measure of the degree of coherence of photon states in terms of the variance of the photon statistics in the beam under study.

 * $$~g^2(0) =1+\frac{{\rm Var}\left(\hat a^\dagger \hat a\right)-\langle \hat a^\dagger \hat a \rangle}{(\langle \hat a^\dagger \hat a \rangle)^2} = 1+\frac{{\rm Var}(n)-\bar{n}}{\bar{n}^2} $$

In Glauber's development, it is seen that the coherent states are distributed according to a Poisson distribution. In the case of a Poisson distribution, the variance is equal to the mean, i.e.
 * $${\rm Var}(n) =\bar{n}$$
 * $$g^2(0) = 1$$.

A second-order correlation coefficient of 1 means that photons in coherent states are uncorrelated.

Hanbury Brown and Twiss studied the correlation behavior of photons emitted from a thermal, incoherent source described by Bose–Einstein statistics. The variance of the Bose–Einstein distribution is
 * $${\rm Var(n)}=\bar{n}+\bar{n}^2$$
 * $$g^2(0) = 2$$.

This corresponds to the correlation measurements of Hanbury Brown and Twiss, and illustrates that photons in incoherent Bose–Einstein states are correlated or bunched.

Quanta that obey Fermi–Dirac statistics are anti-correlated. In this case the variance is
 * $${\rm Var}(n)=\bar{n}-\bar{n}^2$$
 * $$g^2(0) = 0$$.

Anti-correlation is characterized by a second-order correlation coefficient =0.

Roy J. Glauber's work was prompted by the results of Hanbury-Brown and Twiss that produced long-range (hundreds or thousands of miles) first-order interference patterns through the use of intensity fluctuations (lack of second order coherence), with narrow band filters (partial first order coherence) at each detector. (One can imagine, over very short durations, a near-instantaneous interference pattern from the two detectors, due to the narrow band filters, that dances around randomly due to the shifting relative phase difference. With a coincidence counter, the dancing interference pattern would be stronger at times of increased intensity [common to both beams], and that pattern would be stronger than the background noise.)  Almost all of optics had been concerned with first order coherence. The Hanbury-Brown and Twiss results prompted Glauber to look at higher order coherence, and he came up with a complete quantum-theoretic description of coherence to all orders in the electromagnetic field (and a quantum-theoretic description of signal-plus-noise). He coined the term coherent state and showed that they are produced when a classical electric current interacts with the electromagnetic field.

At $α ≫ 1$,  from Figure 5, simple geometry gives Δθ |α | = 1/2. From this, it appears that there is a tradeoff between number uncertainty and phase uncertainty,  Δθ  Δn  = 1/2,  which is sometimes interpreted as a number-phase uncertainty relation; but  this is not a formal strict uncertainty relation: there is no uniquely defined phase operator in quantum mechanics.

The wavefunction of a coherent state
To find the wavefunction of the coherent state, the minimal uncertainty Schrödinger wave packet, it is easiest to start with the Heisenberg picture of the quantum harmonic oscillator for the coherent state $$|\alpha\rangle$$. Note that
 * $$~a(t)|\alpha\rangle =e^{-i\omega t}a(0)|\alpha\rangle$$

The coherent state is an eigenstate of the annihilation operator in the Heisenberg picture.

It is easy to see that, in the Schrödinger picture, the same eigenvalue
 * $$~ \alpha(t) = e^{-i\omega t}\alpha(0)~$$

occurs,
 * $$~a|\alpha(t)\rangle=\alpha(t)|\alpha(t)\rangle$$.

In the coordinate representations resulting from operating by $$\langle x|$$, this amounts to the  differential equation,
 * $$~\sqrt{\frac{m \omega}{2 \hbar}}\left(x+\frac{\hbar}{m\omega}\frac{\partial }{\partial x}\right)\psi^\alpha(x,t)=\alpha(t)\psi^\alpha(x,t) ~, $$

which is easily solved to yield
 * $$~\psi^{(\alpha)}(x,t)=\left(\frac{m\omega}{\pi\hbar}\right)^{1/4} \exp \Bigg( -\frac{m\omega}{2\hbar}\left(x-\sqrt{\frac{2\hbar}{m\omega}}\Re[\alpha(t)]\right)^2+i\sqrt{\frac{2m\omega}{\hbar}}\Im[\alpha(t)]x+i\theta(t) \Bigg) ~ ,$$

where $θ(t)$ is a yet undetermined phase, to be fixed by demanding that the wavefunction satisfies the Schrödinger equation.

It follows that
 * $$~\theta(t)=-\frac{\omega t}{2}+\frac{|\alpha(0)|^2\sin(2\omega t-2\sigma)}{2}   ~, \text{where} \qquad        \alpha(0)\equiv|\alpha(0)|\exp(i\sigma)  ~,         $$

so that $n$ is the initial phase of the eigenvalue.

The mean position and momentum of this "minimal Schrödinger wave packet" $ψ^{(α)}$ are thus oscillating just like a classical system,

The probability density remains a Gaussian centered on this oscillating mean,
 * $$|\psi^{(\alpha)}(x,t)|^2=\sqrt{\frac{m\omega}{\pi\hbar} } e^{-\frac{m\omega}{\hbar}\left(x- \langle \hat{x}(t) \rangle  \right)^2  }  .$$

Mathematical features of the canonical coherent states
The canonical coherent states described so far have three properties that are mutually equivalent, since each of them completely specifies the state $$|\alpha\rangle$$, namely,
 * 1) They are eigenvectors of the annihilation operator:   $$ \hat{a}|\alpha\rangle=\alpha|\alpha\rangle \,$$.
 * 2) They are obtained from the vacuum by application of a unitary displacement operator:  $$|\alpha\rangle=e^{\alpha \hat a^\dagger - \alpha^*\hat a}|0\rangle  = D(\alpha)|0\rangle\,$$.
 * 3) They are states of (balanced) minimal uncertainty:   $$\Delta X = \Delta P= \sqrt{\frac{\hbar}{2}}\,$$.

Each of these properties may lead to generalizations, in general different from each other (see the article "Coherent states in mathematical physics" for some of these). We emphasize that coherent states have mathematical features that are very different from those of a Fock state; for instance, two different coherent states are not orthogonal,
 * $$\langle\beta|\alpha\rangle=e^{-{1\over2}(|\beta|^2+|\alpha|^2-2\beta^*\alpha)}\neq\delta(\alpha-\beta)$$

(linked to the fact that they are eigenvectors of the non-self-adjoint annihilation operator $â$).

Thus, if the oscillator is in the quantum state $$|\alpha \rangle$$ it is also with nonzero probability in the other quantum state $$|\beta \rangle$$ (but the farther apart the states are situated in phase space, the lower the probability is). However, since they obey a closure relation, any state can be decomposed on the set of coherent states. They hence form an overcomplete basis, in which one can diagonally decompose any state. This is the premise for the Glauber–Sudarshan P representation.

This closure relation can be expressed by the resolution of the identity operator $σ$ in the vector space of quantum states,
 * $$\frac{1}{\pi} \int |\alpha\rangle\langle\alpha| d^2\alpha = I

\qquad d^2\alpha \equiv d\Re(\alpha) \, d\Im(\alpha) ~.$$ This resolution of the identity is intimately connected to the Segal–Bargmann transform.

Another peculiarity is that $$\hat a^\dagger $$ has no eigenket (while $â$ has no eigenbra). The following equality is the closest formal substitute, and turns out to be useful for technical computations,

a^{\dagger}|\alpha\rangle \langle \alpha |=\left({\partial\over\partial\alpha}+\alpha^*\right)|\alpha\rangle \langle \alpha | ~. $$ This last state is known as an "Agarwal state" or photon-added coherent state and denoted as $$|\alpha,1\rangle.$$

Normalized Agarwal states of order $I$ can be expressed as $$|\alpha,n\rangle=[{\hat{a}^{\dagger}]}^n|\alpha\rangle / \| [{\hat{a}^{\dagger}]}^n|\alpha\rangle \|     ~.$$

The above resolution of the identity may be derived (restricting to one spatial dimension for simplicity) by taking matrix elements between eigenstates of position, $$ \langle x | \cdots | y \rangle $$, on both sides of the equation. On the right-hand side, this immediately gives $δ(x-y)$. On the left-hand side, the same is obtained by inserting

\psi^\alpha(x,t) = \langle x | \alpha(t)\rangle $$ from the previous section (time is arbitrary), then integrating over $$ \Im (\alpha) $$ using the Fourier representation of the delta function, and then performing a Gaussian integral over $$ \Re (\alpha) $$.

In particular, the Gaussian Schrödinger wave-packet state follows from the explicit value
 * $$\langle x | \alpha\rangle= \frac {1} {\pi^{1/4}}{e^{-\frac{1}{2}{(x-\sqrt{2}\Re(\alpha))^2} +ix \sqrt{2} \Im (\alpha)-i\Re(\alpha)\Im(\alpha)}}                   ~.$$

The resolution of the identity may also be expressed in terms of particle position and momentum. For each coordinate dimension (using an adapted notation with new meaning for $$x$$),

|\alpha\rangle \equiv |x,p\rangle \qquad \qquad x \equiv \langle \hat{x} \rangle \qquad\qquad p \equiv \langle \hat{p} \rangle $$ the closure relation of coherent states reads

I = \int |x,p\rangle \, \langle x,p| ~ \frac{\mathrm{d}x\,\mathrm{d}p}{2\pi\hbar}  ~. $$ This can be inserted in any quantum-mechanical expectation value, relating it to some quasi-classical phase-space integral and explaining, in particular, the origin of normalisation factors $$ (2\pi\hbar)^{-1} $$ for classical partition functions, consistent with quantum mechanics.

In addition to being an exact eigenstate of annihilation operators, a coherent state is an approximate common eigenstate of particle position and momentum. Restricting to one dimension again,

\hat{x} |x,p\rangle \approx x |x,p\rangle \qquad \qquad \hat{p} |x,p\rangle \approx p |x,p\rangle $$ The error in these approximations is measured by the uncertainties of position and momentum,

\langle x, p | \left(\hat{x} - x \right)^2 |x,p\rangle = \left(\Delta x\right)^2 \qquad \qquad \langle x, p | \left(\hat{p} - p \right)^2 |x,p\rangle = \left(\Delta p\right)^2 ~. $$

Thermal coherent state
A single mode thermal coherent state is produced by displacing a thermal mixed state in phase space, in direct analogy to the displacement of the vacuum state in view of generating a coherent state. The density matrix of a coherent thermal state in operator representation reads

\rho(\alpha, \beta)=\frac{1}{Z}D(\alpha)e^{-\hbar \beta\omega a^{\dagger} a}D^{\dagger}(\alpha), $$

where $$D(\alpha)$$ is the displacement operator, which generates the coherent state $$D(\alpha)|0\rangle=|\alpha\rangle$$ with complex amplitude $$\alpha$$, and $$\beta=1/(k_B T)$$. The partition function is equal to

Z=\text{tr}\left\{ \displaystyle e^{-\hbar \beta\omega a^{\dagger} a}\right\}=\sum_{n=0}^{\infty}e^{-n\beta \hbar\omega}=\frac{1}{1-e^{-\hbar\beta\omega}}. $$

Using the expansion of the identity operator in Fock states, $$I\equiv \sum_{n=0}^{\infty}|n\rangle\langle n|$$, the density operator definition can be expressed in the following form

\rho(\alpha, \beta)= \frac{1}{Z}\sum_{n=0}^{\infty}e^{-n\hbar\beta\omega} D(\alpha)|n\rangle\langle n| D^{\dagger}(\alpha)=\frac{1}{Z}\sum_{n=0}^{\infty}e^{-n\hbar\beta\omega}|\alpha,n\rangle\langle \alpha,n|, $$

where $$|\alpha,n\rangle$$ stands for the displaced Fock state. We remark that if temperature goes to zero we have

\lim_{\beta\to\infty}\rho(\alpha,\beta)=\lim_{\beta\to\infty}\sum_{n=0}^{\infty}e^{-n\hbar\beta\omega} (1-e^{-\hbar\beta\omega})|\alpha,n\rangle\langle \alpha,n|=\sum_{n=0}^{\infty} \delta_{n,0}|\alpha,n\rangle\langle\alpha,n|=|\alpha,0\rangle\langle\alpha,0|, $$

which is the density matrix for a coherent state. The average number of photons in that state can be calculated as below

\langle n\rangle =\text{Tr}\{\rho a^{\dagger}a\}=\frac{1}{Z}\text{Tr}\{D^{\dagger}(\alpha)a^{\dagger}D({\alpha})D^{\dagger}(\alpha) a D(\alpha) e^{-\beta\hbar\omega a^{\dagger}a}\}=\frac{1}{Z}\text{Tr}\{(a^{\dagger} + \alpha^{*})(a + \alpha)e^{-\beta\hbar\omega a^{\dagger}a}\}=$$

=|\alpha|^2\frac{1}{Z}\text{Tr}\{e^{-\beta\hbar\omega a^{\dagger}a} \} + \frac{1}{Z}\text{Tr}\{a^{\dagger}a e^{-\beta\hbar\omega a^{\dagger}a}\}=|\alpha|^2 + \frac{1}{Z} \sum_{n=0}^{\infty}ne^{-n\beta\hbar\omega}, $$

where for the last term we can write

\sum_{n=0}^{\infty}ne^{-n\beta\hbar\omega}=-\frac{\partial}{\partial (\beta\hbar\omega)} \left( \sum_{n=0}^{\infty}e^{-n\beta\hbar\omega}\right)=\frac{e^{-\beta\hbar\omega}}{(1-e^{-\beta\hbar\omega})^2}. $$

As a result, we find

\langle n\rangle=|\alpha|^2 +\langle n\rangle_{\text{th}}, $$

where $$\langle n\rangle_{\text{th}}$$ is the average of the photon number calculated with respect to the thermal state. Here we have defined, for ease of notation,

\langle O\rangle_{\text{th}}=\frac{1}{Z}\text{tr}\{e^{-\beta\hbar\omega a^{\dagger}a}O\}, $$

and we write explicitly

\langle n\rangle_{\text{th}}=\frac{1}{e^{\beta\hbar\omega}-1}. $$

In the limit $$\beta \to \infty$$ we obtain $$\langle n\rangle=|\alpha|^2$$, which is consistent with the expression for the density matrix operator at zero temperature. Likewise, the photon number variance can be evaluated as

\sigma^2=\langle n^2\rangle-\langle n\rangle^2=\sigma_{\text{th}}^2+|\alpha|^2\left(1+2\langle a^{\dagger}a\rangle_{\text{th}}\right), $$ with $$\sigma_{\text{th}}^2=\langle n^2\rangle_{\text{th}}-\langle n\rangle_{\text{th}}^2$$. We deduce that the second moment cannot be uncoupled to the thermal and the quantum distribution moments, unlike the average value (first moment). In that sense, the photon statistics of the displaced thermal state is not described by the sum of the Poisson statistics and the Boltzmann statistics. The distribution of the initial thermal state in phase space broadens as a result of the coherent displacement.

Coherent states of Bose–Einstein condensates

 * A Bose–Einstein condensate (BEC) is a collection of boson atoms that are all in the same quantum state. In a thermodynamic system, the ground state becomes macroscopically occupied below a critical temperature — roughly when the thermal de Broglie wavelength is longer than the interatomic spacing. Superfluidity in liquid Helium-4 is believed to be associated with the Bose–Einstein condensation in an ideal gas. But 4He has strong interactions, and the liquid structure factor (a 2nd-order statistic) plays an important role.  The use of a coherent state to represent the superfluid component of 4He provided a good estimate of the condensate / non-condensate fractions in superfluidity, consistent with results of slow neutron scattering.   Most of the special superfluid properties follow directly from the use of a coherent state to represent the superfluid component — that acts as a macroscopically occupied single-body state with well-defined amplitude and phase over the entire volume.  (The superfluid component of 4He goes from zero at the transition temperature to 100% at absolute zero. But the condensate fraction is about 6% at absolute zero temperature, T=0K.)
 * Early in the study of superfluidity, Penrose and Onsager proposed a metric ("order parameter") for superfluidity. It was represented by a macroscopic factored component (a macroscopic eigenvalue) in the first-order reduced density matrix.  Later, C. N. Yang proposed a more generalized measure of macroscopic quantum coherence, called "Off-Diagonal Long-Range Order" (ODLRO), that included fermion as well as boson systems.  ODLRO exists whenever there is a macroscopically large factored component (eigenvalue) in a reduced density matrix of any order.  Superfluidity corresponds to a large factored component in the first-order reduced density matrix.  (And, all higher order reduced density matrices behave similarly.)  Superconductivity involves a large factored component in the 2nd-order ("Cooper electron-pair") reduced density matrix.
 * The reduced density matrices used to describe macroscopic quantum coherence in superfluids are formally the same as the correlation functions used to describe orders of coherence in radiation. Both are examples of macroscopic quantum coherence.  The macroscopically large coherent component, plus noise, in the electromagnetic field, as given by Glauber's description of signal-plus-noise, is formally the same as the macroscopically large superfluid component plus normal fluid component in the two-fluid model of superfluidity.
 * Every-day electromagnetic radiation, such as radio and TV waves, is also an example of near coherent states (macroscopic quantum coherence). That should "give one pause" regarding the conventional demarcation between quantum and classical.
 * The coherence in superfluidity should not be attributed to any subset of helium atoms; it is a kind of collective phenomena in which all the atoms are involved (similar to Cooper-pairing in superconductivity, as indicated in the next section).

Coherent electron states in superconductivity

 * Electrons are fermions, but when they pair up into Cooper pairs they act as bosons, and so can collectively form a coherent state at low temperatures. This pairing is not actually between electrons, but in the states available to the electrons moving in and out of those states. Cooper pairing refers to the first model for superconductivity.
 * These coherent states are part of the explanation of effects such as the Quantum Hall effect in low-temperature superconducting semiconductors.

Generalizations

 * According to Gilmore and Perelomov, who showed it independently, the construction of coherent states may be seen as a problem in group theory, and thus coherent states may be associated to groups different from the Heisenberg group, which leads to the canonical coherent states discussed above.   Moreover, these coherent states may be generalized to quantum groups. These topics, with references to original work, are discussed in detail in Coherent states in mathematical physics.

 * In quantum field theory and string theory, a generalization of coherent states to the case where infinitely many degrees of freedom are used to define a vacuum state with a different vacuum expectation value from the original vacuum.
 * In one-dimensional many-body quantum systems with fermionic degrees of freedom, low energy excited states can be approximated as coherent states of a bosonic field operator that creates particle-hole excitations. This approach is called bosonization.
 * The Gaussian coherent states of nonrelativistic quantum mechanics can be generalized to relativistic coherent states of Klein-Gordon and Dirac particles.
 * Coherent states have also appeared in works on loop quantum gravity or for the construction of (semi)classical canonical quantum general relativity.

title: Coherent state


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Talk:Autonomous Community of Region of Murcia

Moved because I have no idea what the hell it is supposed to mean:

 * Also, a famous bird of spain is a parakeet. Wild ferrits and rabied squirels are often found roaming around the rural parts of murcia

- Montréalais 17:29 25 Jul 2003 (UTC)

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Flat-headed cat

The flat-headed cat (Prionailurus planiceps) is a small wild cat with short reddish-brown fur. Its head is elongated, and its ears are rounded. Its slender body is 41 to 50 cm long with a tail of 13 to 15 cm, and it weighs 1.5 to 2.5 kg.

The flat-headed cat was first described in 1827 based on a zoological specimen collected in Sumatra. It is also native to the Thai-Malay Peninsula and Borneo, and inhabits wetlands like mixed freshwater swamp forests and lowland freshwater swamps near rivers and coastal areas. It preys foremost on fish, frogs and crustaceans.

The flat-headed cat is threatened by habitat destruction due to conversion for settlements, agriculture and oil palm plantations. It has been listed on the IUCN Red List as Endangered since 2008.

Taxonomy
The scientific name Felis planiceps was proposed by Nicholas Aylward Vigors and Thomas Horsfield in 1827, who first described a skin of a flat-headed cat specimen collected in Sumatra. Prionailurus was proposed by Nikolai Severtzov in 1858 as generic name for spotted wild cats native to Asia. He proposed the generic name Ictailurus for the flat-headed cat.

In 1951, Ellerman and Morrison-Scott grouped the flat-headed cat with the fishing cat (P. viverrinus), assuming it occurs in Lower Siam, Patani, the Malay States, Sumatra and Borneo. It was subordinated to the genus Prionailurus by Ingrid Weigel in 1961 who compared fur patterns of wild and domestic cats. It was grouped into Ictailurus in 1997 following a study on mitochondrial genes of cat species. Today, it is still considered to be a Prionailurus species.

Phylogeny
Phylogenetic analysis of the nuclear DNA in tissue samples from all Felidae species revealed that their evolutionary radiation began in Asia in the Miocene around. Analysis of mitochondrial DNA of Felidae species indicates a radiation at around. Both models agree in the rusty-spotted cat (P. rubiginosus) having been the first cat of the Prionailurus lineage that genetically diverged, followed by the flat-headed cat and then the fishing cat. It is estimated to have diverged together with the leopard cat (P. bengalensis) between and.

The following cladogram shows their phylogenetic relationship as derived through analysis of nuclear DNA:

Characteristics
The flat-headed cat is reddish-brown on top of the head, dark roan brown on the body with a mottled white underbelly. Its face is lighter in colour than the body, and its muzzle and chin are white. Two prominent buff whitish streaks run on either side of the nose between the eyes. It has rounded ears, and its eyes are set close together, giving it improved stereoscopic vision. The teeth together with well-muscled jaws facilitate catching and retaining slippery prey. The legs are fairly short, and the sheaths of its retractile claws are reduced in size so that about two-thirds are protruding. The depressed skull extends along the nose to the extremity of the muzzle, the sides of which are laterally distended. The head itself is lengthened and cylindrical, and the teeth are unusually long. The body is slender, and the extremities are delicate and lengthened. The anterior upper premolars are larger and sharper relative to other cats. The interdigital webs on its paws help the cat gain better traction in muddy environments and water.

It has a head-and-body length of 41 to 50 cm and a short tail of 13 to 15 cm. It weighs 1.5 to 2.5 kg.

Distribution and habitat
The flat-headed cat's distribution is restricted to lowland tropical rainforests in extreme southern Thailand, Peninsular Malaysia, Sumatra and Borneo. It primarily inhabits freshwater habitats near coastal and lowland areas. More than 70% of records were collected less than 3 km away from water.

In peninsular Malaysia, flat-headed cats were recorded in Pasoh Forest Reserve in 2013 less than 1.5 km away from oil palm plantations. This detection suggests that the flat-headed cat is more tolerant of changes in its surrounding environment than previously assumed. As Pasoh Forest Reserve contains no major rivers or lakes and is generally covered by hill dipterocarp forest, this detection provides new evidence of the flat-headed cat's potential habitat range. The reserve ranks as low probability of occurrence in a previously published species distribution model.

In Sumatra, flat-headed cats were recorded by camera traps on the Kampar Peninsula for the first time in 2015; until 2019, a total of 11 records were obtained in this peat swamp forest at an elevation of 1-7 m, and at most 1.2 km away from a river or canal.

In Kalimantan, flat-headed cats were recorded in mixed swamp forest and tall interior forest at elevations below 20 m in the vicinity of Sabangau National Park. In Sarawak, a flat-headed cat was sighted and photographed on the bank of Maludam River in Maludam National Park in 2013. In Ulu Sebuyau National Park, flat-headed cats were recorded less than 80 m away from Sarawak River.

Ecology and behavior
It is difficult to study the cat, which lives in thick vegetation in wet areas. Camera-trap photos are rare. Flat-headed cats recorded in Kalimantan were foremost active by night. They are presumably solitary, and probably maintain their home ranges by scent marking. In captivity, both females and males spray urine by walking forward in a crouching position, leaving a trail on the ground. Anecdotal historical accounts report that they are nocturnal, but an adult captive female was crepuscular and most active between 8:00 and 11:30 and between 18:00 and 22:00 hours. The stomach contents of an adult shot on a Malaysian riverbank consisted only of fish. They have been observed to wash objects, raccoon-style. Live fish are readily taken, with full submergence of the head, and the fish were usually carried at least 2 m away, suggesting a feeding strategy to avoid letting aquatic prey escape back into water. Captive specimens show much greater interest in potential prey in the water than on dry land, suggesting a strong preference for riverine hunting in their natural habitat. Their morphological specializations suggest that their diet is mostly composed of fish, but they are reported to hunt for frogs, and are thought to catch crustaceans. They also catch rats and chickens.

Vocalizations of a flat-headed cat kitten resembled those of a domestic cat. The vocal repertoire of adults has not been analyzed completely, but they purr and give other short-ranged vocalizations.

Their gestation period lasts about 56 days. Of three litters recorded in captivity, one consisted of two kittens; the other two were singletons. Two captive individuals have lived for 14 years.

Threats
The flat-headed cat is foremost threatened by destruction of wetlands and lowland forests, and environmental degradation. This habitat destruction is caused by conversion of natural habitats for settlements, plantations, agriculture and aquaculture, and clearing of mangroves. Overfishing in wetlands and expansion of oil palm plantations are considered to be significant threats. Incidental trapping is also a threat, as some flat-headed cats are reported to have been caught in traps set to protect domestic fowl. Collision with cars and competition with domestic cats could pose serious threats as well.

Conservation
The flat-headed cat is included on CITES Appendix I. It is fully protected by national legislations in all range countries, with hunting and trade prohibited in Thailand, Malaysia and Indonesia.

In captivity
Songkhla Zoo kept three flat-headed cats as of 2017, which were all born in this zoo in 2009 by captive parents.

title: Flat-headed cat


text:
Talk:Dialogue Concerning the Two Chief World Systems

Comments
"Some of this is to show what Galileo considered good science, such as the discussion of William Gilbert's work on magnetism." The point of the section on magnetism is to show that there are bodies in nature that have two natural motions -- in the case of the magnet, weight and magnetism. This is a refutation of the Aristotelian argument that bodies can only have one natural motion (such as downwards, or circular), which would disallow the double motion of the Earth (rotation and revolution). Thus, this section is by no means a digression from astronomy. (I think this was pointed out by Finocchiaro, but I don't have the reference handy.) 67.186.28.212 02:19, 20 July 2006 (UTC)

It is not clear to me why the popularity of Tycho's system in the Church is especially relevant. The Church, after all, was not the only group with a vested interest in non-Copernican ideas.

 * It is relevant because it makes the story of Galileo versus the Church less good and evil than the popular rendition.

It's fine for the fight not to be Martyr Galileo versus the Evil Church Empire; but if the idea is that Galileo was attacking the Church and being dishonest with the issues, I've already argued against that point. Those points. The concept that Galileo was not launching an assault on the Church deserves a better hearing than it has got from either "side" since White's time; the possibility that he meant what he said can explain a lot. Dandrake 21:07 25 Jul 2003 (UTC) Gack! Did I fail to sign my original posting? Mea culpa,

BTW I've heard many times the statement that the Church had moved from Ptolemy to Tycho by that time, but it seems always to have been people citing each other or giving no citation at all. I wouldn't mind seeing a development of this claim based on primary sources, or a citation of a good secondary source as distinct from all the tertiary and n-ary sources. And null-ary source, if I may say so.

In any case, I don't know of any strong Tychonic tendencies among the academic philosophers. Theories that Simplicio was a taunt at the Pope should not distract us from the fact that his role in the Dialogue is that of a conservative philosopher, follower of Simplicius, and not a churchman at all. Colombe and Cremonini, the real-life models for Simplicio, appear to have been much concerned with the Aristotelean perfection of the Heavens, which the Tychonian system tends to disrupt. In fact, Simplicio quotes many times, with approval, a philosopher who was no fan of Tycho: Scipio Chiaramonte, author of the Anti-Tycho.

Galileo's directing no arguments directly at Tycho's system can be thought a sort of intellectual dishonesty, a straw-man approach, if his purpose was to attack the Church. It's another matter if, as the text pretty clearly indicates, he was aiming at fuddy-duddy philosophers who insisted on the inerrancy of Aristotle. (And see the paragraph I'm about to add to the main entry.)

By the way, during the entire difficulty process of negotiating a license for publication of the book, did any Church authority whatever raise any problem about its supposedly treating the Church unfairly by omitting Tycho's system? I've never run into evidence of that, but I could be wrong.

If not, then the whole idea that G omitted T's system in order to make the Church look bad (I think that's the idea) looks like a red herring. It's not as if the problem, if it were one, could have been overlooked because nobody thought of Tycho; the "T" section of an index to the Dialogue will confirm this. Dandrake 21:19 25 Jul 2003 (UTC)

Adding the pointer to on-line text was a well-intentioned idea, but that page appears to be a copyright violation; specifically, it's lifted from Dartmouth, which I think has a license to use the text privately, but not to post it or let it be posted publicly. Zapped it pending clarification. Dandrake 22:45, Jul 24, 2004 (UTC)

Does it state how long it took Galileo to write it? VLCMSTNHXXXC 10:26, Nov 26, 2006

Written in Italian, not Latin
Shouldn't there be a mention that the book was written in Italian, which the general population could read? Had it been written in Latin, the norm for scholarly work of the period, I believe, the general population would not have been able to read it. This was supposed to have been a particular irritation to the Church.  Sagredo Discussione? 18:22, 24 December 2007 (UTC)

 * Who did the supposing? —Preceding unsigned comment added by 81.158.207.21 (talk) 13:25, 21 March 2009 (UTC)

 * I believe I have seen something like this claim made in print, but—as far as I can recall—not in any source I would regard as authoritative. In all the surviving documents from Galileo's trial, there is only one brief comment about his having written the Dialogue in Italian. This was in the report of Melchior Inchofer, one of the theologians tasked with providing an opinion on the orthodoxy of its contents.  Inchofer argued that by using Italian Galileo showed that he had written the Dialogue not to appeal either to foreigners, as he had stated in his preface, nor to other learned men, but to induce the common people to adopt the Copernican opinion.

 * I agree that the article should mention the fact that the Dialogue was written in Italian. Just at the moment, though, I can't see any obviously natural place to slot it in. I'll therefore put off doing it until either someone else beats me to it, or I can come up with a wording that will smoothly fit somewhere into the article.
 * —David Wilson (talk · cont) 15:00, 21 March 2009 (UTC)

Note 6
The site given in Note 6 omits anything that puts Galileo or Einstein in a poor light. The original Preface should be consulted. —Preceding unsigned comment added by 81.158.207.21 (talk) 13:16, 21 March 2009 (UTC)
 * Note 6 has now become Note 14.

Full text?
Sorry, I know Wikipedia is not a forum but still, where can I find the full text of the Dialogue? "Acceptable" languages for me would be German, English or Russian.--141.20.53.121 (talk) 11:22, 24 April 2009 (UTC)

 * The reference desks—either humanities or science in this case—are the proper places to ask questions like this.  I have copied your question to a new section at the humanities reference desk and answered it there.
 * —David Wilson (talk · cont) 15:45, 24 April 2009 (UTC)
 * The full text would be appropriate content for Wikisource, at which point, an infobox template could be added to this article which links to it. That'd be rather nice actually. -Verdatum (talk) 20:08, 24 April 2009 (UTC)

 * A copy of the original Italian text is available on the Italian wikisource. Please feel free to add an infobox with a link to it.
 * —David Wilson (talk · cont) 09:48, 25 April 2009 (UTC)

Sunspots
A recent edit added text which maintained that Galileo's argument in favour of Copernicus's system from the apparent motion of sunspots "is wrong, as the relative motions of sunspots are the same whether in both the Ptolemaic and Copernican system". No source was provided for this assertion, and it is contradicted by numerous modern reliable sources. I have therefore provided three such sources and rewritten the relevant passage to conform to what is contained in those sources.

It does seem to me to be true—if I may be permitted to peddle a modest piece of original reseach of my own—that the explanation of the apparent motion of sunspots in geostatic systems doesn't need to be quite as complicated as the cited sources appear to make out. It could be accounted for by having the Sun's axis of rotation precess uniformly about another axis, parallel to the Earth's, with a period of precisely one sidereal day. However, this is still rather more implausible than the simple explanation available in Copernicus's system, and, in any case, I don't see how it can be included in the article until someone has found a reliable source which contains it. David Wilson (talk · cont) 18:19, 22 June 2010 (UTC)

Galileo was convicted of "grave suspicion of heresy" based on the book
If I remember it well, what was concerned was not the conclusions of Galileo himself (as at the time, and because nobody had thought about the yet to come Foucault pendulum experiment, both systems could be seen as functionally identical, one of them allowing just much easier computations as Copernic had signalled it), but the fact that he explained the reasons to consider geocentrism as superior were "irrelevant". They were, but that statement seemed at the time an insult towards Aristotes and the university teachers (as well as a blasphemy, for the Church). Galileo, overestimating the intelligence of his peers, has already irritated most university professors at the time by showing Aristote's affirmation that heaviest objects always fell faster was incorrect (by dropping weights from the tower of Pisa). 212.198.146.177 (talk) 16:17, 26 August 2011 (UTC)

Origin of the Name "Simplicio"
The origin given in the article for the "Simplicio" character appears to be contrary to the one Galileo gave in the prologue of the Dialogue itself.

The Dialogue's prologue (I'm using the English translation by Stillman Drake, which can be found at various places on the Net) has this to say:

 * I often talked with these two [Sagredo & Salviati] of such matters in the presence of a certain Peripatetic philosopher whose greatest obstacle in apprehending the truth seemed to be the reputation he had acquired by his interpretations of Aristotle.

A little further on it adds (parenthetically):

 * Nor shall the good Peripatetic lack a place; because of his excessive affection toward the Commentaries of Simplicius, I have thought fit to leave him under the name of the author he so much revered, without mentioning his own.

In other words ALL THREE characters derive from real individuals, not just Sagredo & Salviati. In the case of "Simplicio", however, Galileo chose to conceal the name of the real individual behind a nom-de-guerre, doubtless for good reason!

So what does the article have to say? First of all it does not state that the origin of all of three names (let alone Simplicio's) is given by Galileo himself in the Dialogue.

Bearing that failure in mind, it makes the following claims:

 * He is supposedly named after Simplicius of Cilicia, a sixth-century commentator on Aristotle, but it was suspected the name was a double entendre, as the Italian for "simple" (as in "simple minded") is "semplice".

That word "supposedly" conveys doubt as to the truth of the origin of that name. Had Galileo been cited as the origin of the name, that word would have conveyed doubt as to the truth of Galileo's account. However, since Galileo has not been mentioned in that context it instead carries the imputation that the Simplicius origin is a mere hypothesis by some OTHER individual trying to work out where Galileo got these names from--which in turn would make the double entendre alternative a competing hypothesis by yet another individual.

All of which is kind of weird given that the statement in the article which follows would appear to itself be an hypothesis despite being presented as undisputed fact:

 * Simplicio is modeled on two contemporary conservative philosophers, Ludovico delle Colombe (1565-1616?), Galileo's fiercest detractor, and Cesare Cremonini (1550–1631), a Paduan colleague who had refused to look through the telescope. Colombe was the leader of a group of Florentine opponents of Galileo's, which some of the latter's friends referred to as "the pigeon league".

So instead of Galileo's one anonymous individual masquerading as "Simplicio" we now have an amalgam of two!

So who do we believe? Galileo or the hypothesizers?

I have no idea and the article is itself short on details as to why Colombe and Cremonini are suspected to be Simplicio.

For my own part I can't argue one way or the other, nor do I want to. I merely point out that on such matters the article would seem to be deficient and misleading and needs to be modified. --114.73.70.230 (talk) 11:19, 29 September 2011 (UTC)

Omissions
About four passages have been omitted from Einstein's remarks. — Preceding unsigned comment added by 86.145.205.224 (talk) 15:07, 19 July 2013 (UTC)
 * There is no actual deception, as the word "condensed" is used. — Preceding unsigned comment added by 86.145.205.224 (talk) 15:15, 19 July 2013 (UTC)
 * Even the word "condensed" is deceptive, as it gives the impression that the effort was to save time only, when it is actually for ideological reasons. The omissions are made to put Einstein and Galileo in a good light. — Preceding unsigned comment added by 86.145.205.224 (talk) 10:38, 24 July 2013 (UTC)

Letter
Prince Cesi's letter to Galileo was written in 1612. This mentioned Kepler's two laws of 1609 as common knowledge, at that time. — Preceding unsigned comment added by 146.247.25.51 (talk) 10:46, 20 August 2014 (UTC)
 * This is not quite accurate. The letter in which Cesi mentions Kepler's elliptical orbits is this one of July 21, 1612.  Neither Kepler's second law nor the location of the Sun at one of the foci of a planet's elliptical orbit is mentioned anywhere in the letter.   Nor does it appear to me that Cesi's choice of language gives much of a clue as to whether he thought Kepler's proposal for elliptical orbits was  "common knowledge at that time".  Here's my own translation of the relevant passage:
 * " ... and because, just like Your Lordship, I knew of motions concentric with neither the Sun nor the Earth, of some [concentric] with the Earth, and of some—or perhaps all, if the paths of the planets are elliptical, as Kepler would have it—[concentric] with the Sun."
 * Whether the expression enclosed in square brackets should be "concentric" or "not concentric" is not entirely clear, because the expression is not repeated in Cesi's Italian, and has to be inferred from its first occurrence. But, as is the standard practice in Italian, Cesi uses a double negative, in which the first occurrence of the expression is actually "not concentric", and this makes the Italian ambiguous.   My reading agrees with that of Shea and Artigas (also adopted by Edward Rosen), which seems to me to make more sense.  But there are at least two scholars (James Voelkel and Erwin Panofsky) who adopt the alternative interpretation.
 * The view that Cesi's letter shows Kepler's first two laws to have been "common knowledge" doesn't appear to be widely held by respected authorites in the history of science.  While I have found two such authorities who have held that view (James Voelkel, in the reference linked to above,  is one; Max Caspar, in his biography of Kepler, is another), most I have seen discuss Cesi's letter say merely that it mentions elliptical orbits.   According to this paper, Kepler's ideas were "rather slow in establishing themselves, and until about 1630 there are few references to them in the literature of the time."  I therefore don't believe Wikipedia should be stating as fact that Cesi's letter mentions either of Kepler's laws as being "common knowledge" at the time.
 * David Wilson (talk · cont) 16:25, 11 April 2015 (UTC)
 * Galileo might have heard about Kepler's laws by a variety of methods. A group of supporters of Kepler formed in England, including William Crabtree, Jeremiah Horrocks and William Gascoigne. This was in 1636. — Preceding unsigned comment added by 119.237.177.5  (talk • contribs) 09:47, April 15, 2015
 * Caspar, in his biography of Kepler, implies the Galileo might have read Kepler's 1609 book. See pages 192-197. — Preceding unsigned comment added by 119.237.177.5  (talk • contribs) 10:09, April 15, 2015.
 * In general, Galileo and Kepler were on good terms with one another from 1610. — Preceding unsigned comment added by 119.237.177.5  (talk • contribs) 10:11, April 15, 2015
 * Galileo and Kepler could be said to have been on good terms from August 1597 when the former wrote to the latter complimenting him on his Mysterium Cosmographicum.  Kepler replied enthusiastically the following October.
 * As far as I can tell, most respected authorities on Galileo consider it very likely that Kepler's proposal of elliptical orbits was known to him.  However, if he ever made any comments on any of Kepler's laws, no record of them seems to have survived.  Thus, all the evidence of what Galileo knew about Kepler's laws, of which Cesi's letter of 1612 is apparently one of the most important pieces, is indirect, and there is apparently none to indicate that his knowledge went any further than the mere proposal of elliptic orbits.
 * I can't find any implication on pp.192–97 of Caspar's Kepler that "Galileo might have read" Kepler's Astronomia nova of 1609.  All Caspar says (p.195) is that Kepler longed for Galileo's acknowledgement of what he had achieved in Astronomia nova—an acknowledgement which he never got.  Nevertheless,  It is true that Galileo might have read it, especially since he is known to have owned a copy.   By all accounts, however, it was very obscure and difficult to read, even by experts, and no hint of the conclusion that Mars's orbit is elliptical appears before the end of chapter 58.  In this article (p.105) Stillman Drake says "Galileo's familiarity with Kepler's tidal explanation suggests that he had read the Astronomia nova".  But Kepler's tidal theory appears in the introduction.  I have seen at least one other scholar assert that that's probably the only part of the book that Galileo might have read, and here, Hans Blumenthal Blumenberg  asserts that Galileo had not read the book at all before he made  his telescopic discoveries (how Blumenthal Blumenberg thinks he knows this is anybody's guess).
 * David Wilson (talk · cont) 12:13, 17 April 2015 (UTC)
 * Hans Blumenberg's name seems to have been Blumenberg, not Blumenthal. Hans might have thought that his claim put Galileo in a good light, as independent of Kepler.
 * How far Seleucus of Seleucia had come to the notice of Galileo is hard to ascertain.

Ship Argument
I should point out that the "Ship" thought experiment for frame of reference did not originate with Galileo as the text suggests, it appears centuries earlier in Oresme's "Livre du ciel et du monde" and also in Cusa's "Learned Ignorance". — Preceding unsigned comment added by 202.160.98.101 (talk) 00:02, 31 March 2015 (UTC)
 * Nicholas of Cusa made several remarks that are close to modern ideas. Kepler might have mentioned him. — Preceding unsigned comment added by 185.43.110.15 (talk) 09:38, 28 April 2015 (UTC)
 * Kepler mentioned Nicholas of Cusa in 1597. — Preceding unsigned comment added by 185.43.110.15 (talk) 09:44, 28 April 2015 (UTC)
 * Nicholas of Cusa might have come to the notice of Galileo in 1597. — Preceding unsigned comment added by 103.245.205.174 (talk) 13:29, 11 May 2015 (UTC)

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Einstein's opinion
In 1953, in his Preface to Stillman Drake's translation of the "Dialogue", Einstein said, "To be sure, he wanted to avoid an open commitment in these controversial questions that would have delivered him to destruction by the Inquisition." — Preceding unsigned comment added by 87.10.31.162 (talk) 12:17, 14 June 2018 (UTC)
 * Galileo actually said, "A questo fine ho presa nel discorso la parte Copernicana, procedendo in pura ipotesi matematica". "I have taken the Copernican side". — Preceding unsigned comment added by 87.10.31.162 (talk) 12:25, 14 June 2018 (UTC)
 * Galileo's house arrest was spent in five places. He visited the French ambassador in Poggibonsi, in Italy. — Preceding unsigned comment added by 87.10.31.162 (talk) 12:33, 14 June 2018 (UTC)
 * Galileo was in Rome, Siena, Poggibonsi, Florence and Arcetri during his house arrest. — Preceding unsigned comment added by 78.62.50.39 (talk) 12:46, 14 June 2018 (UTC)

Copernicus was actually wrong
Because it predicted a parallex, and none could be observed, even with Galeleo's telescope. Theories need to be tested against the data available at the time.

I think Ptolemy was aware of that and is why he rejected heliocentric. Anyway, a note along those lines by someone more knowledgeable would be interesting. Tuntable (talk) 07:36, 31 January 2019 (UTC)

 * If a theory predicts an effect which is small or difficult to observe, then not observing it is not evidence against the theory. For example, Einstein's General Relativity predicts gravitational waves, yet it took nearly 100 years to actually detect them, because doing so is really difficult. Copernicus argued that the non-observation of stellar parallax meant that the stars are far away. He was, in fact, right. It took nearly 300 years for stellar parallax to finally be observed, but scientists accepted heliocentrism long before that. -Thucydides411 (talk) 07:17, 5 March 2019 (UTC)


text:
Swarthmore

Swarthmore may refer to:
 * Swarthmore Lecture, an annual lecture given during the Britain Yearly Meeting
 * Swarthmore, Pennsylvania, a borough in Pennsylvania
 * Swarthmore College, a liberal arts college in Pennsylvania
 * List of Swarthmore College people, individuals associated with the above college
 * Swarthmore station, a railroad station in Swarthmore, Pennsylvania
 * Swarthmore High School, a secondary school (now closed) in Swarthmore, Pennsylvania

title: Swarthmore


text:
Coptic Catholic Church

The Coptic Catholic Church is an Eastern Catholic particular Church in full communion with the Catholic Church. Along with the Ethiopian Catholic Church and Eritrean Catholic Church, it belongs to the Alexandrian liturgical tradition. Uniquely among the Alexandrian Rite Eastern Catholic liturgies, the Coptic Catholic Church uses the Coptic Rite and the Coptic language (derived from Ancient Egyptian) in its liturgy; the Ethiopian Catholic Church and Eritrean Catholic Church use the Ge'ez Rite.

The current Coptic Catholic Patriarch of Alexandria is Ibrahim Isaac Sidrak, who replaced Antonios Naguib in 2013. The offices of the patriarchate are located in Cairo. The patriarchal Cathedral of Our Lady of Egypt is in Nasr City, a suburb of Cairo.

Beginnings
Since the Council of Chalcedon in the 5th century and the official separation of the Coptic Church from the Western Christian and Eastern Orthodox Churches, the Catholic Church has attempted to achieve reunion with the Copts in Egypt many times. During the Council of Florence in 1442, the Coptic delegates present agreed to a reunion with the papacy in Rome, but the Coptic populace was opposed to the idea, and the union did not take effect. Further failed attempts at reunion were undertaken by Coptic delegates in 1560 and 1582.

In the 17th century, at the behest of Pope Urban VIII, Catholic missionaries (primarily Franciscans) started to come to Egypt. In 1630, a number of missions of the Capuchin Order were founded in the Levant by Joseph of Paris, including in Cairo. Although the mission in Cairo initially faced setbacks, tensions with the local Coptic priesthood were minimized with the arrival of Capuchin priest Agathangelo of Vendome to the city in 1633.

Initial relations between Catholics and Copts in Egypt were poor. One Coptic councilor in 1637 referred to the Roman Church in Egypt as "a brothel". Attempts to excommunicate Catholic offenders in the city were seemingly fruitless. Agathangelo would later be hanged as a martyr in Ethiopia by order of the Ethiopian king in 1638, and the mission in Cairo would start to decline. The Jesuits came in 1675.

In 1741, the Coptic bishop, Anba Athanasius of Jerusalem, became a Catholic. In 1781, he was appointed by Pope Benedict XIV as vicar apostolic of the less than 2,000 Egyptian Coptic Catholics. Eventually, Athanasius returned to the Coptic Orthodox Church and others served as Catholic vicar apostolic.

Patriarchate
Under the assumption that the Ottoman viceroy wanted a Catholic patriarch for the Coptic Catholics in 1824, the Pope established the Patriarchate of Alexandria from the Apostolic Vicariate of Syria, Egypt, Arabia and Cyprus but it was basically titular. The Ottomans in 1829 allowed the Coptic Catholics to build their own churches.

The number of Catholics of this rite increased to the point that Leo XIII in 1895 restored the Catholic patriarchate. He initially named Bishop Cyril Makarios as patriarchal vicar. Makarios then presided over a synod, which led to the introduction of some Latin practices. In 1899, Leo appointed Makarios as patriarch of Alexandria of the Copts, taking the name Cyril II. He resigned in 1908 at the request of the Pope over a controversy. The patriarchate seat remained vacant until an election in 1947 and was administered by an apostolic administrator.

Hierarchy
The Coptic Catholic Church sui juris comprises a single ecclesiastical province, covering Egypt alone. The Patriarch is the sole Metropolitan Archbishop, retaining the ancient title Alexandria but his actual seat is in Egypt's modern capital Cairo.

Coptic Catholic Church has eight suffragan bishops, throughout Egypt, comprising the only Coptic Catholic ecclesiastical province: Abu Qurqas, Alexandria (Patriarch's original home seat), Assiut, Giza, Ismailia, Luxor, Minya and Sohag.

Religious orders
The Coptic Catholic Church does not have Coptic monasteries. Instead the Church has religious congregations such as the three communities for women: the Sisters of the Sacred Heart, the Coptic Sisters of Jesus and Mary (both based in Egypt) and the Egyptian Province of the Little Sisters of Jesus. There is also a community of male Franciscans and Jesuits.

Educational and health services
Most candidates for the priesthood are trained at St. Leo's Patriarchal Seminary, in suburban Cairo. More than 100 Coptic Catholic parishes administer primary schools, and some have secondary schools as well. The church maintains a hospital, a number of medical dispensaries and clinics, and several orphanages.

Ecumenism
Relations between the Coptic Catholic Church and the larger Coptic Orthodox Church are generally very good.

Sources and external links

 * Coptic Catholic Church page at Fellowship and Aid to the Christians of the East
 * Article on "Life in a Coptic Catholic Village"
 * "Italian-language video on the Coptic Catholic Church"
 * "Video of the ordination of Coptic Catholic deacons"
 * Article on the Coptic Catholic Church by Ronald Roberson on the CNEWA web site
 * Common Declaration of Pope Paul VI and Pope Alexandria Shenouda III, 1973
 * GigaCatholic
 * GigaCatholic

title: Coptic Catholic Church


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Talk:List of Chinese writers

Untitled
should we add the Chinese characters here? --Samuel 17:42 25 Jul 2003 (UTC)

 * That's not the common practise. The Chinese characters are -- or at least, should be -- on the individual biography-articles. For nouns without articles, then their Hanzi are placed beside them within parenthesis, for the time being anyway. --Menchi 18:10 25 Jul 2003 (UTC)

The French version of the article describes some of the poets better. 大将军, 都督中外诸军事 (talk) 08:31, 6 Jan 2004 (UTC)

XUE MO
Xue Mo (Chinese: 雪漠； 陈开红 1963- Now) is a Chinese writer.

XUE MO was born in Liangzhou, Gansu province.

He was earlier known as Chen kaihong.  he used the pen names Xue Mo.

His work explores themes such as love, freedom, life and death, and humanity’s relationship with nature.

So far, Xue Mo’s works have been translated into more than 20 foreign languages, including English, German, French, Russian, Korean, Japanese, Italian, Nepali, Spanish, Arabic, etc. RT-WORLD-PEASE (talk) 02:04, 13 March 2024 (UTC)

title: Talk:List of Chinese writers


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User talk:Timaeusj

John Timaeus is a Network and Computer Instructor, currently living in Little Rock, Arkansas.

title: User talk:Timaeusj


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Demonym

A demonym or gentilic  is a word that identifies a group of people (inhabitants, residents, natives) in relation to a particular place. Demonyms are usually derived from the name of the place (hamlet, village, town, city, region, province, state, country, and continent). Demonyms are used to designate all people (the general population) of a particular place, regardless of ethnic, linguistic, religious or other cultural differences that may exist within the population of that place. Examples of demonyms include Cochabambino, for someone from the city of Cochabamba; Tunisian for a person from Tunisia; and Swahili, for a person of the Swahili coast.

As a sub-field of anthroponymy, the study of demonyms is called demonymy or demonymics.

Since they are referring to territorially defined groups of people, demonyms are semantically different from ethnonyms (names of ethnic groups). In the English language, there are many polysemic words that have several meanings (including demonymic and ethnonymic uses), and therefore a particular use of any such word depends on the context. For example, the word Thai may be used as a demonym, designating any inhabitant of Thailand, while the same word may also be used as an ethnonym, designating members of the Thai people. Conversely, some groups of people may be associated with multiple demonyms. For example, a native of the United Kingdom may be called a British person, a Briton or, informally, a Brit.

Some demonyms may have several meanings. For example, the demonym Macedonians may refer to the population of North Macedonia, or more generally to the entire population of the region of Macedonia, a portion of which is in Greece. In some languages, a demonym may be borrowed from another language as a nickname or descriptive adjective for a group of people: for example, Québécois, Québécoise (female) is commonly used in English for a native of the province or city of Quebec (though Quebecer, Quebecker are also available).

In English, demonyms are always capitalized.

Often, demonyms are the same as the adjectival form of the place, e.g. Egyptian, Japanese, or Greek. However, they are not necessarily the same, as exemplified by Spanish instead of Spaniard or British instead of Briton.

English commonly uses national demonyms such as Brazilian or Algerian, while the usage of local demonyms such as Chicagoan, Okie or Parisian is less common. Many local demonyms are rarely used and many places, especially smaller towns and cities, lack a commonly used and accepted demonym altogether.

Etymology
National Geographic attributes the term demonym to Merriam-Webster editor Paul Dickson in a work from 1990. The word did not appear for nouns, adjectives, and verbs derived from geographical names in the Merriam-Webster Collegiate Dictionary nor in prominent style manuals such as the Chicago Manual of Style. It was subsequently popularized in this sense in 1997 by Dickson in his book Labels for Locals. However, in ''What Do You Call a Person From...? A Dictionary of Resident Names (the first edition of Labels for Locals) Dickson attributed the term to George H. Scheetz, in his Names' Names: A Descriptive and Prescriptive Onymicon'' (1988), which is apparently where the term first appears. The term may have been fashioned after demonymic, which the Oxford English Dictionary defines as the name of an Athenian citizen according to the deme to which the citizen belongs, with its first use traced to 1893.

Suffixation
Several linguistic elements are used to create demonyms in the English language. The most common is to add a suffix to the end of the location name, slightly modified in some instances. These may resemble Late Latin, Semitic, Celtic, or Germanic suffixes, such as -(a)n, -ian, -anian, -nian, -in(e), -a(ñ/n)o/a, -e(ñ/n)o/a, -i(ñ/n)o/a, -ite, -(e)r, -(i)sh, -ene, -ensian, -ard, -ese, -nese, -lese, -i(e), -i(ya), -iot, -iote, -k, -asque, -(we)gian, -onian, -vian, -ois(e), or -ais(e).

Prefixation
It is much rarer to find demonyms created with a prefix. Mostly they are from Africa and the Pacific, and are not generally known or used outside the country concerned. In much of East Africa, a person of a particular ethnic group will be denoted by a prefix. For example, a person of the Luba people would be a Muluba, the plural form Baluba, and the language, Kiluba or Tshiluba. Similar patterns with minor variations in the prefixes exist throughout on a tribal level. And Fijians who are indigenous Fijians are known as Kaiviti (Viti being the Fijian name for Fiji). On a country level:
 * Botswana → Motswana (singular), Batswana (plural)
 * Burundi → Umurundi (singular), Abarundi (plural)
 * Eswatini → Liswati (singular), Emaswati (plural)
 * Lesotho → Mosotho (singular), Basotho (plural)

Non-standard examples
Demonyms may also not conform to the underlying naming of a particular place, but instead arise out of historical or cultural particularities that become associated with its denizens. In the United States such demonyms frequently become associated with regional pride such as "Burqueño" and the feminine "Burqueña" of Albuquerque, or with the mascots of intercollegiate sports teams of the state university system, take for example the sooner of Oklahoma and the Oklahoma Sooners.

Ethnonyms
Since names of places, regions and countries (toponyms) are morphologically often related to names of ethnic groups (ethnonyms), various ethnonyms may have similar, but not always identical, forms as terms for general population of those places, regions or countries (demonyms).

Fiction
Literature and science fiction have created a wealth of gentilics that are not directly associated with a cultural group. These will typically be formed using the standard models above. Examples include Martian for hypothetical people of Mars (credited to scientist Percival Lowell), Gondorian for the people of Tolkien's fictional land of Gondor, and Atlantean for Plato's island Atlantis.

Other science fiction examples include Jovian for those of Jupiter or its moons and Venusian for those of Venus. Fictional aliens refer to the inhabitants of Earth as Earthling (from the diminutive -ling, ultimately from Old English -ing meaning "descendant"), as well as Terran, Terrene, Tellurian, Earther, Earthican, Terrestrial, and Solarian (from Sol, the sun).

Fantasy literature which involves other worlds or other lands also has a rich supply of gentilics. Examples include Lilliputians and Brobdingnagians, from the islands of Lilliput and Brobdingnag in the satire Gulliver's Travels.

In a few cases, where a linguistic background has been constructed, non-standard gentilics are formed (or the eponyms back-formed). Examples include Tolkien's Rohirrim (from Rohan), the Star Trek franchise's Klingons (with various names for their homeworld), and the Sangheili from the Halo franchise, (also known as Elites in the game by humans, as well as players) named after their homeworld of Sanghelios.

title: Demonym


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Buñol

Buñol is a town and municipality in the province of Valencia, Spain. The municipality has an area of some 112 km2, and is situated approximately 38 km west of the provincial and autonomous community capital city, Valencia. It lies along the Buñol River and is surrounded by the mountain ranges La Sierra de Las Cabrillas, la Sierra de Dos Aguas and la Sierra de Malacara y Martés.

The local economic base is a mixture of the industrial and the agricultural (carob trees, almond trees, fruit trees, olive trees and grapes).

Buñol's population is about 9,000 people.

Buñol has regional rail passenger service to Valencia via the Renfe system.

Every year upwards of 40,000 people gather in the town to throw over 115,000 kilograms of tomatoes at each other in the yearly Tomatina festival. The festival is a popular sight and tourists from across Europe flock to the town to join in the fun.

Near the town are the training grounds and facilities of football club Levante UD.

title: Buñol