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Quantum harmonic oscillator

The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary smooth potential can usually be approximated as a harmonic potential at the vicinity of a stable equilibrium point, it is one of the most important model systems in quantum mechanics. Furthermore, it is one of the few quantum-mechanical systems for which an exact, analytical solution is known.

Hamiltonian and energy eigenstates

The Hamiltonian of the particle is: $$\hat H = \frac{{\hat p}^2}{2m} + \frac{1}{2} k {\hat x}^2 = \frac{{\hat p}^2}{2m} + \frac{1}{2} m \omega^2 {\hat x}^2 \, ,$$ where $m$ is the particle's mass, $k$ is the force constant, $\omega = \sqrt{k / m}$ is the angular frequency of the oscillator, $$\hat{x}$$ is the position operator (given by $x$ in the coordinate basis), and $$\hat{p}$$  is the momentum operator (given by $$\hat p = -i \hbar \, \partial / \partial x$$ in the coordinate basis). The first term in the Hamiltonian represents the kinetic energy of the particle, and the second term represents its potential energy, as in Hooke's law.

The time-independent Schrödinger equation is, $$ \hat H \left| \psi \right\rangle = E \left| \psi \right\rangle ~,$$ where $$E$$ denotes a real number (which needs to be determined) that will specify a time-independent energy level, or eigenvalue, and the solution $$| \psi \rangle$$ denotes that level's energy eigenstate.

Then solve the differential equation representing this eigenvalue problem in the coordinate basis, for the wave function $$\langle x | \psi \rangle = \psi (x) $$, using a spectral method. It turns out that there is a family of solutions. In this basis, they amount to Hermite functions, $$ \psi_n(x) = \frac{1}{\sqrt{2^n\,n!}} \left(\frac{m\omega}{\pi \hbar}\right)^{1/4}  e^{ - \frac{m\omega x^2}{2 \hbar}} H_n\left(\sqrt{\frac{m\omega}{\hbar}} x \right), \qquad n = 0,1,2,\ldots. $$

The functions Hn are the physicists' Hermite polynomials, $$H_n(z)=(-1)^n~ e^{z^2}\frac{d^n}{dz^n}\left(e^{-z^2}\right).$$

The corresponding energy levels are $$ E_n = \hbar \omega\bigl(n + \tfrac{1}{2}\bigr).$$The expectation values of position and momentum combined with variance of each variable can be derived from the wavefunction to understand the behavior of the energy eigenkets. They are shown to be $\langle \hat{x} \rangle = 0 $ and $\langle \hat{p} \rangle = 0  $  owing to the symmetry of the problem, whereas:

$$\langle \hat{x}^2 \rangle = (2n+1)\frac{\hbar}{2m\omega} = \sigma_x^2 $$

$$\langle \hat{p}^2 \rangle = (2n+1)\frac{m\hbar\omega}{2} = \sigma_p^2  $$

The variance in both position and momentum are observed to increase for higher energy levels. The lowest energy level has value of $\sigma_x \sigma_p = \frac{\hbar}{2}  $  which is its minimum value due to uncertainty relation and also corresponds to a gaussian wavefunction.

This energy spectrum is noteworthy for three reasons. First, the energies are quantized, meaning that only discrete energy values (integer-plus-half multiples of $ħω$) are possible; this is a general feature of quantum-mechanical systems when a particle is confined. Second, these discrete energy levels are equally spaced, unlike in the Bohr model of the atom, or the particle in a box. Third, the lowest achievable energy (the energy of the $n = 0$ state, called the ground state) is not equal to the minimum of the potential well, but $ħω/2$ above it; this is called zero-point energy. Because of the zero-point energy, the position and momentum of the oscillator in the ground state are not fixed (as they would be in a classical oscillator), but have a small range of variance, in accordance with the Heisenberg uncertainty principle.

The ground state probability density is concentrated at the origin, which means the particle spends most of its time at the bottom of the potential well, as one would expect for a state with little energy. As the energy increases, the probability density peaks at the classical "turning points", where the state's energy coincides with the potential energy. (See the discussion below of the highly excited states.) This is consistent with the classical harmonic oscillator, in which the particle spends more of its time (and is therefore more likely to be found) near the turning points, where it is moving the slowest. The correspondence principle is thus satisfied. Moreover, special nondispersive wave packets, with minimum uncertainty, called coherent states oscillate very much like classical objects, as illustrated in the figure; they are not eigenstates of the Hamiltonian.

Ladder operator method

The "ladder operator" method, developed by Paul Dirac, allows extraction of the energy eigenvalues without directly solving the differential equation. It is generalizable to more complicated problems, notably in quantum field theory. Following this approach, we define the operators $x$ and its adjoint $a^{†}$, $$\begin{align} a &=\sqrt{m\omega \over 2\hbar} \left(\hat x + {i \over m \omega} \hat p \right) \\ a^\dagger &=\sqrt{m\omega \over 2\hbar} \left(\hat x - {i \over m \omega} \hat p \right) \end{align}$$Note these operators classically are exactly the generators of normalized rotation in the phase space of $$x$$ and $$m\frac{dx}{dt}$$, i.e they describe the forwards and backwards evolution in time of a classical harmonic oscillator.

These operators lead to the following representation of $$\hat{x}$$ and $$\hat{p}$$, $$\begin{align} \hat x &= \sqrt{\frac{\hbar}{2 m\omega}}(a^\dagger + a) \\ \hat p &= i\sqrt{\frac{\hbar m \omega}{2}}(a^\dagger - a) ~. \end{align}$$

The operator $a$ is not Hermitian, since itself and its adjoint $a^{†}$ are not equal. The energy eigenstates $|n\rangle$, when operated on by these ladder operators, give $$\begin{align} a^\dagger|n\rangle &= \sqrt{n + 1} | n + 1\rangle \\ a|n\rangle &= \sqrt{n} | n - 1\rangle. \end{align}$$

From the relations above, we can also define a number operator $a$, which has the following property: $$\begin{align} N &= a^\dagger a \\ N\left| n \right\rangle &= n\left| n \right\rangle. \end{align}$$

The following commutators can be easily obtained by substituting the canonical commutation relation, $$[a, a^\dagger] = 1,\qquad[N, a^\dagger] = a^{\dagger},\qquad[N, a] = -a, $$

and the Hamilton operator can be expressed as $$\hat H = \hbar\omega\left(N + \frac{1}{2}\right),$$

so the eigenstate of $N$ is also the eigenstate of energy.

The commutation property yields $$\begin{align} Na^{\dagger}|n\rangle &= \left(a^\dagger N + [N, a^\dagger]\right)|n\rangle \\ &= \left(a^\dagger N + a^\dagger\right)|n\rangle \\ &= (n + 1)a^\dagger|n\rangle, \end{align} $$

and similarly, $$Na|n\rangle = (n - 1)a | n \rangle.$$

This means that $N$ acts on $|n⟩$  to produce, up to a multiplicative constant,   $|n–1⟩$, and $a^{†}$ acts on   $|n⟩$ to produce $|n+1⟩$. For this reason, $a$ is called an annihilation operator ("lowering operator"), and $a^{†}$ a creation operator ("raising operator"). The two operators together are called ladder operators.

Given any energy eigenstate, we can act on it with the lowering operator, $a$, to produce another eigenstate with $ħω$ less energy. By repeated application of the lowering operator, it seems that we can produce energy eigenstates down to $E = −∞$. However, since $$n = \langle n | N | n \rangle = \langle n | a^\dagger a | n \rangle = \Bigl(a | n \rangle \Bigr)^\dagger a | n \rangle \geqslant 0,$$

the smallest eigenvalue of the number operator is 0, and $$a \left| 0 \right\rangle = 0. $$

In this case, subsequent applications of the lowering operator will just produce zero, instead of additional energy eigenstates. Furthermore, we have shown above that $$\hat H \left|0\right\rangle = \frac{\hbar\omega}{2} \left|0\right\rangle$$

Finally, by acting on |0⟩ with the raising operator and multiplying by suitable normalization factors, we can produce an infinite set of energy eigenstates $$\left\{\left| 0 \right\rangle, \left| 1 \right\rangle, \left| 2 \right\rangle, \ldots, \left| n \right\rangle, \ldots\right\},$$

such that $$\hat H \left| n \right\rangle = \hbar\omega \left( n + \frac{1}{2} \right) \left| n \right\rangle, $$ which matches the energy spectrum given in the preceding section.

Arbitrary eigenstates can be expressed in terms of |0⟩, $$|n\rangle = \frac{(a^\dagger)^n}{\sqrt{n!}} |0\rangle. $$ $a$

Analytical questions
The preceding analysis is algebraic, using only the commutation relations between the raising and lowering operators. Once the algebraic analysis is complete, one should turn to analytical questions. First, one should find the ground state, that is, the solution of the equation $$a\psi_0 = 0$$. In the position representation, this is the first-order differential equation $$\left(x+\frac{\hbar}{m\omega}\frac{d}{dx}\right)\psi_0 = 0,$$ whose solution is easily found to be the Gaussian $$\psi_0(x)=Ce^{-\frac{m\omega x^2}{2\hbar}}.$$ Conceptually, it is important that there is only one solution of this equation; if there were, say, two linearly independent ground states, we would get two independent chains of eigenvectors for the harmonic oscillator. Once the ground state is computed, one can show inductively that the excited states are Hermite polynomials times the Gaussian ground state, using the explicit form of the raising operator in the position representation. One can also prove that, as expected from the uniqueness of the ground state, the Hermite functions energy eigenstates $$\psi_n$$ constructed by the ladder method form a complete orthonormal set of functions.

Explicitly connecting with the previous section, the ground state |0⟩  in the position representation is determined by $$ a| 0\rangle =0$$, $$ \left\langle x \mid a \mid 0 \right\rangle = 0 \qquad \Rightarrow \left(x + \frac{\hbar}{m\omega}\frac{d}{dx}\right)\left\langle x\mid 0\right\rangle = 0 \qquad \Rightarrow          $$ $$ \left\langle x\mid 0\right\rangle = \left(\frac{m\omega}{\pi\hbar}\right)^\frac{1}{4} \exp\left( -\frac{m\omega}{2\hbar}x^2 \right) = \psi_0 ~,$$ hence $$ \langle x \mid a^\dagger \mid 0 \rangle = \psi_1 (x) ~,$$ so that $$\psi_1(x,t)=\langle x \mid e^{-3i\omega t/2} a^\dagger \mid 0 \rangle $$, and so on.

Natural length and energy scales
The quantum harmonic oscillator possesses natural scales for length and energy, which can be used to simplify the problem. These can be found by nondimensionalization.

The result is that, if  energy is measured in units of  $ħω$ and distance in units of $√ħ/(mω)$, then the Hamiltonian simplifies to $$ H = -\frac{1}{2} {d^2 \over dx^2} +\frac{1}{2}  x^2 ,$$ while the energy eigenfunctions and eigenvalues simplify to Hermite functions and integers offset by a half, $$\psi_n(x)= \left\langle x \mid n \right\rangle = {1 \over \sqrt{2^n n!}}~ \pi^{-1/4} \exp(-x^2 / 2)~ H_n(x),$$ $$E_n = n + \tfrac{1}{2} ~,$$ where $H_{n}(x)$ are the Hermite polynomials.

To avoid confusion, these "natural units"   will mostly not be adopted in this article. However, they frequently come in handy when performing calculations, by bypassing clutter.

For example, the fundamental solution (propagator) of $H − i∂_{t}$, the time-dependent Schrödinger operator for this oscillator, simply boils down to the Mehler kernel, $$\langle x \mid \exp (-itH) \mid y \rangle \equiv K(x,y;t)= \frac{1}{\sqrt{2\pi i \sin t}} \exp \left(\frac{i}{2\sin t}\left ((x^2+y^2)\cos t - 2xy\right )\right )~,$$ where $K(x,y;0) = δ(x − y)$. The most general solution for a given initial configuration $ψ(x,0)$ then is simply $$\psi(x,t)=\int dy~ K(x,y;t) \psi(y,0) \,.$$

Coherent states

The coherent states (also known as Glauber states) of the harmonic oscillator are special nondispersive wave packets, with minimum uncertainty $σ_{x} σ_{p} = ℏ/2$, whose observables' expectation values evolve like a classical system. They are eigenvectors of the annihilation operator, not the Hamiltonian, and form an overcomplete basis which consequentially lacks orthogonality.

The coherent states are indexed by $$\alpha \in \mathbb{C}$$ and expressed in the $⟨$ basis as

$$|\alpha\rangle = \sum_{n=0}^\infty |n\rangle \langle n | \alpha \rangle = e^{-\frac{1}{2} |\alpha|^2} \sum_{n=0}^\infty\frac{\alpha^n}{\sqrt{n!}} |n\rangle = e^{-\frac{1}{2} |\alpha|^2} e^{\alpha a^\dagger} e^{-{\alpha^* a}} |0\rangle.$$

Since coherent states are not energy eigenstates, their time evolution is not a simple shift in wavefunction phase. The time-evolved states are, however, also coherent states but with phase-shifting parameter $$ instead: $$\alpha(t) = \alpha(0) e^{-i\omega t} = \alpha_0 e^{-i\omega t}$$.$$|\alpha(t)\rangle = \sum_{n=0}^\infty e^{-i\left(n+\frac{1}{2}\right) \omega t}|n\rangle \langle n | \alpha \rangle = e^{\frac{-i\omega t}{2}}e^{-\frac{1}{2} |\alpha|^2} \sum_{n=0}^\infty\frac{(\alpha e^{-i\omega t})^n}{\sqrt{n!}} |n\rangle = e^{-\frac{i\omega t}{2}}|\alpha e^{-i\omega t}\rangle $$

Because $$a \left| 0 \right\rangle = 0 $$ and via the Kermack-McCrae identity, the last form is equivalent to a unitary displacement operator acting on the ground state: $$|\alpha\rangle=e^{\alpha \hat a^\dagger - \alpha^*\hat a}|0\rangle = D(\alpha)|0\rangle$$. Calculating the expectation values:

$$\langle \hat{x} \rangle_{\alpha(t)} = \sqrt{\frac{2\hbar}{m\omega}}|\alpha_0|\cos{(\omega t - \phi)} $$

$$\langle \hat{p} \rangle_{\alpha(t)} = -\sqrt{2m\hbar \omega}|\alpha_0|\sin{(\omega t - \phi)} $$

where $$\phi $$ is the phase contributed by complex $α$. These equations confirm the oscillating behavior of the particle.

The uncertainties calculated using the numeric method are:

$$\sigma_x(t)=\sqrt{\frac{\hbar}{2m\omega}} $$

$$\sigma_p(t) = \sqrt{\frac{m\hbar\omega}{2}} $$

which gives $\sigma_x(t)\sigma_p(t) = \frac{\hbar}{2} $. Since the only wavefunction that can have lowest position-momentum uncertainty, $\frac{\hbar}{2} $, is a gaussian wavefunction, and since the coherent state wavefunction has minimum position-momentum uncertainty, we note that the general gaussian wavefunction in quantum mechanics has the form:$$\psi_\alpha(x')= \left(\frac{m\omega}{\pi\hbar}\right)^{\frac{1}{4}} e^{\frac{i}{\hbar} \langle\hat{p}\rangle_\alpha (x' - \frac{\langle\hat{x}\rangle_\alpha}{2}) - \frac{m\omega}{2\hbar}(x' - \langle\hat{x}\rangle_\alpha)^2} .$$Substituting the expectation values as a function of time, gives the required time varying wavefunction.

The probability of each energy eigenstates can be calculated to find the energy distribution of the wavefunction:

$$P(E_n)=|\langle n | \alpha \rangle|^2 = \frac{e^{-|\alpha|^2}|\alpha|^{2n}}{n!}$$

which corresponds to Poisson distribution.

Highly excited states
When $α$ is large, the eigenstates are localized into the classical allowed region, that is, the region in which a classical particle with energy $n = 30$ can move. The eigenstates are peaked near the turning points: the points at the ends of the classically allowed region where the classical particle changes direction. This phenomenon can be verified through asymptotics of the Hermite polynomials, and also through the WKB approximation.

The frequency of oscillation at $n$ is proportional to the momentum $E_{n}$ of a classical particle of energy $p(x)$ and position $x$. Furthermore, the square of the amplitude (determining the probability density) is inversely proportional to $E_{n}$,  reflecting the length of time the classical particle spends near $x$. The system behavior in a small neighborhood of the turning point does not have a simple classical explanation, but can be modeled using an Airy function. Using properties of the Airy function, one may estimate the probability of finding the particle outside the classically allowed region,  to be approximately $$\frac{2}{n^{1/3}3^{2/3}\Gamma^2(\tfrac{1}{3})}=\frac{1}{n^{1/3}\cdot 7.46408092658...}$$ This is also given, asymptotically, by the integral $$\frac{1}{2\pi}\int_{0}^{\infty}e^{(2n+1)\left (x-\tfrac{1}{2}\sinh(2x) \right )}dx ~.$$

Phase space solutions
In the phase space formulation of quantum mechanics, eigenstates of the quantum harmonic oscillator in several different representations of the quasiprobability distribution can be written in closed form. The most widely used of these is for the Wigner quasiprobability distribution.

The Wigner quasiprobability distribution for the energy eigenstate $p(x)$ is, in the natural units described above, $$F_n(x, p) = \frac{(-1)^n}{\pi \hbar} L_n\left(2(x^2 + p^2)\right) e^{-(x^2 + p^2)} \,,$$ where Ln are the Laguerre polynomials. This example illustrates how the Hermite and Laguerre polynomials are linked through the Wigner map.

Meanwhile, the Husimi Q function of the harmonic oscillator eigenstates have an even simpler form. If we work in the natural units described above, we have $$Q_n(x,p)=\frac{(x^2+p^2)^n}{n!}\frac{e^{-(x^2+p^2)}}{\pi}$$ This claim can be verified using the Segal–Bargmann transform. Specifically, since the raising operator in the Segal–Bargmann representation is simply multiplication by $$z=x+ip$$ and the ground state is the constant function 1, the normalized harmonic oscillator states in this representation are simply $$z^n/\sqrt{n!}$$. At this point, we can appeal to the formula for the Husimi Q function in terms of the Segal–Bargmann transform.

N-dimensional isotropic harmonic oscillator
The one-dimensional harmonic oscillator is readily generalizable to $|n⟩$ dimensions, where $N$. In one dimension, the position of the particle was specified by a single coordinate, $N = 1, 2, 3, …$. In $x$ dimensions, this is replaced by $N$ position coordinates, which we label $N$. Corresponding to each position coordinate is a momentum; we label these $x_{1}, …, x_{N}$. The canonical commutation relations between these operators are $$\begin{align} {[}x_i, p_j{]} &= i\hbar\delta_{i,j} \\ {[}x_i, x_j{]} &= 0                 \\ {[}p_i, p_j{]} &= 0 \end{align}$$

The Hamiltonian for this system is $$ H = \sum_{i=1}^N \left( {p_i^2 \over 2m} + {1\over 2} m \omega^2 x_i^2 \right).$$

As the form of this Hamiltonian makes clear, the $p_{1}, …, p_{N}$-dimensional harmonic oscillator is exactly analogous to $N$ independent one-dimensional harmonic oscillators with the same mass and spring constant. In this case, the quantities $N$ would refer to the positions of each of the $x_{1}, ..., x_{N}$ particles. This is a convenient property of the $N$ potential, which allows the potential energy to be separated into terms depending on one coordinate each.

This observation makes the solution straightforward. For a particular set of quantum numbers $$\{n\}\equiv \{n_1, n_2, \dots, n_N\}$$ the energy eigenfunctions for the $r^{2}$-dimensional oscillator are expressed in terms of the 1-dimensional eigenfunctions as:

$$\langle \mathbf{x}|\psi_{\{n\}}\rangle = \prod_{i=1}^N\langle x_i\mid \psi_{n_i}\rangle$$

In the ladder operator method, we define $N$ sets of ladder operators,

$$\begin{align} a_i &= \sqrt{m\omega \over 2\hbar} \left(x_i + {i \over m \omega} p_i \right), \\ a^{\dagger}_i &= \sqrt{m \omega \over 2\hbar} \left( x_i - {i \over m \omega} p_i \right). \end{align}$$

By an analogous procedure to the one-dimensional case, we can then show that each of the $N$ and $a_{i}$ operators lower and raise the energy by $a^{†}_{i}$ respectively. The Hamiltonian is $$H = \hbar \omega \, \sum_{i=1}^N \left(a_i^\dagger \,a_i + \frac{1}{2}\right).$$ This Hamiltonian is invariant under the dynamic symmetry group $ℏω$ (the unitary group in $U(N)$ dimensions), defined by $$ U\, a_i^\dagger \,U^\dagger = \sum_{j=1}^N a_j^\dagger\,U_{ji}\quad\text{for all}\quad U \in U(N),$$ where $$U_{ji}$$ is an element in the defining matrix representation of $N$.

The energy levels of the system are $$ E = \hbar \omega \left[(n_1 + \cdots + n_N) + {N\over 2}\right].$$ $$n_i = 0, 1, 2, \dots \quad (\text{the energy level in dimension } i).$$

As in the one-dimensional case, the energy is quantized. The ground state energy is $U(N)$ times the one-dimensional ground energy, as we would expect using the analogy to $N$ independent one-dimensional oscillators. There is one further difference: in the one-dimensional case, each energy level corresponds to a unique quantum state. In $N$-dimensions, except for the ground state, the energy levels are degenerate, meaning there are several states with the same energy.

The degeneracy can be calculated relatively easily. As an example, consider the 3-dimensional case: Define $N$. All states with the same $n = n_{1} + n_{2} + n_{3}$ will have the same energy. For a given $n$, we choose a particular $n$. Then $n_{1}$. There are $n_{2} + n_{3} = n − n_{1}$ possible pairs $n − n_{1} + 1$. $\{n_{2}, n_{3}\}$ can take on the values $n_{2}$ to $0$, and for each $n − n_{1}$ the value of $n_{2}$ is fixed. The degree of degeneracy therefore is: $$g_n = \sum_{n_1=0}^n n - n_1 + 1 = \frac{(n+1)(n+2)}{2}$$ Formula for general $n_{3}$ and $N$ [$n$ being the dimension of the symmetric irreducible $g_{n}$-th power representation of the unitary group $n$]: $$g_n = \binom{N+n-1}{n}$$ The special case $U(N)$ = 3, given above, follows directly from this general equation. This is however, only true for distinguishable particles, or one particle in $N$ dimensions (as dimensions are distinguishable). For the case of $N$ bosons in a one-dimension harmonic trap, the degeneracy scales as the number of ways to partition an integer $N$ using integers less than or equal to $n$.

$$g_n = p(N_{-},n)$$

This arises due to the constraint of putting $N$ quanta into a state ket where $\sum_{k=0}^\infty k n_k = n $    and $ \sum_{k=0}^\infty  n_k = N $, which are the same constraints as in integer partition.

Example: 3D isotropic harmonic oscillator
The Schrödinger equation for a particle in a spherically-symmetric three-dimensional harmonic oscillator can be solved explicitly by separation of variables; see this article for the present case. This procedure is analogous to the separation performed in the hydrogen-like atom problem, but with a different spherically symmetric potential $$V(r) = {1\over 2} \mu \omega^2 r^2,$$ where $x$ is the mass of the particle. Because $μ$ will be used below for the magnetic quantum number, mass is indicated by $m$, instead of $μ$, as earlier in this article.

The solution to the equation is: $$\psi_{klm}(r,\theta,\phi) = N_{kl} r^{l}e^{-\nu r^2}L_k^{\left(l+{1\over 2}\right)}(2\nu r^2) Y_{lm}(\theta,\phi)$$ where
 * $$N_{kl}=\sqrt{\sqrt{\frac{2\nu^3}{\pi }}\frac{2^{k+2l+3}\;k!\;\nu^l}{(2k+2l+1)!!}}$$ is a normalization constant; $$\nu \equiv {\mu \omega \over 2 \hbar}~$$;
 * $${L_k}^{(l+{1\over 2})}(2\nu r^2)$$

are generalized Laguerre polynomials; The order $m$ of the polynomial is a non-negative integer;
 * $$Y_{lm}(\theta,\phi)\,$$ is a spherical harmonic function;
 * $k$ is the reduced Planck constant: $$\hbar\equiv\frac{h}{2\pi}~.$$

The energy eigenvalue is $$E=\hbar \omega \left(2k + l + \frac{3}{2}\right) .$$ The energy is usually described by the single quantum number $$n\equiv 2k+l \,.$$

Because $ħ$ is a non-negative integer, for every even $k$ we have $N$ and for every odd $n$ we have $ℓ = 0, 2, …, n − 2, n$. The magnetic quantum number $n$ is an integer satisfying $ℓ = 1, 3, …, n − 2, n$, so for every $m$ and ℓ there are 2ℓ + 1 different quantum states, labeled by $n$. Thus, the degeneracy at level $m$ is $$\sum_{l=\ldots,n-2,n} (2l+1) = {(n+1)(n+2)\over 2} \,,$$ where the sum starts from 0 or 1, according to whether $n$ is even or odd. This result is in accordance with the dimension formula above, and amounts to the dimensionality of a symmetric representation of $−ℓ ≤ m ≤ ℓ$, the relevant degeneracy group.

Harmonic oscillators lattice: phonons
The notation of a harmonic oscillator can be extended to a one-dimensional lattice of many particles. Consider a one-dimensional quantum mechanical harmonic chain of N identical atoms. This is the simplest quantum mechanical model of a lattice, and we will see how phonons arise from it. The formalism that we will develop for this model is readily generalizable to two and three dimensions.

As in the previous section, we denote the positions of the masses by $SU(3)$, as measured from their equilibrium positions (i.e.  $x_{1}, x_{2}, …$ if the particle $n$ is at its equilibrium position). In two or more dimensions, the $x_{i} = 0$ are vector quantities. The Hamiltonian for this system is

$$\mathbf{H} = \sum_{i=1}^N {p_i^2 \over 2m} + {1\over 2} m \omega^2 \sum_{\{ij\} (nn)} (x_i - x_j)^2 \,,$$ where $i$ is the (assumed uniform) mass of each atom, and $x_{i}$ and  $x_{i}$ are the position and momentum operators for the i th atom and the sum is made over the nearest neighbors (nn). However, it is customary to rewrite the Hamiltonian in terms of the normal modes of the wavevector rather than in terms of the particle coordinates so that one can work in the more convenient Fourier space. We introduce, then, a set of $m$ "normal coordinates" $p_{i}$, defined as the discrete Fourier transforms of the $N$s, and $x$ "conjugate momenta" $N$ defined as the Fourier transforms of the $Π$s, $$Q_k = {1\over\sqrt{N}} \sum_{l} e^{ikal} x_l$$ $$\Pi_{k} = {1\over\sqrt{N}} \sum_{l} e^{-ikal} p_l \,.$$

The quantity $Q_{k}$ will turn out to be the wave number of the phonon, i.e. 2π divided by the wavelength. It takes on quantized values, because the number of atoms is finite.

This preserves the desired commutation relations in either real space or wave vector space $$ \begin{align} \left[x_l, p_m \right]&=i\hbar\delta_{l,m} \\ \left[ Q_k, \Pi_{k'} \right] &={1\over N} \sum_{l,m} e^{ikal} e^{-ik'am} [x_l , p_m ] \\ &= {i \hbar\over N} \sum_{m} e^{iam(k-k')} = i\hbar\delta_{k,k'} \\ \left[ Q_k, Q_{k'} \right] &= \left[ \Pi_k , \Pi_{k'} \right] = 0 ~. \end{align}$$

From the general result $$ \begin{align} \sum_{l}x_l x_{l+m}&={1\over N}\sum_{kk'}Q_k Q_{k'}\sum_{l} e^{ial\left(k+k'\right)}e^{iamk'}= \sum_{k}Q_k Q_{-k}e^{iamk} \\ \sum_{l}{p_l}^2 &= \sum_{k}\Pi_k \Pi_{-k}  ~, \end{align}$$ it is easy to show, through elementary trigonometry, that the potential energy term is $$ {1\over 2} m \omega^2 \sum_{j} (x_j - x_{j+1})^2= {1\over 2} m \omega^2\sum_{k}Q_k Q_{-k}(2-e^{ika}-e^{-ika})= {1\over 2} m \sum_{k}{\omega_k}^2Q_k Q_{-k} ~ ,$$ where $$\omega_k = \sqrt{2 \omega^2 (1 - \cos(ka))} ~.$$

The Hamiltonian may be written in wave vector space as $$\mathbf{H} = {1\over {2m}}\sum_k \left( { \Pi_k\Pi_{-k} } + m^2 \omega_k^2 Q_k Q_{-k} \right) ~.$$

Note that the couplings between the position variables have been transformed away; if the $p$s and $Q$s were hermitian (which they are not), the transformed Hamiltonian would describe $Π$ uncoupled harmonic oscillators.

The form of the quantization depends on the choice of boundary conditions; for simplicity, we impose periodic boundary conditions, defining the $k_{n}$-th atom as equivalent to the first atom. Physically, this corresponds to joining the chain at its ends. The resulting quantization is

$$k=k_n = {2n\pi \over Na} \quad \hbox{for}\ n = 0, \pm1, \pm2, \ldots, \pm {N \over 2}. $$

The upper bound to $N$ comes from the minimum wavelength, which is twice the lattice spacing $n$, as discussed above.

The harmonic oscillator eigenvalues or energy levels for the mode $(N + 1)$ are $$E_n = \left({1\over2}+n\right)\hbar\omega_k  \quad\hbox{for}\quad n=0,1,2,3,\ldots$$

If we ignore the zero-point energy then the levels are evenly spaced at $$\tfrac32\hbar\omega, \ \tfrac52\hbar\omega ,\ \tfrac72\hbar\omega \ \cdots $$

So an exact amount of energy $ω_{k}$, must be supplied to the harmonic oscillator lattice to push it to the next energy level. In analogy to the photon case when the electromagnetic field is quantised, the quantum of vibrational energy is called a phonon.

All quantum systems show wave-like and particle-like properties. The particle-like properties of the phonon are best understood using the methods of second quantization and operator techniques described elsewhere.

In the continuum limit, $ħω$, $a → 0$, while $N → ∞$ is held fixed. The canonical coordinates $Na$ devolve to the decoupled momentum modes of a scalar field, $$\phi_k$$, whilst the location index $a$ (not the displacement dynamical variable) becomes the parameter $i$ argument of the scalar field, $$\phi (x,t)$$.

Molecular vibrations

 * The vibrations of a diatomic molecule are an example of a two-body version of the quantum harmonic oscillator. In this case, the angular frequency is given by $$\omega = \sqrt{\frac{k}{\mu}} $$ where $$\mu = \frac{m_1 m_2}{m_1 + m_2}$$ is the reduced mass and $$m_1$$ and $$m_2$$ are the masses of the two atoms.
 * The Hooke's atom is a simple model of the helium atom using the quantum harmonic oscillator.
 * Modelling phonons, as discussed above.
 * A charge $$q$$ with mass $$m$$ in a uniform magnetic field $$\mathbf{B}$$ is an example of a one-dimensional quantum harmonic oscillator: Landau quantization.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Quantum harmonic oscillator


text:
Buccaneer

Buccaneers were a kind of privateer or free sailors particular to the Caribbean Sea during the 17th and 18th centuries. First established on northern Hispaniola as early as 1625, their heyday was from the Restoration in 1660 until about 1688, during a time when governments in the Caribbean area were not strong enough to suppress them.

Originally the name applied to the landless hunters of wild boars and cattle in the largely uninhabited areas of Tortuga and Hispaniola. The meat they caught was smoked over a slow fire in little huts the French called boucans to make viande boucanée – jerked meat or jerky – which they sold to the corsairs who preyed on the (largely Spanish) shipping and settlements of the Caribbean. Eventually the term was applied to the corsairs and (later) privateers themselves, also known as the Brethren of the Coast. Although corsairs, also known as filibusters or freebooters, were largely lawless, privateers were nominally licensed by the authorities – first the French, later the English and Dutch – to prey on the Spanish, until their depredations became so severe they were suppressed.

Etymology
The term buccaneer derives from the Caribbean Arawak word buccan, which refers to a wooden frame on which Tainos and Caribs slowly roasted or smoked meat, commonly manatee. The word was adopted into French as boucan, hence the name boucanier for French hunters who used such frames to smoke meat from feral cattle and pigs on Hispaniola. English colonists anglicised boucanier to buccaneer.

History
About 1630, French interlopers were driven away from the island of Hispaniola and fled to nearby Tortuga. French buccaneers were established on northern Hispaniola as early as 1625, but lived at first mostly as hunters rather than robbers; their transition to full-time piracy was gradual and motivated in part by Spanish efforts to wipe out both the buccaneers and the prey animals on which they depended. The buccaneers' migration from Hispaniola's mainland to the more defensible offshore island of Tortuga limited their resources and accelerated their piratical raids. According to Alexandre Exquemelin, the Tortuga buccaneer Pierre Le Grand pioneered the settlers' attacks on galleons making the return voyage to Spain. The Spaniards also tried to drive them out of Tortuga, but the buccaneers were joined by many more French, Dutch, and English adventurers who turned to piracy. They set their eyes on Spanish shipping, generally using small craft to attack galleons in the vicinity of the Windward Passage. With the support and encouragement of rival European powers, they became strong enough to sail for the mainland of Spanish America, known as the Spanish Main, and sacked cities.

Perhaps what distinguished the buccaneers from earlier Caribbean sailors was their use of permanent bases in the West Indies. During the mid 17th century, the Bahama Islands attracted many lawless people who had taken over New Providence. Encouraged by its large harbour, they were joined by several pirates who made their living by raiding the Spanish on the coast of Cuba. They called this activity buccaneering. Their principal station was Tortuga, but from time to time they seized other strongholds, like Providence, and they were welcomed with their booty in ports like Port Royal in Jamaica. At first they were international. In 1663 it was estimated that there were fifteen of their ships with nearly a thousand men, English, French, and Dutch, belonging to Jamaica and Tortuga. As time went on and the European governments asserted their authority, the buccaneers first became separated by nationalities and then in time were suppressed altogether, leaving behind only dispersed bands of pirates.

English settlers occupying Jamaica began to spread the name buccaneers with the meaning of pirates. The name became universally adopted later in 1684 when the first English translation of Alexandre Exquemelin's book The Buccaneers of America was published.

Viewed from London, buccaneering was a budget way to wage war on England's rival, Spain. The English crown licensed buccaneers with letters of marque, legalising their operations in return for a share of their profits. The buccaneers were invited by Jamaica's Governor Thomas Modyford to base ships at Port Royal. The buccaneers robbed Spanish shipping and colonies, and returned to Port Royal with their plunder, making the city the most prosperous in the Caribbean. There even were Royal Navy officers sent to lead the buccaneers, such as Christopher Myngs. Their activities went on irrespective of whether England happened to be at war with Spain or France.

Among the leaders of the buccaneers were two Frenchmen, Jean-David Nau, better known as François l'Ollonais, and Daniel Montbars, who destroyed so many Spanish ships and killed so many Spaniards that he was called "the Exterminator".

Another noted leader was Welshman Henry Morgan, who sacked Maracaibo, Portobello, and Panama City, stealing a huge amount from the Spanish. Morgan became rich and went back to England, where he was knighted by Charles II.

While the buccaneers were powerful it was not only hostility to Spain, but also lack of authority, that prevented the other states from ending the old state of affairs in which, even when they were at peace with Spain and Portugal in Europe, there was 'no peace beyond the Line'. The West Indies were beyond the range of the European international system. Sometimes this was for their advantage but on the whole, with the intermingled possessions, trade rivalries, and disputes about territorial rights, the local conditions led to conflicts. The West Indies continued to be one of the centres of international strife throughout the eighteenth century although by that time it was regulated in the same way as in Europe, and had become inseparable from the European wars.

During the Second Anglo-Dutch War in 1665, de Ruyter attacked Barbados with a strong squadron, and the English had no choice but to base their defence on the buccaneers whom the governor of Jamaica had previously been trying to suppress. They were unmanageable and destroyed where they conquered, but they mastered the Dutch colonies of St. Eustatius and Tobago. In 1666, however, when the French joined the Dutch in the war, the weakness of this policy was proved. The English hoped to capture the French plantations of St. Kitts, where there were new settlers of both nations, and so they declined to make a new agreement for neutrality. They made what was intended to be a surprise attack, but was an ignominious failure, and the English settlers in the island had to surrender unconditionally. More than 8,000 of them were shipped away, and their property was seized by the French. Lord Willoughby, the able governor of Barbados, got together an expedition for a counter-stroke, but his fleet was broken up by a hurricane in which he perished. The French captured one island after another. In 1667 naval ships from England regained the command of the sea and made various conquests, but the Peace of Breda re-established the status quo in March of that year.

Henry Morgan was knighted in 1674 and became lieutenant-governor of Jamaica. In the late 1670s there was a succession of raids on Spanish ports. In 1680 a party made its way across the Isthmus of Panama and, sailing in captured Spanish ships, pillaged the coasts and commerce of the Pacific. They had not been long on their journey when the Anglo-Spanish treaty of 1680 was signed, which at last stipulated for a real peace beyond the Line and indirectly recognised the right of the English to trade in West Indian waters. When the buccaneers returned by way of Cape Horn in 1682, the survivors found themselves treated as pirates. The French, within a very few years, also controlled their buccaneers, and in the Nine Years' War (1688-1697) they were no longer an important factor. Until about 1688 the governments were not strong enough, and did not consistently attempt, to suppress the buccaneers.

In January 1684, Havana responded to the attacks by the buccaneers of the Bahamas in the event known as the Raid on Charles Town.

In the 1690s, the old buccaneering ways began to die out, as European governments began to discard the policy of "no peace beyond the Line". Buccaneers were hard to control; some even embroiled their colonies in unwanted wars. Notably, at the 1697 joint French-buccaneer siege of Cartagena, led by Bernard Desjean, Baron de Pointis, the buccaneers and the French regulars parted on extremely bitter terms. Less tolerated by local Caribbean officials, buccaneers increasingly turned to legal work or else joined regular pirate crews who sought plunder in the Indian Ocean, the east coast of North America, or West Africa as well as in the Caribbean.

Legal status
Sometimes the buccaneers held more or less regular commissions as privateers, and they always preyed upon the Spaniards; but often they became mere pirates and plundered any nation. As a rule, the buccaneers called themselves privateers, and many sailed under the protection of a letter of marque granted by British, French or Dutch authorities. For example, Henry Morgan had some form of legal cover for all of his attacks, and expressed great indignation at being called a "corsair" by the governor of Panama. Nevertheless, these rough men had little concern for legal niceties, and exploited every opportunity to pillage Spanish targets, whether or not a letter of marque was available. Many of the letters of marque used by buccaneers were legally invalid, and any form of legal paper in that illiterate age might be passed off as a letter of marque. Furthermore, even those buccaneers who had valid letters of marque often failed to observe their terms. The legal status of buccaneers was still further obscured by the practice of the Spanish authorities, who regarded them as heretics and interlopers, and thus hanged or garroted captured buccaneers entirely without regard to whether their attacks were licensed by French or English monarchs.

Simultaneously, French and English governors tended to turn a blind eye to the buccaneers' depredations against the Spanish, even when unlicensed. But as Spanish power waned toward the end of the 17th century, the buccaneers' attacks began to disrupt France and England's merchant traffic with Spanish America, such that merchants who had previously regarded the buccaneers as a defence against Spain now saw them as a threat to commerce, and colonial authorities grew hostile. This change in political atmosphere, more than anything else, put an end to buccaneering.

Lifestyle
A hundred years before the French Revolution, the buccaneer companies were run on lines in which liberty, equality and fraternity were the rule. In a buccaneer camp, the captain was elected and could be deposed by the votes of the crew. The crew, and not the captain, decided whether to attack a particular ship, or a fleet of ships. Spoils were evenly divided into shares; the captain received an agreed amount for the ship, plus a portion of the share of the prize money, usually five or six shares.

Crews generally had no regular wages, being paid only from their shares of the plunder, a system called "no purchase, no pay" by Modyford or "no prey, no pay" by Enqueueing. There was a strong esprit among buccaneers. This, combined with overwhelming numbers, allowed them to win battles and raids. There was also, for some time, a social insurance system guaranteeing compensation for battle wounds at a worked-out scale.

Naval
Buccaneers initially used small boats to attack Spanish galleons surreptitiously, often at night, and climb aboard before the alarm could be raised. Buccaneers were expert marksmen and would quickly kill the helmsman and any officers aboard. Buccaneers' reputation as cruel pirates grew to the point that, eventually, most victims would surrender, hoping they would not be killed.

Land
When buccaneers raided towns, they did not sail into port and bombard the defences, as naval forces typically did. Instead, they secretly beached their ships out of sight of their target, marched overland, and attacked the towns from the landward side, which was usually less fortified. Their raids relied mainly on surprise and speed. The sack of Campeche was considered the first such raid and many others that followed replicated the same techniques including the attack on Veracruz in 1683 and the raid on Cartagena later that same year.

Downturn
Spanish authorities always viewed buccaneers as trespassers and a threat to their hegemony in the Caribbean basin, and over the second half of the 17th century, other European powers learned to perceive them in the same way. These new powers had appropriated and secured territories in the area and needed to protect them. Buccaneers who did not settle down on agriculture or some other acceptable business after the so-called Golden Age of Piracy proved a nuisance to them, too. Spanish anti-pirate practices became thus a model for all recently arrived colonial governments. Some expanded them.

Punishments
When caught by anti-pirate English authorities, 17th and 18th century buccaneers received justice in a summary fashion, and many ended their lives by "dancing the hempen jig", a euphemism for hanging. Public executions were a form of entertainment, and people came out to watch them as they would for a sporting event today. Newspapers reported details such as condemned men's last words, the prayers said by the priests, and descriptions of their final moments in the gallows. In England, most executions took place at Execution Dock on the River Thames in London.

In the cases of more famous prisoners, usually captains, their punishments extended beyond death. Their bodies were enclosed in iron cages (for which they were measured before their execution) and left to swing in the air until the flesh rotted off them—a process that could take as long as two years. The bodies of captains such as William "Captain" Kidd, Charles Vane, William Fly, and Jack Rackham ("Calico Jack") were all treated this way.

It is doubtful many buccaneers got off with just a time in the pillory. However, a pirate who was flogged could very well spend some time in the pillory after being beaten. "The most common shaming punishment was confinement in the pillory often with symbols of their crimes."

In literature
After the threat began to abate, literature brought buccaneers to glory as example of virility and self-reliance. Daniel Defoe’s works like Robinson Crusoe (1719), Captain Singleton (1720), and A General History of the Pyrates (1724) (purportedly written by Defoe) set the tone for the glamorous ways in which later generations would perceive them.

title: Buccaneer


text:
Talk:Interstate Highway System

Translation into Chinese Wikipedia
Part of the version 21:57, 20 July 2022 92.22.184.183 of this article is translated into Chinese Wikipedia to update an existing older translation.--Wing (talk) 10:51, 28 July 2022 (UTC)

I-96
Why is Interstate 96 a different shade of red than the rest of them? NintendoTTTEfan2005 (talk) 15:58, 6 September 2023 (UTC)
 * I am not finding a mention of I-96 in this article. Do you mean in the I-96 article itself or where? [Nevermind I-96 shown in main Intertstate map for Infobox.] -Fnlayson (talk) 17:50, 6 September 2023 (UTC)
 * That data is pulled from a couple dozen different GeoJSON files. It appears whomever made the file for interstate 96, which is  here, used a different color from the rest. If you can compare that file from one from the another interstate highway, maybe you can find the offending code and change the color in that GeoJSON file to be the same as the others. Dave (talk) 00:08, 7 September 2023 (UTC)
 * It looks like it was already fixed, just today. Comparing this map file with the one used for I-70, they are both color #cc0000. However, the wikipedia server must be using a cached copy of the old version. Should be fixed next time the cache is flushed or refreshed. Dave (talk) 00:16, 7 September 2023 (UTC)

Wiki Education assignment: Technology and Culture
— Assignment last updated by Thecanyon (talk) 05:32, 12 December 2023 (UTC)

Listing accidents or incidents
Please provide a permalink to whatever policy or discussion established that incidents or accidents cannot be tabulated on this page, as you claimed in your edit summary. Einsof (talk) 06:08, 19 November 2023 (UTC)
 * Additionally, claimed on my talk page that the table cannot be added without citations, but the corresponding table on the Amtrak page contains no such citations, so I find this rationale to be dubious. Since every entry in the table is linked to an article or article section that contains citations, I see no WP:V issue here. Einsof (talk) 06:15, 19 November 2023 (UTC)
 * For reasons I don't fully understand, the policy enforcers and article taggers have been much harder on road articles than train articles. So far train articles have been routinely given a pass, and mass transit articles even more so. They seem to be almost untouchable by the policy enforcers. However, policy is still policy, regardless of if the policy enforcers are paying attention or not. Also, I suspect that won't last forever. The policy enforcers tend to focus on one set of articles at a time, and while rail and mass transit articles have so far escaped the microscope, I suspect it will someday be their turn to be placed under a microscope and orange tagged to death.
 * Setting policy aside, let's focus on article quality, which IMHO is more important. Where I agree with Imzadi is road accidents are much more common than train accidents. Every accident involving an Amrtak Train makes national news, as they are relatively rare. Except for the most local of coverage, accidents involving roads almost never make the news, as there are dozens every day. As such I agree the threshold of notability would have to be higher for a list of road accidents verses a list of train accidents.
 * I am open to the existence of such a list, this is where I would disagree with Imzadi. However, in a practical sense I think Imzadi is right. Until we could find an agreeable threshold of notability, the list of Interstate Highway accidents would become so bloated that it would be unmanagable. It would also be a "crap magnet" of people adding "hey you forgot about this one car rollover in the Arizona desert on I-40 in 1989". So probably better to decide that threshold first before creating such a list to avoid that magnet. That's my $.02.Dave (talk) 07:10, 19 November 2023 (UTC)
 * Seems like "more than 10 fatalities in a single incident" is a suitably high threshold. If that proves too unmanageable we could always raise it. Einsof (talk) 00:24, 22 November 2023 (UTC)
 * I didn't hear an objection to this proposed threshold, and never substantiated the putative editorial consensus underlying Special:Diff/1185816757 (explicitly, that at some point in the past it was decided that unless incidents have had a lasting impact on the design of a roadway, we do NOT include them in the history of the individual highways). In any case, this article is not about individual highways, so one has to wonder about the relevance of such a discussion anyway. I added an updated table with references. Einsof (talk) 01:52, 28 November 2023 (UTC)
 * @Einsof: let me reiterate/restate my previous comments a different way: my threshold is no table as I consider it unnecessary. There is no injury/fatality count that I would consider as appropriate to mention unless it resulted in specific changes to the Interstate Highway standards.  Imzadi 1979  →   05:13, 28 November 2023 (UTC)

 * It's simply excessive to list every incident on a nationwide system regardless of their lasting impact. Imagine if every car crash with a fatality was listed in a state's article. No one would think to do that, so why add it here?  Sounder Bruce  03:00, 28 November 2023 (UTC)
 * You've misunderstood. The table does not list every incident on the system. It only lists incidents with more than 10 fatalities. Furthermore, each incident in that table either has its own article or a substantial subsection of an article, so they obviously meet our notability criteria (conversely: if you think any of these incidents don't meet our notability criteria, feel free to put them up at WP:AFD). Einsof (talk) 04:12, 28 November 2023 (UTC)
 * There are some state level articles that mention accidents, Interstate 70 in Colorado comes to mind. While I'm doing this blind deliberately, I'm guessing U.S. Route 466 would have the accident where James Dean died. However, I agree there's just too many to list at the system level. The more I think about this the more I come back to my original comment. Notable accidents should be on the individual articles for the highways, or segments of highways. To have a list on the article of the Interstate Highway System, there's just too many. I understand there is a equivalent list for Amtrak, but again, that's because Amtrak accidents are relatively rare. They are rare enough that when there is one, it makes national news.Dave (talk) 05:52, 28 November 2023 (UTC)
 * Concur with User:Moabdave, User:SounderBruce, and User:Imzadi1979 on this one and disagree with User:Einsof. The appropriate place for listing truly notable car crashes on Interstate Highways is in the articles for the relevant Interstate Highways and the locale (city, town, or CDP) where the crash actually occurred.  A table of incidents on Interstate Highways with 10 or more fatalities in this article is analogous to listing all large-scale plane crashes in the United States in Aviation in the United States or all large-scale ship collisions on rivers in Inland waterways of the United States.   Which, of course, we don't do.  See WP:INDISCRIMINATE. --Coolcaesar (talk) 17:06, 28 November 2023 (UTC)

Reverting incorrect edit by User:Kehkou on 19 October 2023
I just reverted this incorrect edit on 19 October 2023, which was not supported by a source as required by WP:NOR, WP:V, and WP:NOT.

I skim articles in most major American English news publications every week and have done so for over 25 years. I have never seen "the Interstate" used to refer to the system as a whole. Unless someone can produce one or more sources verifying that unusual usage, it should stay out of the article. --- Coolcaesar (talk) 18:13, 19 November 2023 (UTC)

 * @Coolcaesar Thank you for your edit (too old and busy to go source hunting all the time, reversion is easier). Incidentally, on this side of the country, "the Interstate" is indeed quite popularly used to refer to any or all parts of the system (i.e., as a whole); in fact, you are the first and only person I have witnessed to object to this colloquialism, online or otherwise. At any rate, it may serve better to omit it from the lead and include this info elsewhere in the article if/as appropriate. Happy editing! Kehkou (talk) 01:20, 20 November 2023 (UTC)

title: Talk:Interstate Highway System


text:
Sully Prudhomme

René François Armand "Sully" Prudhomme (16 March 1839 – 6 September 1907) was a French poet and essayist. He was the first winner of the Nobel Prize in Literature in 1901.

Born in Paris, Prudhomme originally studied to be an engineer, but turned to philosophy and later to poetry; he declared it as his intention to create scientific poetry for modern times. In character sincere and melancholic, he was linked to the Parnassus school, although, at the same time, his work displays characteristics of its own.

Early life
Prudhomme's parents were M. Sully Prudhomme and Clotilde Caillat. They had been engaged for 10 years before they had felt financially able to marry. When Prudhomme was two, his father, a shopkeeper, died. His mother and he relocated to Prudhomme's uncle's house. Prudhomme joined his father's name "Sully" with his surname Prudhomme, becoming Sully-Prudhomme. He was interested in classic literature and mathematics in school. He also considered entering the Dominican order, but decided against it.

Prudhomme attended the Lycée Bonaparte, but eye trouble interrupted his studies. He worked for a while in the Creusot region for the Schneider steel foundry, and then began studying law in a notary's office. The favourable reception of his early poems by the Conférence La Bruyère (a student society) encouraged him to begin a literary career.

Writing
His first collection, Stances et Poèmes ("Stanzas and Poems", 1865), was praised by Sainte-Beuve. It included his most famous poem, Le vase brisé. He published more poetry before the outbreak of the Franco-Prussian War. This war, which he discussed in Impressions de la guerre (1872) and La France (1874), permanently damaged his health.

During his career, Prudhomme gradually shifted from the sentimental style of his first books towards a more personal style which unified the formality of the Parnassus school with his interest in philosophical and scientific subjects. One of his inspirations was clearly Lucretius's De rerum natura, whose first book he translated into verse. His philosophy was expressed in La Justice (1878) and Le Bonheur (1888). The extreme economy of means employed in these poems has, however, usually been judged as compromising their poetical quality without advancing their claims as works of philosophy. He was elected to the Académie française in 1881. Another distinction, Chevalier de la Légion d’honneur, was to follow in 1895.

After Le Bonheur, Prudhomme turned from poetry to write essays on aesthetics and philosophy. He published two important essays: L'Expression dans les beaux-arts (1884) and Réflexions sur l'art des vers (1892), a series of articles on Blaise Pascal in La Revue des Deux Mondes (1890), and an article on free will (La Psychologie du Libre-Arbitre, 1906) in the Revue de métaphysique et de morale.

Nobel Prize
The first writer to receive the Nobel Prize for Literature (given "in special recognition of his poetic composition, which gives evidence of lofty idealism, artistic perfection and a rare combination of the qualities of both heart and intellect"), he devoted the bulk of the money he received to the creation of a poetry prize awarded by the Société des gens de lettres. He also founded, in 1902, the Société des poètes français with Jose-Maria de Heredia and Leon Dierx.

Death
At the end of his life, his poor health (which had troubled him ever since 1870) forced him to live almost as a recluse at Châtenay-Malabry, suffering attacks of paralysis while continuing to work on essays. He died suddenly on 6 September 1907, and was buried at Père-Lachaise in Paris.

Poetry

 * 1865: Stances et poèmes
 * 1866: Les épreuves
 * 1868: Croquis italiens
 * 1869: Les solitudes: poésies [Les écuries d’Augias]
 * 1872: Les destins
 * 1874: La révolte des fleurs
 * 1874: La France
 * 1875: Les vaines tendresses
 * 1876: Le zénith, previously published in Revue des deux mondes
 * 1878: La justice
 * 1865–1888: Poésie
 * 1886: Le prisme, poésies diverses
 * 1888: Le bonheur
 * 1908: Épaves

Prose

 * 1883–1908: Œuvres de Sully Prudhomme (poetry and prose), 8 volumes, A. Lemerre
 * 1896: Que sais-je? (philosophy)
 * 1901: Testament poétique (essays)
 * 1905: La vraie religion selon Pascal (essays)
 * 1922: Journal intime: lettres-pensée

title: Sully Prudhomme


text:
Convergence of random variables

In probability theory, there exist several different notions of convergence of sequences of random variables, including convergence in probability, convergence in distribution, and almost sure convergence. The different notions of convergence capture different properties about the sequence, with some notions of convergence being stronger than others. For example, convergence in distribution tells us about the limit distribution of a sequence of random variables. This is a weaker notion than convergence in probability, which tells us about the value a random variable will take, rather than just the distribution.

The concept is important in probability theory, and its applications to statistics and stochastic processes. The same concepts are known in more general mathematics as stochastic convergence and they formalize the idea that certain properties of a sequence of essentially random or unpredictable events can sometimes be expected to settle down into a behavior that is essentially unchanging when items far enough into the sequence are studied. The different possible notions of convergence relate to how such a behavior can be characterized: two readily understood behaviors are that the sequence eventually takes a constant value, and that values in the sequence continue to change but can be described by an unchanging probability distribution.

Background
"Stochastic convergence" formalizes the idea that a sequence of essentially random or unpredictable events can sometimes be expected to settle into a pattern. The pattern may for instance be
 * Convergence in the classical sense to a fixed value, perhaps itself coming from a random event
 * An increasing similarity of outcomes to what a purely deterministic function would produce
 * An increasing preference towards a certain outcome
 * An increasing "aversion" against straying far away from a certain outcome
 * That the probability distribution describing the next outcome may grow increasingly similar to a certain distribution

Some less obvious, more theoretical patterns could be These other types of patterns that may arise are reflected in the different types of stochastic convergence that have been studied.
 * That the series formed by calculating the expected value of the outcome's distance from a particular value may converge to 0
 * That the variance of the random variable describing the next event grows smaller and smaller.

While the above discussion has related to the convergence of a single series to a limiting value, the notion of the convergence of two series towards each other is also important, but this is easily handled by studying the sequence defined as either the difference or the ratio of the two series.

For example, if the average of n independent random variables Yi, i = 1, ..., n, all having the same finite mean and variance, is given by

 * $$X_n = \frac{1}{n}\sum_{i=1}^n Y_i\,,$$

then as n tends to infinity, $X_{n}$ converges in probability (see below) to the common mean, μ, of the random variables Yi. This result is known as the weak law of large numbers. Other forms of convergence are important in other useful theorems, including the central limit theorem.

Throughout the following, we assume that (Xn) is a sequence of random variables, and X is a random variable, and all of them are defined on the same probability space $$(\Omega, \mathcal{F}, \mathbb{P} )$$.

Convergence in distribution
Loosely, with this mode of convergence, we increasingly expect to see the next outcome in a sequence of random experiments becoming better and better modeled by a given probability distribution. More precisely, the distribution of the associated random variable in the sequence becomes arbitrarily close to a specified fixed distribution.

Convergence in distribution is the weakest form of convergence typically discussed, since it is implied by all other types of convergence mentioned in this article. However, convergence in distribution is very frequently used in practice; most often it arises from application of the central limit theorem.

Definition
A sequence $$X_1, X_2, \ldots $$ of real-valued random variables, with cumulative distribution functions $$F_1, F_2, \ldots $$, is said to converge in distribution, or converge weakly, or converge in law to a random variable $X_{n}$ with cumulative distribution function $n$ if

 * $$\lim_{n\to\infty} F_n(x) = F(x),$$

for every number $$x \in \mathbb{R}$$ at which $n$ is continuous.

The requirement that only the continuity points of $Z_{n}$ should be considered is essential. For example, if $n$ are distributed uniformly on intervals $X_{1}$, then this sequence converges in distribution to the degenerate random variable $μ = 0.5$. Indeed, $σ^{2} = 0.25$ for all n when $X_{2}, X_{3}, ...$, and $X_{n}$ for all ${X_{i}}$ when $U(−1, 1)$. However, for this limiting random variable $N(0, 1⁄3)$, even though $(0, 1⁄n)$ for all $X$. Thus the convergence of cdfs fails at the point $X = 0$ where $F$ is discontinuous.

Convergence in distribution may be denoted as

where $$\scriptstyle\mathcal{L}_X$$ is the law (probability distribution) of $F$. For example, if $F$ is standard normal we can write.

For random vectors $F_{n}(x) = 0$ the convergence in distribution is defined similarly. We say that this sequence converges in distribution to a random $X_{n}$-vector $n$ if
 * $$\lim_{n\to\infty} \mathbb{P}(X_n\in A) = \mathbb{P}(X\in A)$$

for every $x ≤ 0$ which is a continuity set of $F$.

The definition of convergence in distribution may be extended from random vectors to more general random elements in arbitrary metric spaces, and even to the “random variables” which are not measurable — a situation which occurs for example in the study of empirical processes. This is the “weak convergence of laws without laws being defined” — except asymptotically.

In this case the term weak convergence is preferable (see weak convergence of measures), and we say that a sequence of random elements $F_{n}(x) = 1$ converges weakly to $$ (denoted as $x ≥ 1⁄n$) if
 * $$\mathbb{E}^*h(X_n) \to \mathbb{E}\,h(X)$$

for all continuous bounded functions $X$. Here E* denotes the outer expectation, that is the expectation of a “smallest measurable function $X$ that dominates $n > 0$”.

Properties

 * Since $$F(a) = \mathbb{P}(X \le a)$$, the convergence in distribution means that the probability for $k$ to be in a given range is approximately equal to the probability that the value of $X$ is in that range, provided $X$ is sufficiently large.
 * In general, convergence in distribution does not imply that the sequence of corresponding probability density functions will also converge. As an example one may consider random variables with densities $F(0) = 1$. These random variables converge in distribution to a uniform U(0, 1), whereas their densities do not converge at all.
 * However, according to Scheffé’s theorem, convergence of the probability density functions implies convergence in distribution.
 * The portmanteau lemma provides several equivalent definitions of convergence in distribution. Although these definitions are less intuitive, they are used to prove a number of statistical theorems. The lemma states that $F_{n}(0) = 0$ converges in distribution to $X$ if and only if any of the following statements are true:
 * $$\mathbb{P}(X_n \le x) \to \mathbb{P}(X \le x)$$ for all continuity points of $$x\mapsto \mathbb{P}(X \le x)$$;
 * $$\mathbb{E}f(X_n) \to \mathbb{E}f(X)$$ for all bounded, continuous functions $$f$$ (where $$\mathbb{E}$$ denotes the expected value operator);
 * $$\mathbb{E}f(X_n) \to \mathbb{E}f(X)$$ for all bounded, Lipschitz functions $$f$$;
 * $$\lim\inf \mathbb{E}f(X_n) \ge \mathbb{E}f(X)$$ for all nonnegative, continuous functions $$f$$;
 * $$\lim\inf \mathbb{P}(X_n \in G) \ge \mathbb{P}(X \in G)$$ for every open set $$G$$;
 * $$\lim\sup \mathbb{P}(X_n \in F) \le \mathbb{P}(X \in F)$$ for every closed set $$F$$;
 * $$\mathbb{P}(X_n \in B) \to \mathbb{P}(X \in B)$$ for all continuity sets $$B$$ of random variable $$X$$;
 * $$\limsup \mathbb{E}f(X_n) \le \mathbb{E}f(X)$$ for every upper semi-continuous function $$f$$ bounded above;
 * $$\liminf \mathbb{E}f(X_n) \ge \mathbb{E}f(X)$$ for every lower semi-continuous function $$f$$ bounded below.
 * The continuous mapping theorem states that for a continuous function $h$, if the sequence $x = 0$ converges in distribution to $g$, then ${X_{1}, X_{2}, ...} ⊂ R^{k}$ converges in distribution to $A ⊂ R^{k}$.
 * Note however that convergence in distribution of ${X_{n}}$ to $X_{n}$ and $X_{n} ⇒ X$ to $X$ does in general not imply convergence in distribution of $h(X_{n})$ to $f_{n}(x) = (1 + cos(2πnx))1_{(0,1)}$ or of ${X_{n}}$ to $n$.
 * Lévy’s continuity theorem: The sequence ${X_{n}}$ converges in distribution to $X$ if and only if the sequence of corresponding characteristic functions ${g(X_{n})}$ converges pointwise to the characteristic function $g$ of $X$.
 * Convergence in distribution is metrizable by the Lévy–Prokhorov metric.
 * A natural link to convergence in distribution is the Skorokhod's representation theorem.

Convergence in probability
The basic idea behind this type of convergence is that the probability of an “unusual” outcome becomes smaller and smaller as the sequence progresses.

The concept of convergence in probability is used very often in statistics. For example, an estimator is called consistent if it converges in probability to the quantity being estimated. Convergence in probability is also the type of convergence established by the weak law of large numbers.

Definition
A sequence {Xn} of random variables converges in probability towards the random variable X if for all ε > 0

 * $$\lim_{n\to\infty}\mathbb{P}\big(|X_n-X| > \varepsilon\big) = 0.$$

More explicitly, let Pn(ε) be the probability that Xn is outside the ball of radius ε centered at X. Then $X$ is said to converge in probability to X if for any $g(X)$ and any δ > 0 there exists a number N (which may depend on ε and δ) such that for all n ≥ N, Pn(ε) < δ (the definition of limit).

Notice that for the condition to be satisfied, it is not possible that for each n the random variables X and Xn are independent (and thus convergence in probability is a condition on the joint cdf's, as opposed to convergence in distribution, which is a condition on the individual cdf's), unless X is deterministic like for the weak law of large numbers. At the same time, the case of a deterministic X cannot, whenever the deterministic value is a discontinuity point (not isolated), be handled by convergence in distribution, where discontinuity points have to be explicitly excluded.

Convergence in probability is denoted by adding the letter p over an arrow indicating convergence, or using the "plim" probability limit operator:

For random elements {Xn} on a separable metric space ${X_{n}}$, convergence in probability is defined similarly by
 * $$\forall\varepsilon>0, \mathbb{P}\big(d(X_n,X)\geq\varepsilon\big) \to 0.$$

Properties

 * Convergence in probability implies convergence in distribution.[proof]
 * In the opposite direction, convergence in distribution implies convergence in probability when the limiting random variable X is a constant.[proof]
 * Convergence in probability does not imply almost sure convergence.[proof]
 * The continuous mapping theorem states that for every continuous function $$g$$, if $X_n \xrightarrow{p} X$, then also $g(X_n)\xrightarrow{p}g(X)$.
 * Convergence in probability defines a topology on the space of random variables over a fixed probability space. This topology is metrizable by the Ky Fan metric: or alternately by this metric $$d(X,Y)=\mathbb E\left[\min(|X-Y|, 1)\right].$$

Counterexamples
Not every sequence of random variables which converges to another random variable in distribution also converges in probability to that random variable. As an example, consider a sequence of standard normal random variables $$X_n$$ and a second sequence $$Y_n = (-1)^nX_n$$. Notice that the distribution of $$Y_n$$ is equal to the distribution of $$X_n$$ for all $$n$$, but: $$P(|X_n - Y_n| \geq \epsilon) = P(|X_n|\cdot|(1 - (-1)^n)| \geq \epsilon)$$

which does not converge to $$0$$. So we do not have convergence in probability.

Almost sure convergence
This is the type of stochastic convergence that is most similar to pointwise convergence known from elementary real analysis.

Definition
To say that the sequence $Y$ converges almost surely or almost everywhere or with probability 1 or strongly towards X means that $$\mathbb{P}\!\left( \lim_{n\to\infty}\! X_n = X \right) = 1.$$

This means that the values of $XY$ approach the value of X, in the sense that events for which $X$ does not converge to X have probability 0 (see Almost surely). Using the probability space $$(\Omega, \mathcal{F}, \mathbb{P} )$$ and the concept of the random variable as a function from Ω to R, this is equivalent to the statement $$\mathbb{P}\Bigl( \omega \in \Omega: \lim_{n \to \infty} X_n(\omega) = X(\omega) \Bigr) = 1.$$

Using the notion of the limit superior of a sequence of sets, almost sure convergence can also be defined as follows: $$\mathbb{P}\Bigl( \limsup_{n\to\infty} \bigl\{\omega \in \Omega: | X_n(\omega) - X(\omega) | > \varepsilon \bigr\} \Bigr) = 0 \quad\text{for all}\quad \varepsilon>0.$$

Almost sure convergence is often denoted by adding the letters a.s. over an arrow indicating convergence:

For generic random elements {Xn} on a metric space $$(S,d)$$, convergence almost surely is defined similarly: $$\mathbb{P}\Bigl( \omega\in\Omega\colon\, d\big(X_n(\omega),X(\omega)\big)\,\underset{n\to\infty}{\longrightarrow}\,0 \Bigr) = 1$$

Properties

 * Almost sure convergence implies convergence in probability (by Fatou's lemma), and hence implies convergence in distribution. It is the notion of convergence used in the strong law of large numbers.
 * The concept of almost sure convergence does not come from a topology on the space of random variables. This means there is no topology on the space of random variables such that the almost surely convergent sequences are exactly the converging sequences with respect to that topology. In particular, there is no metric of almost sure convergence.

Sure convergence or pointwise convergence
To say that the sequence of random variables (Xn) defined over the same probability space (i.e., a random process) converges surely or everywhere or pointwise towards X means

$$\forall \omega \in \Omega \colon \ \lim_{n\to\infty} X_n(\omega) = X(\omega),$$

where Ω is the sample space of the underlying probability space over which the random variables are defined.

This is the notion of pointwise convergence of a sequence of functions extended to a sequence of random variables. (Note that random variables themselves are functions).

$$\left\{\omega \in \Omega : \lim_{n \to \infty}X_n(\omega) = X(\omega) \right\} = \Omega.$$

Sure convergence of a random variable implies all the other kinds of convergence stated above, but there is no payoff in probability theory by using sure convergence compared to using almost sure convergence. The difference between the two only exists on sets with probability zero. This is why the concept of sure convergence of random variables is very rarely used.

Counterexamples
Consider a sequence $$\{X_n\}$$ of independent random variables such that $$P(X_n=1)=\frac{1}{n}$$ and $$P(X_n=0)=1-\frac{1}{n}$$. For $$0<\varepsilon<1/2$$ we have $$P(|X_n|\geq \varepsilon)=\frac{1}{n}$$ which converges to $$0$$ hence $$X_n\to 0$$ in probability.

Since $$\sum_{n\geq 1}P(X_n=1)\to\infty$$ and the events $$\{X_n=1\}$$ are independent, second Borel Cantelli Lemma ensures that $$P(\limsup_n \{X_n=1\})=1$$ hence the sequence $$\{X_n\}$$ does not converge to $$0$$ almost everywhere (in fact the set on which this sequence does not converge to $$0$$ has probability $$1$$).

Convergence in mean
Given a real number ${Y_{n}}$, we say that the sequence $φ$ converges in the r-th mean (or in the Lr-norm) towards the random variable X, if the $X$-th absolute moments $$\mathbb{E}$$(|Xn|r ) and $$\mathbb{E}$$(|X|r ) of $X$ and X exist, and
 * $$\lim_{n\to\infty} \mathbb{E}\left( |X_n-X|^r \right) = 0,$$

where the operator E denotes the expected value. Convergence in $X_{n}$-th mean tells us that the expectation of the $n$-th power of the difference between $$X_n$$ and $$X$$ converges to zero.

This type of convergence is often denoted by adding the letter Lr over an arrow indicating convergence:

The most important cases of convergence in r-th mean are:
 * When $X_{n}$ converges in r-th mean to X for r = 1, we say that $X$ converges in mean to X.
 * When $X$ converges in r-th mean to X for r = 2, we say that $X_{n}$ converges in mean square (or in quadratic mean) to X.

Convergence in the r-th mean, for r ≥ 1, implies convergence in probability (by Markov's inequality). Furthermore, if r > s ≥ 1, convergence in r-th mean implies convergence in s-th mean. Hence, convergence in mean square implies convergence in mean.

Additionally,
 * $$\overset{}{X_n \xrightarrow{L^r} X} \quad\Rightarrow\quad  \lim_{n \to \infty} \mathbb{E}[|X_n|^r] = \mathbb{E}[|X|^r]. $$

The converse is not necessarily true, however it is true if $$\overset{}{X_n \, \xrightarrow{p} \, X}$$ (by a more general version of Scheffé's lemma).

Properties
Provided the probability space is complete: The chain of implications between the various notions of convergence are noted in their respective sections. They are, using the arrow notation:
 * If $$X_n\ \xrightarrow{\overset{}{p}}\ X$$ and $$X_n\ \xrightarrow{\overset{}{p}}\ Y$$, then $$X=Y$$ almost surely.
 * If $$X_n\ \xrightarrow{\overset{}\text{a.s.}}\ X$$ and $$X_n\ \xrightarrow{\overset{}\text{a.s.}}\ Y$$, then $$X=Y$$ almost surely.
 * If $$X_n\ \xrightarrow{\overset{}{L^r}}\ X$$ and $$X_n\ \xrightarrow{\overset{}{L^r}}\ Y$$, then $$X=Y$$ almost surely.
 * If $$X_n\ \xrightarrow{\overset{}{p}}\ X$$ and $$Y_n\ \xrightarrow{\overset{}{p}}\ Y$$, then $$aX_n+bY_n\ \xrightarrow{\overset{}{p}}\ aX+bY$$ (for any real numbers $n$ and $X_{n}$) and $$X_n Y_n\xrightarrow{\overset{}{p}}\ XY$$.
 * If $$X_n\ \xrightarrow{\overset{}\text{a.s.}}\ X$$ and $$Y_n\ \xrightarrow{\overset{}\text{a.s.}}\ Y$$, then $$aX_n+bY_n\ \xrightarrow{\overset{}\text{a.s.}}\ aX+bY$$ (for any real numbers $X$ and $X_{n}$) and $$X_n Y_n\xrightarrow{\overset{}\text{a.s.}}\ XY$$.
 * If $$X_n\ \xrightarrow{\overset{}{L^r}}\ X$$ and $$Y_n\ \xrightarrow{\overset{}{L^r}}\ Y$$, then $$aX_n+bY_n\ \xrightarrow{\overset{}{L^r}}\ aX+bY$$ (for any real numbers $X$ and $X_{n}$).
 * None of the above statements are true for convergence in distribution.

 * $$\begin{matrix}

\xrightarrow{\overset{}{L^s}} & \underset{s>r\geq1}{\Rightarrow} &  \xrightarrow{\overset{}{L^r}}  &             & \\ &                                 &     \Downarrow                  &             & \\ \xrightarrow{\text{a.s.}}     &            \Rightarrow           &  \xrightarrow{p}                & \Rightarrow & \xrightarrow{d} \end{matrix}$$

These properties, together with a number of other special cases, are summarized in the following list:

{{NumBlk|*::|$$\left. \begin{matrix} X_n\xrightarrow{\overset{}\text{a.s.}} X \\ |X_n| < Y \\ \mathbb{E}[Y] < \infty \end{matrix}\right\} \quad\Rightarrow \quad X_n\xrightarrow X $$|$$}}
 * Almost sure convergence implies convergence in probability: [proof]
 * $$X_n\ \xrightarrow{\text{a.s.}}\ X \quad\Rightarrow\quad  X_n\ \xrightarrow{\overset{}{p}}\ X$$
 * Convergence in probability implies there exists a sub-sequence $$(n_k)$$ which almost surely converges:
 * $$X_n\ \xrightarrow{\overset{}{p}}\ X \quad\Rightarrow\quad  X_{n_k}\ \xrightarrow{\text{a.s.}}\ X$$
 * Convergence in probability implies convergence in distribution: [proof]
 * $$X_n\ \xrightarrow{\overset{}{p}}\ X \quad\Rightarrow\quad X_n\ \xrightarrow{\overset{}{d}}\ X$$
 * Convergence in r-th order mean implies convergence in probability:
 * $$X_n\ \xrightarrow{\overset{}{L^r}}\ X \quad\Rightarrow\quad  X_n\ \xrightarrow{\overset{}{p}}\ X$$
 * Convergence in r-th order mean implies convergence in lower order mean, assuming that both orders are greater than or equal to one:
 * $$X_n\ \xrightarrow{\overset{}{L^r}}\ X \quad\Rightarrow\quad  X_n\ \xrightarrow{\overset{}{L^s}}\ X,$$ provided r ≥ s ≥ 1.
 * If Xn converges in distribution to a constant c, then Xn converges in probability to c: [proof]
 * $$X_n\ \xrightarrow{\overset{}{d}}\ c \quad\Rightarrow\quad X_n\ \xrightarrow{\overset{}{p}}\ c,$$ provided c is a constant.
 * If $X_{n}$ converges in distribution to X and the difference between Xn and Yn converges in probability to zero, then Yn also converges in distribution to X: [proof]
 * $$X_n\ \xrightarrow{\overset{}{d}}\ X,\ \ |X_n-Y_n|\ \xrightarrow{\overset{}{p}}\ 0\ \quad\Rightarrow\quad  Y_n\ \xrightarrow{\overset{}{d}}\ X$$
 * If $X_{n}$ converges in distribution to X and Yn converges in distribution to a constant c, then the joint vector ${X_{n} + Y_{n}}$ converges in distribution to $X_{n}$: [proof]
 * $$X_n\ \xrightarrow{\overset{}{d}}\ X,\ \ Y_n\ \xrightarrow{\overset{}{d}}\ c\ \quad\Rightarrow\quad (X_n,Y_n)\ \xrightarrow{\overset{}{d}}\ (X,c)$$ provided c is a constant.
 * Note that the condition that $$ converges to a constant is important, if it were to converge to a random variable Y then we wouldn't be able to conclude that $X + Y$ converges to $X_{n}$.
 * If Xn converges in probability to X and Yn converges in probability to Y, then the joint vector ${X_{n}Y_{n}}$ converges in probability to ${X_{n}}$: [proof]
 * $$X_n\ \xrightarrow{\overset{}{p}}\ X,\ \ Y_n\ \xrightarrow{\overset{}{p}}\ Y\ \quad\Rightarrow\quad (X_n,Y_n)\ \xrightarrow{\overset{}{p}}\ (X,Y)$$
 * If $r$ converges in probability to X, and if ${φ_{n}}$ for all n and some b, then $X_{n}$ converges in rth mean to X for all $ε > 0$. In other words, if $r$ converges in probability to X and all random variables $r$ are almost surely bounded above and below, then $$ converges to X also in any rth mean.
 * Almost sure representation. Usually, convergence in distribution does not imply convergence almost surely. However, for a given sequence {Xn} which converges in distribution to X0 it is always possible to find a new probability space (Ω, F, P) and random variables {Yn, n = 0, 1, ...} defined on it such that Yn is equal in distribution to $X_{n}$ for each $(S, d)$, and Yn converges to Y0 almost surely.
 * If for all ε > 0,
 * $$\sum_n \mathbb{P} \left(|X_n - X| > \varepsilon\right) < \infty,$$
 * then we say that $X_{n}$ converges almost completely, or almost in probability towards X. When $X_{n}$ converges almost completely towards X then it also converges almost surely to X. In other words, if $X_{n}$ converges in probability to X sufficiently quickly (i.e. the above sequence of tail probabilities is summable for all $r ≥ 1$), then $a$ also converges almost surely to X. This is a direct implication from the Borel–Cantelli lemma.
 * If $b$ is a sum of n real independent random variables:
 * $$S_n = X_1+\cdots+X_n \, $$
 * then $a$ converges almost surely if and only if $b$ converges in probability.
 * The dominated convergence theorem gives sufficient conditions for almost sure convergence to imply L1-convergence:

 * A necessary and sufficient condition for L1 convergence is $$X_n\xrightarrow{\overset{}{P}} X$$ and the sequence (Xn) is uniformly integrable.
 * If $$X_n\ \xrightarrow{\overset{}{p}}\ X $$, the followings are equivalent
 * $$X_n\ \xrightarrow{\overset{}{L^r}}\ X$$,
 * $$ \mathbb{E}[|X_n|^r] \rightarrow \mathbb{E}[|X|^r] < \infty $$,
 * $$\{|X_n|^r\}$$ is uniformly integrable.
 * If $$X_n$$ are discrete and independent, then $$X_n \stackrel{p}{\rightarrow} X$$ implies that $$X_n \stackrel{a.s.}{\rightarrow} X$$. This is a consequence of the second Borel–Cantelli lemma.

title: Convergence of random variables


text:
Strong convergence

In mathematics, strong convergence may refer to:

 * The strong convergence of random variables of a probability distribution.
 * The norm-convergence of a sequence in a Hilbert space (as opposed to weak convergence).
 * The convergence of operators in the strong operator topology.

title: Strong convergence


text:
Weak convergence

In mathematics, weak convergence may refer to:
 * Weak convergence of random variables of a probability distribution
 * Weak convergence of measures, of a sequence of probability measures
 * Weak convergence (Hilbert space) of a sequence in a Hilbert space
 * more generally, convergence in weak topology in a Banach space or a topological vector space

title: Weak convergence


text:
Extreme value theory

Extreme value theory or extreme value analysis (EVA) is a branch of statistics dealing with the extreme deviations from the median of probability distributions. It seeks to assess, from a given ordered sample of a given random variable, the probability of events that are more extreme than any previously observed. Extreme value analysis is widely used in many disciplines, such as structural engineering, finance, economics, earth sciences, traffic prediction, and geological engineering. For example, EVA might be used in the field of hydrology to estimate the probability of an unusually large flooding event, such as the 100-year flood. Similarly, for the design of a breakwater, a coastal engineer would seek to estimate the 50 year wave and design the structure accordingly.

Data analysis
Two main approaches exist for practical extreme value analysis.

The first method relies on deriving block maxima (minima) series as a preliminary step. In many situations it is customary and convenient to extract the annual maxima (minima), generating an annual maxima series (AMS).

The second method relies on extracting, from a continuous record, the peak values reached for any period during which values exceed a certain threshold (falls below a certain threshold). This method is generally referred to as the peak over threshold method (POT).

For AMS data, the analysis may partly rely on the results of the Fisher–Tippett–Gnedenko theorem, leading to the generalized extreme value distribution being selected for fitting. However, in practice, various procedures are applied to select between a wider range of distributions. The theorem here relates to the limiting distributions for the minimum or the maximum of a very large collection of independent random variables from the same distribution. Given that the number of relevant random events within a year may be rather limited, it is unsurprising that analyses of observed AMS data often lead to distributions other than the generalized extreme value distribution (GEVD) being selected.

For POT data, the analysis may involve fitting two distributions: One for the number of events in a time period considered and a second for the size of the exceedances.

A common assumption for the first is the Poisson distribution, with the generalized Pareto distribution being used for the exceedances. A tail-fitting can be based on the Pickands–Balkema–de Haan theorem.

Novak (2011) reserves the term "POT method" to the case where the threshold is non-random, and distinguishes it from the case where one deals with exceedances of a random threshold.

Applications
Applications of extreme value theory include predicting the probability distribution of:

 * Extreme floods; the size of freak waves
 * Tornado outbreaks
 * Maximum sizes of ecological populations
 * Side effects of drugs (e.g., ximelagatran)
 * The magnitudes of large insurance losses
 * Equity risks; day-to-day market risk
 * Mutation events during evolution
 * Large wildfires
 * Environmental loads on structures
 * Time the fastest humans could ever run the 100 metres sprint and performances in other athletic disciplines
 * Pipeline failures due to pitting corrosion
 * Anomalous IT network traffic, prevent attackers from reaching important data
 * Road safety analysis
 * Wireless communications
 * Epidemics
 * Neurobiology
 * Solar energy
 * Extreme Space weather

History
The field of extreme value theory was pioneered by L. Tippett (1902–1985). Tippett was employed by the British Cotton Industry Research Association, where he worked to make cotton thread stronger. In his studies, he realized that the strength of a thread was controlled by the strength of its weakest fibres. With the help of R.A. Fisher, Tippet obtained three asymptotic limits describing the distributions of extremes assuming independent variables. E.J. Gumbel (1958) codified this theory. These results can be extended to allow for slight correlations between variables, but the classical theory does not extend to strong correlations of the order of the variance. One universality class of particular interest is that of log-correlated fields, where the correlations decay logarithmically with the distance.

Univariate theory
The theory for extreme values of a single variable is governed by the extreme value theorem, also called the Fisher–Tippett–Gnedenko theorem, which describes which of the three possible distributions for extreme values applies for a particular statistical variable $$\ X\ ,$$ which is summarized in this section.

Let $$\ X_1,\ \dots\ X_n\ $$ be a sample of independent and identically distributed random variables with cumulative distribution function $$\ F\ $$ and let $$\ M_n = \max\ \{\ X_1,\ \dots\ X_n\ \}\ $$ denote the sample maximum.

In theory, the exact distribution of the maximum can be derived:

\begin{align} \boldsymbol{\operatorname\mathcal P} \left\{\ M_n \leq z\ \right\} & = \boldsymbol{\operatorname\mathcal P}\left\{\ X_1 \leq z,\ \dots\ ,\ X_n \leq z\ \right\} \\ & = \boldsymbol{\operatorname\mathcal P}\left\{\ X_1 \leq z\ \right\}\times \cdots \times \boldsymbol{\operatorname\mathcal P} \left\{\ X_n \leq z\ \right\} = \bigl(\ F(z)\ \bigr)^n ~. \end{align} $$

The value of the associated indicator function $$\ I_n = \boldsymbol{\operatorname \mathcal I} \left[\ M_n > z\ \right]\ $$ is a Bernoulli process with a success probability $$\ p(z) = 1 - \bigl(\ F(z)\ \bigr)^n\ $$ that depends on the magnitude $$\ z\ $$ of the extreme event. The number of extreme events within $$\ n\ $$ trials thus follows a binomial distribution and the number of trials until an event occurs follows a geometric distribution with expected value and standard deviation of the same order $$\ \boldsymbol{\operatorname\mathcal O}\left( \tfrac{\ 1\ }{ p(z) } \right) ~.$$

In practice, we might not have the distribution function $$\ F\ $$ but the Fisher–Tippett–Gnedenko theorem provides an asymptotic result. If there exist sequences of paired constants $$\ ( a_n, b_n)\ ,$$ with $$\ a_n > 0\ $$ and $$\ b_n \in \mathbb R\ ,$$ such that

 * $$ \boldsymbol{\operatorname\mathcal P} \left\{ \frac{\ M_n - b_n\ }{\ a_n\ } \leq z \right\} \rightarrow G(z) $$

as $$\ n \rightarrow \infty\ $$ then

 * $$ G(z)\ \propto\ \exp \left[ -\bigl( 1 + \gamma \cdot z \bigr)^{ -\frac{\ 1\ }{ \gamma } } \right] $$

where the parameter $$\ \gamma\ $$ depends on how steeply of the distribution's tail(s) diminish (called "ordinary" tail(s), "thin" tail(s), and "fat" tail(s), with the normal distribution put in the "thin" tailed group instead of "ordinary" for this context, at least). When normalized, $$\ G\ $$ belongs to one of the following non-degenerate distribution families:

 * Type 1: Gumbel distribution, for $$\ \gamma = 0\ $$
 * $$ G(z) = \exp\left[ -\exp\left( -\tfrac{\ z\ -\ b\ }{ a } \right) \right] $$
 * when the distribution of $$\ M_n\ $$ has an "ordinary" exponentially diminishing tail.

 * Type 2: Fréchet distribution, for $$\ \gamma < 0\ $$
 * $$ G(z) = \begin{cases} 0 \quad & z \leq b \\

\exp\left[ -\left( \tfrac{\ z\ -\ b\ }{ a } \right)^{-\left| \gamma \right|} \right] & z > b \end{cases} $$
 * when the distribution of $$\ M_n\ $$ has a heavy tail (including polynomial decay).

 * Type 3: Weibull distribution, $$\ \gamma > 0\ $$
 * $$ G(z) = \begin{cases} \exp \left[ -\left( -\tfrac{\ z\ -\ b\ }{ a } \right)^{ \gamma } \right] & z < b \\ 1 & z \geq b \end{cases}\ $$ for $$\ z \in \mathbb{R}\ $$
 * when the distribution of $$\ M_n\ $$ has a thin tail with finite upper bound.

Multivariate theory
Extreme value theory in more than one variable introduces additional issues that have to be addressed. One problem that arises is that one must specify what constitutes an extreme event. Although this is straightforward in the univariate case, there is no unambiguous way to do this in the multivariate case. The fundamental problem is that although it is possible to order a set of real-valued numbers, there is no natural way to order a set of vectors.

As an example, in the univariate case, given a set of observations $$\ x_i\ $$ it is straightforward to find the most extreme event simply by taking the maximum (or minimum) of the observations. However, in the bivariate case, given a set of observations $$\ ( x_i, y_i )\ $$, it is not immediately clear how to find the most extreme event. Suppose that one has measured the values $$\ (3, 4)\ $$ at a specific time and the values $$\ (5, 2)\ $$ at a later time. Which of these events would be considered more extreme? There is no universal answer to this question.

Another issue in the multivariate case is that the limiting model is not as fully prescribed as in the univariate case. In the univariate case, the model (GEV distribution) contains three parameters whose values are not predicted by the theory and must be obtained by fitting the distribution to the data. In the multivariate case, the model not only contains unknown parameters, but also a function whose exact form is not prescribed by the theory. However, this function must obey certain constraints. It is not straightforward to devise estimators that obey such constraints though some have been recently constructed.

As an example of an application, bivariate extreme value theory has been applied to ocean research.

Non-stationary extremes
Statistical modeling for nonstationary time series was developed in the 1990s. Methods for nonstationary multivariate extremes have been introduced more recently. The latter can be used for tracking how the dependence between extreme values changes over time, or over another covariate.

Software

 * — Package for extreme value statistics in R.
 * — Package for extreme value statistics in Julia.

title: Extreme value theory


text:
Prabuty

Prabuty (Riesenburg) is a town in Kwidzyn County within the Pomeranian Voivodeship of northern Poland. It is the seat of Gmina Prabuty.

Geographical location
Prabuty is located between the Liwieniec and Sowica lakes, approximately 18 kilometers east of Kwidzyn, 100 kilometers southeast of Gdańsk, 100 kilometers west of Olsztyn, and 133 kilometers southwest of Kaliningrad.

Prabuty is a rail junction on the Warszawa–Gdynia railway.

History
In 1236, the Teutonic Knights under Henry III, Margrave of Meissen, destroyed an Old Prussian fortress between the lakes Dzierzgoń and Liwieniec. The settlement was first mentioned in 1250 as Riesenburg. The village grew around the castle and received Culm law city rights on 30 October 1330 from bishop Rudolf of Pomerania (1322–1332). In 1379 the town was visited by Lithuanian duke Švitrigaila. In 1410 and 1414 it was captured by the Poles.

Knights and squires of the Prabuty district were co-founders of the anti-Teutonic Prussian Confederation in 1440. Since its establishment, part of the population wanted the town to join the organization. In 1451, the town council eventually joined the Prussian Confederation, but bishop Kaspar Linke expelled the councilors and confiscated their property. The town was accepted again by the organization in February 1454, and upon the request of the organization, in March 1454, Polish King Casimir IV Jagiellon incorporated the region and town to the Kingdom of Poland, and the Thirteen Years' War broke out. The Bishop and canons of Pomesania also pledged allegiance to the Polish King. Around that time, the town was mentioned in documents as Prabuth. After the Battle of Chojnice, in which Polish forces were defeated, the town was forced to side with the Order again. After the war and the Second Peace of Thorn (1466), the town became a part of Poland as a fief, and Pomesanian bishops retained their rule over the area. In 1525 the town became part of Ducal Prussia, a vassal state of Poland. In 1556, a synod was held in the town.

The town suffered during the 17th century Polish-Swedish wars. In 1628, half of it was burnt down, and in 1688 the remainder was burned. In 1722, fire caused destruction once again.

In 1701, as part of Ducal Prussia, the town became a part of the Kingdom of Prussia and part of the newly created province of West Prussia in 1773. Despite this, as of 1789, Polish Protestant church services were still held in the town, and there was a Polish municipal school there. In October 1831, several Polish cavalry and infantry units and honor guards of the November Uprising stopped in the town on the way to their internment places. In 1871, the town became part of the German Empire in the framework of the Prussian-led unification of Germany. Until 1919, Riesenburg belonged to the administrative district of Regierungsbezirk Marienwerder in the Province of West Prussia.

After World War I, a referendum was held concerning the future nationality of the town, which remained part of Weimar Germany. From 1920 to 1939, Riesenburg belonged to the administrative district of Regierungsbezirk Westpreußen in the Province of East Prussia and from 1939 until 1945 to the district of Regierungsbezirk Marienwerder in the province of Reichsgau Danzig-West Prussia.

During World War II Germany operated a prisoner-of-war camp in the town. The town was captured by the Soviet Red Army in 1945 in the final months of the war. It then became again part of Poland. Most of the German inhabitants were expelled in accordance with the Potsdam Agreement and the pre-war Polish population was joined by Poles displaced from former eastern Poland annexed by the Soviet Union.

Heinz Heydrich (1905–44), brother of Reinhard Heydrich, is buried in the local military cemetery, according to the Deutsche Dienststelle (WASt).

Sports
The local football club is Pogoń Prabuty. It competes in the lower leagues.

Notable residents

 * Peter I, Grand Duke of Oldenburg (1755–1829) the Regent of the Duchy of Oldenburg for his incapacitated cousin William I from 1785 to 1823 and then served himself as Duke from 1823 to 1829.
 * Adolf Treichel (1869–1926), German politician, President of the Volkstag (Free City of Danzig)
 * (1886–1962), Polish activist
 * Charlotte Wolff (1897–1986) a German-British physician who worked as a psychotherapist and wrote on sexology and hand analysis. Her writings on lesbianism and bisexuality were influential early works in the field.
 * (1936–2010), Polish sculptor
 * (born 1940), Polish jurist, former member of the State Tribunal of Poland

title: Prabuty