Conservapedia - User contributions [en]

Schrodinger equation

2009-09-29T13:38:06Z

Guardianofrice:

The '''Schrodinger equation''' is a linear [[differential equation]] used in various fields of [[physics]] to describe the time evolution of quantum states. It is a fundamental aspect of [[quantum mechanics]]. The equation is named for its discoverer, [[Erwin Schrodinger]].

==Mathematical forms==
===General time-dependent form===
The Schrodinger equation may generally be written

<math>i\hbar\frac{\partial}{\partial t}|\Psi\rangle=\hat H|\Psi\rangle</math>

where <math>i</math> is the [[complex number|imaginary unit]], <math>\hbar</math> is [[Planck's constant]] divided by <math>2\pi</math>, <math>|\Psi\rangle</math> is the quantum mechanical state or [[wavefunction]] (expressed here in [[Dirac notation]]), and <math>\hat H</math> is the [[Hamiltonian]] operator.

The left side of the equation describes how the wavefunction changes with time; the right side is related to its energy. For the simplest case of a particle of mass m moving in a one-dimensional potential V(x), the Schrodinger equation can be written

<math>
-\frac{\hbar^2}{2m}\frac{\partial^2\psi}{\partial x^2}+V(x)\psi=i\hbar\frac{\partial \psi}{\partial t}
</math>

===Derivation===
The following derivation, likely one that Schrodinger followed himself, is a completely non-rigorous method which takes a more intuitive approach. Whatever ambiguities arose in this derivation because of its questionable assumptions were wiped out by subsequent experiments verifying again and again the equation's ability to predict probabilities of particle location<ref>French, A.P. and Taylor, E.F.. ''An Introduction to Quantum Physics''. CRC Press, Boca Raton, FL. Copyright MIT 1978.</ref>.

To start out, we assume that Planck and Einstein's quantized energy equation, <math>E = h\nu</math> where E is the energy, h is planck's constant, and nu is the frequency, is correct. We also assume that DeBroglie's wavelength of particles equation, <math>\lambda_{dB} = \frac{h}{p}</math> where lambda is the wavelength, h is planck's constant, and p is the momentum of the particle, is correct (these were indeed questionable assumptions during Schrodinger's time). Now we can rewrite the energy equation by multiplying and dividing the right hand side by <math>2 \pi</math> and turn it into

<math>E = \frac{h\omega}{2\pi}=\hbar\omega</math>

where omega is now the angular frequency. We can also rewrite DeBroglie's equation in the same way when divided by <math>2\pi</math>, turning it into

<math>\frac{\lambda}{2\pi} = \frac{\hbar}{p} or \frac{1}{k} = \frac{\hbar}{p}</math>

where k is the [[wavenumber]] of wavelength <math>\lambda_dB</math>.

Now, we can derive energy equations using classical Newtonian mechanics and plug our results in from the new developments above. Energy in classical terms is

<math>E = K_E + V_E = \frac{p^2}{2m} + V</math>

where
E = energy, <math>K_E = \frac{1}{2}mv^2</math> (kinetic energy), <math>V_E = V</math> (potential energy)

Note that we've rewritten kinetic energy in terms of momentum, <math>p = mv</math>. Subbing in energy from the quantized energy equation and momentum from DeBroglie's equation, we obtain

<math>\hbar\omega = \frac{\hbar^2k^2}{2m} + V</math>

Notice this looks kind of like a quasi-wave equation. The one-dimensional [[wave equation]] from classical mechanics is given by the equation

<math>\frac{\partial^2y(x,t)}{\partial x^2} = \frac{1}{c^2}\frac{\partial^2y(x,t)}{\partial t^2}</math>

In classical wave mechanics, we assume the solution to be of the form <math>y = Ae^{i(kx-\omega t)}</math>. So the <math>\omega</math> in the left hand of the energy equation looks like a single partial derivative with respect to t, and the <math>k^2</math> on the right hand side looks like two partials with respect to x. If we assume a similar solution to the energy equation as in the classical wave equation but ''retain the imaginary parts'', we can set

<math>\Psi(x,t) = Ae^{i(kx-\omega t)}</math>

and each of its partials with respect to each corresponding side of the energy equation equal to each other. For now, we take ''V = 0'' for simplicity. For the partial with respect to x:

<math>\frac{\partial^2\Psi}{\partial x^2} = -k^2\Psi = \frac{-p^2}{\hbar^2}\Psi = \frac{-2mE}{\hbar^2}\Psi </math>

and with respect to t (stop at one partial derivative):

<math>\frac{\partial\Psi}{\partial t} = -i\omega\Psi = \frac{-iE}{\hbar}\Psi </math>

There are <math>E\Psi</math> terms in both equations, and we can equate those two together and get:

<math>\frac{-\hbar^2}{2m}\frac{\partial^2\Psi}{\partial x^2} = i\hbar\frac{\partial\Psi}{\partial t}</math>

Almost in its final form, we sub in the fact that <math> E = K_E + V_E </math> and realize this will just add an additional <math>V\Psi</math> term to the equation, we obtain the final form

<math>\frac{-\hbar^2}{2m}\frac{\partial^2\Psi}{\partial x^2} + V(x)\Psi = i\hbar\frac{\partial\Psi}{\partial t}</math>

===Eigenvalue problems===
In many instances, steady-state solutions to the equation are of great interest. Physically, these solutions correspond to situations in which the wavefunction has a well-defined [[energy]]. The energy is then said to be an [[eigenvalue]] for the equation, and the wavefunction corresponding to that energy is called an [[eigenfunction]] or [[eigenstate]]. In such cases, the Schrodinger equation is time-independent and is often written

<math>E\psi=\hat H\psi</math>

Here, ''E'' is energy, ''H'' is once again the Hamiltonian operator, and <math>\psi</math> is the energy eigenstate for ''E''.

One example of this type of eigenvalue problem is an electrons bound inside an [[atom]].

==Examples for the time-independent equation==
===Free particle in one dimension===
In this case, <math>V(x)=0</math> and so we see that the solution to the Schrodinger equation must be

<math>\psi=Ae^{-ikx}</math>

with energy given by

<math>E=\frac{\hbar^2 k^2}{2m}</math>

Physically, this corresponds to a wave traveling with a [[momentum]] given by <math>\hbar k</math>, where k can in principle take any value.

===Particle in a box===
Consider a one-dimensional box of width a, where the potential energy is 0 inside the box and infinite outside of it. This means that <math>\psi</math> must be zero outside the box. One can verify (by substituting into the Schrodinger equation) that

<math>\psi=\sin(kx)</math>

is a solution if <math>k=n\pi</math> where n is any integer. Thus, rather than the continuum of solutions for the free particle, for the particle in a box there is a set of discrete solutions with energies given by

<math>E_n=\frac{\hbar^2 k^2}{2m}=\frac{\hbar^2n^2\pi^2}{2m}</math>

==References==
<references/>

[[Category:Physics]]

Schrodinger equation

2009-09-28T23:08:24Z

Guardianofrice:

The '''Schrodinger equation''' is a linear [[differential equation]] used in various fields of [[physics]] to describe the time evolution of quantum states. It is a fundamental aspect of [[quantum mechanics]]. The equation is named for its discoverer, [[Erwin Schrodinger]].

==Mathematical forms==
===General time-dependent form===
The Schrodinger equation may generally be written

<math>i\hbar\frac{\partial}{\partial t}|\Psi\rangle=\hat H|\Psi\rangle</math>

where <math>i</math> is the [[complex number|imaginary unit]], <math>\hbar</math> is [[Planck's constant]] divided by <math>2\pi</math>, <math>|\Psi\rangle</math> is the quantum mechanical state or [[wavefunction]] (expressed here in [[Dirac notation]]), and <math>\hat H</math> is the [[Hamiltonian]] operator.

The left side of the equation describes how the wavefunction changes with time; the right side is related to its energy. For the simplest case of a particle of mass m moving in a one-dimensional potential V(x), the Schrodinger equation can be written

<math>
-\frac{\hbar^2}{2m}\frac{\partial^2\psi}{\partial x^2}+V(x)\psi=i\hbar\frac{\partial \psi}{\partial t}
</math>

===Derivation===
The following derivation, likely one that Schrodinger followed himself, is a completely non-rigorous method which takes a more intuitive approach. Whatever ambiguities arose in this derivation because of its questionable assumptions were wiped out by subsequent experiments verifying again and again the equation's ability to predict probabilities of particle location<ref>French, A.P. and Taylor, E.F.. ''An Introduction to Quantum Physics''. CRC Press, Boca Raton, FL. Copyright MIT 1978.</ref>.

To start out, we assume that Planck and Einstein's quantized energy equation, <math>E = h\nu</math> where E is the energy, h is planck's constant, and nu is the frequency, is correct. We also assume that DeBroglie's wavelength of particles equation, <math>\lambda_{dB} = \frac{h}{p}</math> where lambda is the wavelength, h is planck's constant, and p is the momentum of the particle, is correct (these were indeed questionable assumptions during Schrodinger's time). Now we can rewrite the energy equation by multiplying and dividing the right hand side by <math>2 \pi</math> and turn it into

<math>E = \frac{h\omega}{2\pi}=\hbar\omega</math>

where omega is now the angular frequency. We can also rewrite DeBroglie's equation in the same way when divided by <math>2\pi</math>, turning it into

<math>\frac{\lambda}{2\pi} = \frac{\hbar}{p} or \frac{1}{k} = \frac{\hbar}{p}</math>

where k is the [[wavenumber]] of wavelength <math>\lambda_dB</math>.

Now, we can derive energy equations using classical Newtonian mechanics and plug our results in from the new developments above. Energy in classical terms is

<math>E = K_E + V_E = \frac{p^2}{2m} + V</math>

where
E = energy, <math>K_E = frac{1}{2}mv^2</math> (kinetic energy), <mathV_E = V</math> (potential energy)

Note that we've rewritten kinetic energy in terms of momentum, <math>p = mv</math>. Subbing in energy from the quantized energy equation and momentum from DeBroglie's equation, we obtain

<math>\hbar\omega = \frac{\hbar^2k^2}{2m} + V</math>

Notice this looks kind of like a quasi-wave equation. The one-dimensional [[wave equation]] from classical mechanics is given by the equation

<math>\frac{\partial^2y(x,t)}{\partial x^2} = \frac{1}{c^2}\frac{\partial^2y(x,t)}{\partial t^2}</math>

In classical wave mechanics, we assume the solution to be of the form <math>y = Ae^{i(kx-\omega t)}</math>. So the <math>\omega</math> in the left hand of the energy equation looks like a single partial derivative with respect to t, and the <math>k^2</math> on the right hand side looks like two partials with respect to x. If we assume a similar solution to the energy equation as in the classical wave equation but ''retain the imaginary parts'', we can set

<math>\Psi(x,t) = Ae^{i(kx-\omega t)}</math>

and each of its partials with respect to each corresponding side of the energy equation equal to each other. For now, we take ''V = 0'' for simplicity. For the partial with respect to x:

<math>\frac{\partial^2\Psi}{\partial x^2} = -k^2\Psi = \frac{-p^2}{\hbar^2}\Psi = \frac{-2mE}{\hbar^2}\Psi </math>

and with respect to t (stop at one partial derivative):

<math>\frac{\partial\Psi}{\partial t} = -i\omega\Psi = \frac{-iE}{\hbar}\Psi </math>

There are <math>E\Psi</math> terms in both equations, and we can equate those two together and get:

<math>\frac{-\hbar^2}{2m}\frac{\partial^2\Psi}{\partial x^2} = i\hbar\frac{\partial\Psi}{\partial t}</math>

Almost in its final form, we sub in the fact that <math> E = K_E + V_E </math> and realize this will just add an additional <math>V\Psi</math> term to the equation, we obtain the final form

<math>\frac{-\hbar^2}{2m}\frac{\partial^2\Psi}{\partial x^2} + V(x)\Psi = i\hbar\frac{\partial\Psi}{\partial t}</math>

===Eigenvalue problems===
In many instances, steady-state solutions to the equation are of great interest. Physically, these solutions correspond to situations in which the wavefunction has a well-defined [[energy]]. The energy is then said to be an [[eigenvalue]] for the equation, and the wavefunction corresponding to that energy is called an [[eigenfunction]] or [[eigenstate]]. In such cases, the Schrodinger equation is time-independent and is often written

<math>E\psi=\hat H\psi</math>

Here, ''E'' is energy, ''H'' is once again the Hamiltonian operator, and <math>\psi</math> is the energy eigenstate for ''E''.

One example of this type of eigenvalue problem is an electrons bound inside an [[atom]].

==Examples for the time-independent equation==
===Free particle in one dimension===
In this case, <math>V(x)=0</math> and so we see that the solution to the Schrodinger equation must be

<math>\psi=Ae^{-ikx}</math>

with energy given by

<math>E=\frac{\hbar^2 k^2}{2m}</math>

Physically, this corresponds to a wave traveling with a [[momentum]] given by <math>\hbar k</math>, where k can in principle take any value.

===Particle in a box===
Consider a one-dimensional box of width a, where the potential energy is 0 inside the box and infinite outside of it. This means that <math>\psi</math> must be zero outside the box. One can verify (by substituting into the Schrodinger equation) that

<math>\psi=\sin(kx)</math>

is a solution if <math>k=n\pi</math> where n is any integer. Thus, rather than the continuum of solutions for the free particle, for the particle in a box there is a set of discrete solutions with energies given by

<math>E_n=\frac{\hbar^2 k^2}{2m}=\frac{\hbar^2n^2\pi^2}{2m}</math>

==References==
<references/>

[[Category:Physics]]

Schrodinger equation

2009-09-28T22:41:24Z

Guardianofrice:

The '''Schrodinger equation''' is a linear [[differential equation]] used in various fields of [[physics]] to describe the time evolution of quantum states. It is a fundamental aspect of [[quantum mechanics]]. The equation is named for its discoverer, [[Erwin Schrodinger]].

==Mathematical forms==
===General time-dependent form===
The Schrodinger equation may generally be written

<math>i\hbar\frac{\partial}{\partial t}|\Psi\rangle=\hat H|\Psi\rangle</math>

where <math>i</math> is the [[complex number|imaginary unit]], <math>\hbar</math> is [[Planck's constant]] divided by <math>2\pi</math>, <math>|\Psi\rangle</math> is the quantum mechanical state or [[wavefunction]] (expressed here in [[Dirac notation]]), and <math>\hat H</math> is the [[Hamiltonian]] operator.

The left side of the equation describes how the wavefunction changes with time; the right side is related to its energy. For the simplest case of a particle of mass m moving in a one-dimensional potential V(x), the Schrodinger equation can be written

<math>
-\frac{\hbar^2}{2m}\frac{\partial^2\psi}{\partial x^2}+V(x)\psi=i\hbar\frac{\partial \psi}{\partial t}
</math>

===Derivation===
The following derivation, likely one that Schrodinger followed himself, is a completely non-rigorous method which takes a more intuitive approach. Whatever ambiguities arose in this derivation because of its questionable assumptions were wiped out by subsequent experiments verifying again and again the equation's ability to predict probabilities of particle location<ref>French, A.P. and Taylor, E.F.. ''An Introduction to Quantum Physics''. CRC Press, Boca Raton, FL. Copyright MIT 1978.</ref>.

To start out, we assume that Planck and Einstein's quantized energy equation, <math>E = h\nu</math> where E is the energy, h is planck's constant, and nu is the frequency, is correct. We also assume that DeBroglie's wavelength of particles equation, <math>\lambda_dB = \frac{h}{p}</math> where lambda is the wavelength, h is planck's constant, and p is the momentum of the particle, is correct (these were indeed questionable assumptions during Schrodinger's time). Now we can rewrite the energy equation by multiplying and dividing the right hand side by <math>2 \pi</math> and turn it into

<math>E = \frac{h\omega}{2\pi}=\hbar\omega</math>

where omega is now the angular frequency. We can also rewrite DeBroglie's equation in the same way when divided by <math>2\pi</math>, turning it into

<math>\frac{\lambda}{2\pi} = \frac{\hbar}{p} or \frac{1}{k} = \frac{\hbar}{p}</math>

where k is the [[wavenumber]] of wavelength <math>\lambda_dB</math>.

Now, we can derive energy equations using classical Newtonian mechanics and plug our results in from the new developments above. Energy in classical terms is

<math>E = K_E + V_E = \frac{p^2}{2m} + V</math>

where
E = energy, <math>K_E = frac{1}{2}mv^2</math> (kinetic energy), <mathV_E = V</math> (potential energy)

Note that we've rewritten kinetic energy in terms of momentum, <math>p = mv</math>. Subbing in energy from the quantized energy equation and momentum from DeBroglie's equation, we obtain

<math>\hbar\omega = \frac{\hbar^2k^2}{2m} + V</math>

Notice this looks kind of like

===Eigenvalue problems===
In many instances, steady-state solutions to the equation are of great interest. Physically, these solutions correspond to situations in which the wavefunction has a well-defined [[energy]]. The energy is then said to be an [[eigenvalue]] for the equation, and the wavefunction corresponding to that energy is called an [[eigenfunction]] or [[eigenstate]]. In such cases, the Schrodinger equation is time-independent and is often written

<math>E\psi=\hat H\psi</math>

Here, ''E'' is energy, ''H'' is once again the Hamiltonian operator, and <math>\psi</math> is the energy eigenstate for ''E''.

One example of this type of eigenvalue problem is an electrons bound inside an [[atom]].

==Examples for the time-independent equation==
===Free particle in one dimension===
In this case, <math>V(x)=0</math> and so we see that the solution to the Schrodinger equation must be

<math>\psi=Ae^{-ikx}</math>

with energy given by

<math>E=\frac{\hbar^2 k^2}{2m}</math>

Physically, this corresponds to a wave traveling with a [[momentum]] given by <math>\hbar k</math>, where k can in principle take any value.

===Particle in a box===
Consider a one-dimensional box of width a, where the potential energy is 0 inside the box and infinite outside of it. This means that <math>\psi</math> must be zero outside the box. One can verify (by substituting into the Schrodinger equation) that

<math>\psi=\sin(kx)</math>

is a solution if <math>k=n\pi</math> where n is any integer. Thus, rather than the continuum of solutions for the free particle, for the particle in a box there is a set of discrete solutions with energies given by

<math>E_n=\frac{\hbar^2 k^2}{2m}=\frac{\hbar^2n^2\pi^2}{2m}</math>

==References==
<references/>

[[Category:Physics]]

Wave equation

2009-03-05T03:20:52Z

Guardianofrice:

The wave equation is an important differential relationship in physics. It describes how [[waves]] propagate through mediums, whether they be [[transverse waves]] (e.g. electromagnetic radiation) or [[longitudinal waves]] (e.g. sound waves).

==One-Dimensional Wave Equation==
One type of wave equation is the one-dimensional wave equation. The mathematical relation describes a wave whose parts only oscillate in ''one dimension''. This wave, however, can ''propagate'' in all three spacial dimensions. An example would be a vibrating rope or string with both ends fixed.

The one-dimensional wave equation can be written<ref>Pain, H.J. ''The Physics of Vibrations and Waves'' 6th edition. Southern Gate, Chichester, West Sussex, England: John Wiley & Sons, 2005</ref>:

<math> \frac{\partial^2 y}{dx^2} = \frac{1}{v^2} \frac{\partial^2 y}{dt^2} </math>

where y = y(x,t), the y-direction of motion at a point x that changes with time t,
and v = the [[phase velocity]] of the wave.

==References==
<references/>

[[category:physics]]

Wave equation

2009-03-05T03:20:01Z

Guardianofrice:

The wave equation is an important differential relationship in physics. It describes how [[waves]] propagate through mediums, whether they be [transverse waves] (e.g. electromagnetic radiation) or [longitudinal waves] (e.g. sound waves).

==One-Dimensional Wave Equation==
One type of wave equation is the one-dimensional wave equation. The mathematical relation describes a wave whose parts only oscillate in ''one dimension''. This wave, however, can ''propagate'' in all three spacial dimensions. An example would be a vibrating rope or string with both ends fixed.

The one-dimensional wave equation can be written<ref>Pain, H.J. ''The Physics of Vibrations and Waves'' 6th edition. Southern Gate, Chichester, West Sussex, England: John Wiley & Sons, 2005</ref>:

<math> \frac{\partial^2 y}{dx^2} = \frac{1}{v^2} \frac{\partial^2 y}{dt^2} </math>

where y = y(x,t), the y-direction of motion at a point x that changes with time t,
and v = the [[phase velocity]] of the wave.

==References==
<references/>

[[category:physics]]

Wave equation

2009-03-05T03:19:38Z

Guardianofrice: New page: The wave equation is an important differential relationship in physics. It describes how waves propagate through mediums, whether they be [transverse waves] (e.g. electromagnetic radia...

Chain rule

2009-03-01T22:53:11Z

Guardianofrice:

{{Template:Math-h}}

The '''chain rule''' in [[calculus]] is a formula for determining the [[derivative]] of a [[functional composition|composite function]]:

:<math>f(g(x))' = f'(g(x))\times g'(x)</math>

The chain rule can also be expressed as:
:<math>\frac {dy}{dx} = \frac {dy} {du} \times \frac {du}{dx}.</math>

The chain rule can also be applied to multivariable functions. The derivative of a multivariable function is expressed as follows:

<math> \frac {d}{dt}(f(x(t), y(t))) = \frac{\partial f}{dx}\times \frac{dx}{dt} + \frac{\partial f}{dy}\times \frac{dy}{dt} </math>

or in vector notation:

<math> \nabla f \cdot \frac {dr}{dt} </math>

where r is the vector function

<math> r = <x(t), y(t), z(t) ... > </math>

The function r is sometimes called the ''path'' of the particle.

[[category:Calculus]]
[[category:differentiation]]

Chain rule

2009-03-01T17:40:01Z

Guardianofrice:

Differential equation

2009-02-28T18:26:00Z

Guardianofrice:

In [[mathematics]], a '''differential equation''' is an [[equation]] that relates a [[function]] to one or more of its [[derivative]]s. Differential equations are especially applicable when the tools of algebra, which are ideally suited for static systems, are not enough. Many physical systems are modeled by solving differential equations, although their usefulness extends well into other fields of science such as chemistry and economics.

==Types of Differential Equations==
There are two main types of differential equations: Ordinary Differential Equations and Partial Differential Equations.<ref>Edwards, Henry C. and Penney, David E.. ''Differential Equations and Boundary Value Problems'' 4th Edition. Upper Saddle River, NJ: Pearson, 2008</ref>

The former is simpler of the two, as it can be written in the normal form

<math>\frac{d^{(n)}y}{dx^{(n)}} = F(x, y, \frac{dy}{dx}, \frac{d^2y}{dx^2}, \frac{d^3y}{dx^3}, ... , \frac{d^{(n-1)}y}{dx^{(n-1)}})</math>

for a simple function <math>y = g(x)</math>

The function F consists of the function y and its derivatives up to the ''nth'' order. Notice that y is comprised of only one independent variable x. A differential equation is considered ''ordinary'' if the function y in F is dependent on only one variable. It is important to note that, while most ordinary differential equations can be written in the normal form (isolating the highest derivative on one side of the equation and moving all other variables to the other), there are equations in which this cannot be done.

If y were a function of multiple variables, for example

<math>y = F(u,v)</math>

then the derivatives of y in the ordinary equation may be [[partial derivative|partial derivatives]] with respect to either u or v. In that case, any differential equation that has partial derivatives is called a partial differential equation. For example, the 1-dimensional [[wave equation]]<ref>Pain, H.J. ''The Physics of Vibrations and Waves'' 6th edition. Southern Gate, Chichester, West Sussex, England: John Wiley & Sons, 2005</ref> :

<math>\frac{\partial^2 y}{\partial x^2} = \frac{1}{c^2}\frac{\partial^2 y}{\partial t^2}</math>

Obviously, partial differential equations are much more complicated to solve.

==Methods==
There are many ways to find solutions to differential equations.

===Ordinary Differential Equations===
The simplest differential equations to solve are ''separable'' differential equations. A differential equation is separable if it can be written in the form

<math> \frac{d^{(n)}y}{dx^{(n)}} = F(x)G(y) </math>

Then we can separate the two variables, collect the x's on one side and the y's on the other side, then integrate to get the (n-1) derivative, and integrating again to get the (n-2) derivative, until we have found the function y. For example, for the derivative n = 1:

<math> \frac{dy}{dx} = F(x)G(y)</math>

<math>\frac{dy}{G(y)} = F(x)dx </math>

<math>\int\frac{dy}{G(y)} = \int F(x)dx </math>

The solution is then given implicitly by the expression:

<math>\int\frac{dy}{G(y)} - \int F(x)dx = C </math>

where C is an arbitrary constant.

====Linear Differential Equation Solutions====
----
If a differential equation can be written in the form

<math>F_n(x)\frac{d^{(n)}y}{dx^{(n)}} + F_{n-1}(x)\frac{d^{(n-1)}y}{dx^{(n-1)}} + F_{n-2}(x)\frac{d^{(n-2)}y}{dx^{(n-2)}} + ... + F_1(x)\frac{dy}{dx} + F_0(x)y = G(x)</math>

it is considered a ''linear'' differential equation.

=====First Order Linear Equations=====
The ''order'' of a differential equation is equal to the degree of the highest derivative in the equation. For example, the above equations are order n equations. A first order linear equation appears in the form:

<math> \frac{dy}{dx} + F(x)y = G(x) </math>

To solve this type of differential equation, an ''integrating factor''<ref>Edwards, Henry C. and Penney, David E.. ''Differential Equations and Boundary Value Problems'' 4th Edition. Upper Saddle River, NJ: Pearson, 2008</ref> is needed. For the first order equation, the integrating factor is defined as

<math> p = e^{\int F(x)} </math>

Multiplying both sides of the first order equation yields

<math> \frac{dy}{dx}e^{\int F(x)} + F(x)e^{\int F(x)}y = e^{\int F(x)}G(x) </math>

Note the derivative of

<math> e^{\int F(x)} </math>

is

<math> F(x)e^{\int F(x)} </math>

Hence the left hand side of the first order equation now looks like the [[product rule]] expansion for

<math> y </math> and <math> e^{\int F(x)} </math>

The equation can be rewritten

<math> \frac{d(e^{\int F(x)}y)}{dx} = e^{\int F(x)}G(x) </math>

Now we can integrate both sides, yielding the solution y:

<math> \int\frac{d(e^{\int F(x)}y)}{dx} = \int e^{\int F(x)}G(x) </math>

<math> e^{\int F(x)}y = \int e^{\int F(x)}G(x) + C </math>

<math> y = e^{-\int F(x)} \int e^{\int F(x)}G(x) + Ce^{-\int F(x)} </math>

where C is an arbitrary constant.

==References==
<references/>

[[category:calculus]]

Differential equation

2009-02-27T06:03:38Z

Guardianofrice: /* Ordinary Differential Equations */

In [[mathematics]], a '''differential equation''' is an [[equation]] that relates a [[function]] to one or more of its [[derivative]]s. Differential equations are especially applicable when the tools of algebra, which are ideally suited for static systems, are not enough. Many physical systems are modeled by solving differential equations, although their usefulness extends well into other fields of science such as chemistry and economics.

==Types of Differential Equations==
There are two main types of differential equations: Ordinary Differential Equations and Partial Differential Equations.<ref>Edwards, Henry C. and Penney, David E.. ''Differential Equations and Boundary Value Problems'' 4th Edition. Upper Saddle River, NJ: Pearson, 2008</ref>

The former is simpler of the two, as it can be written in the normal form

<math>\frac{d^{(n)}y}{dx^{(n)}} = F(x, y, \frac{dy}{dx}, \frac{d^2y}{dx^2}, \frac{d^3y}{dx^3}, ... , \frac{d^{(n-1)}y}{dx^{(n-1)}})</math>

for a simple function <math>y = g(x)</math>

The function F consists of the function y and its derivatives up to the ''nth'' order. Notice that y is comprised of only one independent variable x. A differential equation is considered ''ordinary'' if the function y in F is dependent on only one variable. It is important to note that, while most ordinary differential equations can be written in the normal form (isolating the highest derivative on one side of the equation and moving all other variables to the other), there are equations in which this cannot be done.

If y were a function of multiple variables, for example

<math>y = F(u,v)</math>

then the derivatives of y in the ordinary equation may be [[partial derivative|partial derivatives]] with respect to either u or v. In that case, any differential equation that has partial derivatives is called a partial differential equation. For example, the 1-dimensional [[wave equation]]<ref>Pain, H.J. ''The Physics of Vibrations and Waves'' 6th edition. Southern Gate, Chichester, West Sussex, England: John Wiley & Sons, 2005</ref> :

<math>\frac{\partial^2 y}{\partial x^2} = \frac{1}{c^2}\frac{\partial^2 y}{\partial t^2}</math>

Obviously, partial differential equations are much more complicated to solve.

==Methods==
There are many ways to find solutions to differential equations.

===Ordinary Differential Equations===
The simplest differential equations to solve are ''separable'' differential equations. A differential equation is separable if it can be written in the form

<math> \frac{d^{(n)}y}{dx^{(n)}} = F(x)G(y) </math>

Then we can separate the two variables, collect the x's on one side and the y's on the other side, then integrate to get the (n-1) derivative, and integrating again to get the (n-2) derivative, until we have found the function y. For example, for the derivative n = 1:

<math> \frac{dy}{dx} = F(x)G(y)</math>

<math>\frac{dy}{G(y)} = F(x)dx </math>

<math>\int\frac{dy}{G(y)} = \int F(x)dx </math>

The solution is then given implicitly by the expression:

<math>\int\frac{dy}{G(y)} - \int F(x)dx = C </math>

where C is an arbitrary constant.

====Linear Differential Equation Solutions====
----
If a differential equation can be written in the form

<math>F_n(x)\frac{d^{(n)}y}{dx^{(n)}} + F_{n-1}(x)\frac{d^{(n-1)}y}{dx^{(n-1)}} + F_{n-2}(x)\frac{d^{(n-2)}y}{dx^{(n-2)}} + ... + F_1(x)\frac{dy}{dx} + F_0(x)y = G(x)</math>

it is considered a ''linear'' differential equation.

=====First Order Linear Equations=====
The ''order'' of a differential equation is equal to the degree of the highest derivative in the equation. For example, the above equations are order n equations. A first order linear equation appears in the form:

<math> \frac{dy}{dx} + F(x)y = G(x) </math>

==References==
<references/>

[[category:calculus]]

Differential equation

2009-02-27T06:02:49Z

Guardianofrice: /* Ordinary Differential Equations */

In [[mathematics]], a '''differential equation''' is an [[equation]] that relates a [[function]] to one or more of its [[derivative]]s. Differential equations are especially applicable when the tools of algebra, which are ideally suited for static systems, are not enough. Many physical systems are modeled by solving differential equations, although their usefulness extends well into other fields of science such as chemistry and economics.

==Types of Differential Equations==
There are two main types of differential equations: Ordinary Differential Equations and Partial Differential Equations.<ref>Edwards, Henry C. and Penney, David E.. ''Differential Equations and Boundary Value Problems'' 4th Edition. Upper Saddle River, NJ: Pearson, 2008</ref>

The former is simpler of the two, as it can be written in the normal form

<math>\frac{d^{(n)}y}{dx^{(n)}} = F(x, y, \frac{dy}{dx}, \frac{d^2y}{dx^2}, \frac{d^3y}{dx^3}, ... , \frac{d^{(n-1)}y}{dx^{(n-1)}})</math>

for a simple function <math>y = g(x)</math>

The function F consists of the function y and its derivatives up to the ''nth'' order. Notice that y is comprised of only one independent variable x. A differential equation is considered ''ordinary'' if the function y in F is dependent on only one variable. It is important to note that, while most ordinary differential equations can be written in the normal form (isolating the highest derivative on one side of the equation and moving all other variables to the other), there are equations in which this cannot be done.

If y were a function of multiple variables, for example

<math>y = F(u,v)</math>

then the derivatives of y in the ordinary equation may be [[partial derivative|partial derivatives]] with respect to either u or v. In that case, any differential equation that has partial derivatives is called a partial differential equation. For example, the 1-dimensional [[wave equation]]<ref>Pain, H.J. ''The Physics of Vibrations and Waves'' 6th edition. Southern Gate, Chichester, West Sussex, England: John Wiley & Sons, 2005</ref> :

<math>\frac{\partial^2 y}{\partial x^2} = \frac{1}{c^2}\frac{\partial^2 y}{\partial t^2}</math>

Obviously, partial differential equations are much more complicated to solve.

==Methods==
There are many ways to find solutions to differential equations.

===Ordinary Differential Equations===
The simplest differential equations to solve are ''separable'' differential equations. A differential equation is separable if it can be written in the form

<math> \frac{d^{(n)}y}{dx^{(n)}} = F(x)G(y) </math>

Then we can separate the two variables, collect the x's on one side and the y's on the other side, then integrate to get the (n-1) derivative, and integrating again to get the (n-2) derivative, until we have found the function y. For example, for the derivative n = 1:

<math> \frac{dy}{dx} = F(x)G(y)

\frac{dy}{G(y)} = F(x)dx

\int\frac{dy}{G(y)} = \int F(x)dx </math>

The solution is then given implicitly by the expression:

<math>\int\frac{dy}{G(y)} - \int F(x)dx = C </math>

where C is an arbitrary constant.

====Linear Differential Equation Solutions====
----
If a differential equation can be written in the form

<math>F_n(x)\frac{d^{(n)}y}{dx^{(n)}} + F_{n-1}(x)\frac{d^{(n-1)}y}{dx^{(n-1)}} + F_{n-2}(x)\frac{d^{(n-2)}y}{dx^{(n-2)}} + ... + F_1(x)\frac{dy}{dx} + F_0(x)y = G(x)</math>

it is considered a ''linear'' differential equation.

=====First Order Linear Equations=====
The ''order'' of a differential equation is equal to the degree of the highest derivative in the equation. For example, the above equations are order n equations. A first order linear equation appears in the form:

<math> \frac{dy}{dx} + F(x)y = G(x) </math>

==References==
<references/>

[[category:calculus]]

Differential equation

2009-02-27T06:01:15Z

Guardianofrice:

In [[mathematics]], a '''differential equation''' is an [[equation]] that relates a [[function]] to one or more of its [[derivative]]s. Differential equations are especially applicable when the tools of algebra, which are ideally suited for static systems, are not enough. Many physical systems are modeled by solving differential equations, although their usefulness extends well into other fields of science such as chemistry and economics.

==Types of Differential Equations==
There are two main types of differential equations: Ordinary Differential Equations and Partial Differential Equations.<ref>Edwards, Henry C. and Penney, David E.. ''Differential Equations and Boundary Value Problems'' 4th Edition. Upper Saddle River, NJ: Pearson, 2008</ref>

The former is simpler of the two, as it can be written in the normal form

<math>\frac{d^{(n)}y}{dx^{(n)}} = F(x, y, \frac{dy}{dx}, \frac{d^2y}{dx^2}, \frac{d^3y}{dx^3}, ... , \frac{d^{(n-1)}y}{dx^{(n-1)}})</math>

for a simple function <math>y = g(x)</math>

The function F consists of the function y and its derivatives up to the ''nth'' order. Notice that y is comprised of only one independent variable x. A differential equation is considered ''ordinary'' if the function y in F is dependent on only one variable. It is important to note that, while most ordinary differential equations can be written in the normal form (isolating the highest derivative on one side of the equation and moving all other variables to the other), there are equations in which this cannot be done.

If y were a function of multiple variables, for example

<math>y = F(u,v)</math>

then the derivatives of y in the ordinary equation may be [[partial derivative|partial derivatives]] with respect to either u or v. In that case, any differential equation that has partial derivatives is called a partial differential equation. For example, the 1-dimensional [[wave equation]]<ref>Pain, H.J. ''The Physics of Vibrations and Waves'' 6th edition. Southern Gate, Chichester, West Sussex, England: John Wiley & Sons, 2005</ref> :

<math>\frac{\partial^2 y}{\partial x^2} = \frac{1}{c^2}\frac{\partial^2 y}{\partial t^2}</math>

Obviously, partial differential equations are much more complicated to solve.

==Methods==
There are many ways to find solutions to differential equations.

===Ordinary Differential Equations===
The simplest differential equations to solve are ''separable'' differential equations. A differential equation is separable if it can be written in the form

<math> \frac{d^{(n)}y}{dx^{(n)}} = F(x)G(y) </math>

Then we can separate the two variables, collect the x's on one side and the y's on the other side, then integrate to get the (n-1) derivative, and integrating again to get the (n-2) derivative, until we have found the function y. For example, for the derivative n = 1:

<math> \frac{dy}{dx} = F(x)G(y)

\frac{dy}{G(y)} = F(x)dx
\int\frac{dy}{G(y)} = \int F(x)dx </math>

The solution is then given implicitly by the expression:

<math>\int\frac{dy}{G(y)} - \int F(x)dx = C </math>

where C is an arbitrary constant.

====Linear Differential Equation Solutions====
----
If a differential equation can be written in the form

<math>F_n(x)\frac{d^{(n)}y}{dx^{(n)}} + F_{n-1}(x)\frac{d^{(n-1)}y}{dx^{(n-1)}} + F_{n-2}(x)\frac{d^{(n-2)}y}{dx^{(n-2)}} + ... + F_1(x)\frac{dy}{dx} + F_0(x)y = G(x)</math>

it is considered a ''linear'' differential equation.

=====First Order Linear Equations=====
The ''order'' of a differential equation is equal to the degree of the highest derivative in the equation. For example, the above equations are order n equations. A first order linear equation appears in the form:

<math> \frac{dy}{dx} + F(x)y = G(x) </math>

==References==
<references/>

[[category:calculus]]

Differential equation

2009-02-27T05:25:04Z

Guardianofrice:

Differential equation

2009-02-12T06:16:37Z

Guardianofrice: /* Types of Differential Equations */

Differential equation

2009-02-12T06:14:02Z

Guardianofrice:

Differential equation

2009-02-12T05:19:55Z

Guardianofrice:

Arianism

2009-02-12T05:02:54Z

Guardianofrice: /* Later developments */

Originating around AD 320 as taught by [[Arius]], '''Arianism''' was the [[theology|theological]] view that [[Jesus]] was divine, but created and lesser than [[God]] the Father. It was Christian, but denounced as [[heresy]] by the orthodox forces [[trinity|trinitarian]] that eventually prevailed.

Arianism was officially condemned as incorrect by the [[Council of Nicaea]] in 325, which gave its seal of authority to the established [[Holy Trinity|trinitarian]] view. On the surface the Arian position appeared to die out, but it simmered under the surface and re-emerged after the death of the Emperor [[Constantine]] being cultivated by key Bishops and a continued rumbling in the East. The acceptance of Constantine's son [[Constantius II]] of the Arian heresy caused many Catholic Bishops under his jurisdiction in the Eastern part of the Empire to be exiled and Catholicism was temporarily suppressed when Constantius took over the Western part of the Roman Empire as well. Even [[Pope Liberius]] was exiled. Under [[Julian the Apostate]], the next Emperor, all exiled Bishops were allowed back, in part to promote infighting among Christians.<ref>http://encyclopedia2.thefreedictionary.com/Arianism</ref> Arianism's sudden rise dissipated when [[Valentinian]], a Catholic became Emperor in the West followed a decade later by [[Theodius]], a Catholic, in the East. Religious uniformity was again restored when another religious council reiterated that Arianism was a heresy (379). Never having the numerical support of Catholicism to begin with, Arianism died out within the Empire almost overnight, but continued to be a threat to the Roman Empire as many of the surrounding tribes had and were accepting Christianity, but were converted with an Arian stamp.
==Barbarians==
The barbarian tribes which invaded [[Rome]], and caused its fall in 476, were often Arians. The [[Visigoths]] and the [[Ostrogoths]], and especially [[Theoderic]], were followers of the Arian heresy. It would take centuries before the trinitarian view prevailed in those regions. The 5th Ecumenical Council in 553 ended the last outreaches of Arianism within greater Europe.<ref>J. Herrin (1987) The Formation of Christendom. Fontana Press, London. </ref>
==Later developments==
Arianism died out as an organized force but its ideas were known to theologians. Arianism Unitarianism in [[Renaissance]] Europe grew out of the teachings of the Italians Lelio Sozzini, or Socinus (1525-1562), and his nephew Fausto (1539-1604) during the period of the [[Protestant Reformation]]. [[Isaac newton]] was an Arian--indeed he modeled himself after Arius--but kept it secret because heresy would cost him his official academic and government appointments.

[[Unitarian]] ideas that emerged in England and the U.S. after 1770 have many similarities to Arianism.

==Bibliography==
*Gonzalez, Justo L. '' A History of Christian Thought: Volume 1: From the Beginnings to the Council of Chalcedon'' (2nd ed. 1987); [http://www.amazon.com/History-Christian-Thought-Beginnings-Chalcedon/dp/0687171822/ref=sr_1_1?ie=UTF8&s=books&qid=1200823941&sr=8-1 excerpt and text search vol 1]
* Williams, Rowan. ''Arius: Heresy and Tradition'' (2002) [http://www.amazon.com/Arius-Heresy-Tradition-Rowan-Williams/dp/0802849695/ref=pd_bbs_sr_1?ie=UTF8&s=books&qid=1233553963&sr=8-1 excerpt and text search]
*''New Schaff-Herzog Encyclopedia of Religious Knowledge'' (1911), major sources of older scholarly articles; mainline Protestant perspective: [http://www.ccel.org/ccel/schaff/encyc01.toc.html Vol. 1: Aachen- Basilians]

==References==
<references/>

[[Category:Heresies]]
[[Category:Christian Denominations]]

Arianism

2009-02-12T04:59:57Z

Guardianofrice:

Originating around AD 320 as taught by [[Arius]], '''Arianism''' was the [[theology|theological]] view that [[Jesus]] was divine, but created and lesser than [[God]] the Father. It was Christian, but denounced as [[heresy]] by the orthodox forces [[trinity|trinitarian]] that eventually prevailed.

Arianism was officially condemned as incorrect by the [[Council of Nicaea]] in 325, which gave its seal of authority to the established [[Holy Trinity|trinitarian]] view. On the surface the Arian position appeared to die out, but it simmered under the surface and re-emerged after the death of the Emperor [[Constantine]] being cultivated by key Bishops and a continued rumbling in the East. The acceptance of Constantine's son [[Constantius II]] of the Arian heresy caused many Catholic Bishops under his jurisdiction in the Eastern part of the Empire to be exiled and Catholicism was temporarily suppressed when Constantius took over the Western part of the Roman Empire as well. Even [[Pope Liberius]] was exiled. Under [[Julian the Apostate]], the next Emperor, all exiled Bishops were allowed back, in part to promote infighting among Christians.<ref>http://encyclopedia2.thefreedictionary.com/Arianism</ref> Arianism's sudden rise dissipated when [[Valentinian]], a Catholic became Emperor in the West followed a decade later by [[Theodius]], a Catholic, in the East. Religious uniformity was again restored when another religious council reiterated that Arianism was a heresy (379). Never having the numerical support of Catholicism to begin with, Arianism died out within the Empire almost overnight, but continued to be a threat to the Roman Empire as many of the surrounding tribes had and were accepting Christianity, but were converted with an Arian stamp.
==Barbarians==
The barbarian tribes which invaded [[Rome]], and caused its fall in 476, were often Arians. The [[Visigoths]] and the [[Ostrogoths]], and especially [[Theoderic]], were followers of the Arian heresy. It would take centuries before the trinitarian view prevailed in those regions. The 5th Ecumenical Council in 553 ended the last outreaches of Arianism within greater Europe.<ref>J. Herrin (1987) The Formation of Christendom. Fontana Press, London. </ref>
==Later developments==
Arianism died out as an organized force but its ideas were known to theologians. Arianism Unitarianism in [[Renaissance]] Europe grew out of the teachings of the Italians Lelio Sozzini, or Socinus (1525-1562), and his nephew Fausto (1539-1604) during the period of the [[Protestant Reformation]]. [[Isaac newton]] was an Arian--indeed he modeled himself after Arius--but kept it secret because heresy would ccost him his official academoc and government appointments.

[[Unitarian]] ideas that emerged in England and the U.S. after 1770 have many similarities to Arianism.
==Bibliography==
*Gonzalez, Justo L. '' A History of Christian Thought: Volume 1: From the Beginnings to the Council of Chalcedon'' (2nd ed. 1987); [http://www.amazon.com/History-Christian-Thought-Beginnings-Chalcedon/dp/0687171822/ref=sr_1_1?ie=UTF8&s=books&qid=1200823941&sr=8-1 excerpt and text search vol 1]
* Williams, Rowan. ''Arius: Heresy and Tradition'' (2002) [http://www.amazon.com/Arius-Heresy-Tradition-Rowan-Williams/dp/0802849695/ref=pd_bbs_sr_1?ie=UTF8&s=books&qid=1233553963&sr=8-1 excerpt and text search]
*''New Schaff-Herzog Encyclopedia of Religious Knowledge'' (1911), major sources of older scholarly articles; mainline Protestant perspective: [http://www.ccel.org/ccel/schaff/encyc01.toc.html Vol. 1: Aachen- Basilians]

==References==
<references/>

[[Category:Heresies]]
[[Category:Christian Denominations]]

Nikolai Rimsky-Korsakov

2008-12-14T19:50:56Z

Guardianofrice: /* Some works */

[[Image:Rimski - Korsakov.jpg|left]]
'''Nikolai Rimsky-Korsakov''' (1844 - 1908) was a [[Russian]] composer and professor. He is noted for his predilection for folk and fairy-tale subjects. Rimsky-Korsakov with Mili Balakirev, Aleksandr Borodín, Modest Músorgski and César Cui formed the grup known as ''The Five''. In 1868, he met [[Pyotr Ilyich Tchaikovsky]]. Rimsky-Korsakov ''taught several students who achieved fame as composers, including [[Sergei Prokofiev]] and [[Igor Stravinsky]].''

<blockquote>
''His symphonic suite Scheherazade (1888) ranks as one of the most popular orchestral works ever written.'' [http://www.geocities.com/Vienna/3606/bio.html]
</blockquote>

In July 1872, Rimsky-Korsakov married Nadezhda Purgold.

== Some works ==
[[Image:Scheherazade.jpg|right]]
* The Maid of Pskov
* Mlada
* '''Capriccio espagnol'''
* The Tsar's Bride
* May Night
* Snowmaiden ('''The Dance of the Tumblers''')
* The Barber of Baghdad
* Mozart and Salieri
* The Tale of Tsar Saltan ('''Flight of the Bumblebee''')
* Kashchei the Immortal
* '''Scheherazade''' (symphonic suite)
* The Legend of the Invisible City of Kitezh and the Maiden Fevroniya
* Christmas Eve
* Heaven and Earth
* The Tempest
* Saul and David
* Russian Easter Overture

== See also ==

*[[Claude Debussy]]
*[[Maurice Ravel]]
*[[Romantic period (music)]]

== External links ==

*[http://www.epdlp.com/compclasico.php?id=1105 Rimsky-Korsakov] In Spanish.
*[http://www.geocities.com/Vienna/3606/start.html The Rimsky-Korsakov Home Page]

[[Category:Composers|Rimsky-Korsakov, Nikolai]]