|
|
| Line 1: |
Line 1: |
| [[Image:Drum vibration mode12.gif|right|frame|This solution of the [[vibrations of a circular drum|vibrating drum problem]] is, at any point in time, an eigenfunction of the [[Laplace operator]] on a disk.]]
| | Nice to be here there, I am Adrianne and I totally get that name. Vermont has always been simple home and I affection every day living here. Gardening is what I do a week. I am a people manager but nevertheless , soon I'll be on my own. You can find my website here: http://prometeu.net<br><br>Look at my webpage: [http://prometeu.net Clash of clans hack cydia] |
| In [[mathematics]], an '''eigenfunction''' of a [[linear operator]], ''A'', defined on some [[function space]], is any non-zero [[function (mathematics)|function]] ''f'' in that space that returns from the operator exactly as is, except for a multiplicative scaling factor. More precisely, one has
| |
| | |
| :<math>
| |
| \mathcal A f = \lambda f
| |
| </math>
| |
| | |
| for some [[scalar (mathematics)|scalar]], λ, the corresponding [[eigenvalue]]. The solution of the differential eigenvalue problem also depends on any boundary conditions required of <math>f</math>. In each case there are only certain eigenvalues <math>\lambda=\lambda_n</math> (<math>n=1,2,3,...</math>) that admit a corresponding solution for <math>f=f_n</math> (with each <math>f_n</math> belonging to the eigenvalue <math>\lambda_n</math>) when combined with the boundary conditions. Eigenfunctions are used to analyze <math>A</math>.
| |
| | |
| For example, <math>f_k(x) = e^{kx}</math> is an eigenfunction for the [[differential operator]]
| |
| | |
| :<math>
| |
| \mathcal A = \frac{d^2}{dx^2} - \frac{d}{dx}
| |
| </math>
| |
| | |
| for any value of <math>k</math>, with corresponding eigenvalue <math>\lambda = k^2 - k</math>. If boundary conditions are applied to this system (e.g., <math>f=0</math> at two physical locations in space), then only certain values of <math>k=k_n</math> satisfy the boundary conditions, generating corresponding discrete eigenvalues <math>\lambda_n=k_n^2-k_n</math>.
| |
| | |
| Specifically, in the study of [[LTI system theory|signals and systems]], the '''eigenfunction''' of a system is the signal <math>f(t)</math> which when input into the system, produces a response <math>y(t) = \lambda f(t)</math> with the complex constant <math>\lambda</math>.<ref>Bernd Girod, Rudolf Rabenstein, Alexander Stenger, ''Signals and systems'', 2nd ed., Wiley, 2001, ISBN 0-471-98800-6 p. 49</ref>
| |
| ==Examples==
| |
| ===Derivative operator===
| |
| A widely used class of linear operators acting on function spaces are the [[differential operator]]s on [[function space]]s. As an example, on the space <math>\mathbf{C^\infty}</math> of infinitely [[derivative|differentiable]] real functions of a real argument <math>t</math>, the process of differentiation is a linear operator since
| |
| | |
| : <math> \displaystyle\frac{d}{dt}(af+bg) = a \frac{df}{dt} + b \frac{dg}{dt},</math>
| |
| | |
| for any functions <math>f</math> and <math>g</math> in <math>\mathbf{C^\infty}</math>, and any real numbers <math>a</math> and <math>b</math>.
| |
| | |
| The eigenvalue equation for a linear differential operator <math>D</math> in <math>\mathbf{C^\infty}</math> is then a [[differential equation]]
| |
| :<math>D f = \lambda f</math>
| |
| The functions that satisfy this equation are commonly called '''eigenfunctions'''. For the derivative operator <math>d/dt</math>, an eigenfunction is a function that, when differentiated, yields a constant times the original function. That is,
| |
| : <math>\displaystyle\frac{d}{dt} f(t) = \lambda f(t)</math>
| |
| for all <math>t</math>. This equation can be solved for any value of <math>\lambda</math>. The solution is an [[exponential function]]
| |
| : <math>f(t) = Ae^{\lambda t}.\ </math>
| |
| | |
| The derivative operator is defined also for complex-valued functions of a complex argument. In the complex version of the space <math>\mathbf{C^\infty}</math>, the eigenvalue equation has a solution for any complex constant <math>\lambda</math>. The spectrum of the operator <math>d/dt</math> is therefore the whole [[complex plane]]. This is an example of a [[continuous spectrum]].
| |
| | |
| ==Applications==
| |
| ===Vibrating strings===
| |
| [[File:Standing wave.gif|thumb|270px|The shape of a standing wave in a string fixed at its boundaries is an example of an eigenfunction of a differential operator. The admissible eigenvalues are governed by the length of the string and determine the frequency of oscillation.]]
| |
| Let <math>h(x,t)</math> denote the sideways displacement of a stressed elastic chord, such as the [[vibrating string]]s of a [[string instrument]], as a function of the position <math>x</math> along the string and of time <math>t</math>. From the laws of mechanics, applied to [[infinitesimal]] portions of the string, one can deduce that the function <math>h</math> satisfies the [[partial differential equation]]
| |
| | |
| : <math>\frac{\partial^2 h}{\partial t^2} = c^2\frac{\partial^2 h}{\partial x^2},</math>
| |
| | |
| which is called the (one-dimensional) [[wave equation]]. Here <math>c</math> is a constant that depends on the tension and mass of the string.
| |
| | |
| This problem is amenable to the method of [[separation of variables]]. If we assume that <math>h(x,t)</math> can be written as the product of the form <math>X(x)T(t)</math>, we can form a pair of ordinary differential equations:
| |
| | |
| : <math>\frac{d^2}{dx^2}X=-\frac{\omega^2}{c^2}X\quad\quad\quad</math> and <math>\quad\quad\quad\displaystyle \frac{d^2}{dt^2}T=-\omega^2 T.\ </math> | |
| | |
| Each of these is an eigenvalue equation, for eigenvalues <math>-\omega^2/c^2</math> and <math>-\omega^2</math>, respectively. For any values of <math>\omega</math> and <math>c</math>, the equations are satisfied by the functions
| |
| | |
| : <math>X(x) = \sin \left(\frac{\omega x}{c} + \phi \right)\quad\quad\quad</math> and <math>\quad\quad\quad T(t) = \sin(\omega t + \psi),\ </math>
| |
| where <math>\phi</math> and <math>\psi</math> are arbitrary real constants. If we impose boundary conditions (that the ends of the string are fixed with <math>X(x) = 0</math> at <math>x = 0 </math> and <math>x = L</math>, for example) we can constrain the eigenvalues. For those [[Boundary value problem|boundary conditions]], we find
| |
| | |
| : <math>\sin(\phi) = 0\ </math>, and so the phase angle <math>\phi=0\ </math>
| |
| | |
| and
| |
| | |
| : <math>\sin\left(\frac{\omega L}{c}\right) = 0.\ </math>
| |
| | |
| Thus, the constant <math>\omega</math> is constrained to take one of the values <math>\omega_n = n c\pi/L</math>, where <math>n</math> is any integer. Thus, the clamped string supports a family of standing waves of the form
| |
| | |
| : <math>h(x,t) = \sin(n\pi x/L)\sin(\omega_n t).\ </math>
| |
| | |
| From the point of view of our musical instrument, the frequency <math>\omega_n\ </math> is the frequency of the <math>n</math>th [[harmonic]], which is called the <math>(n-1)</math>th [[overtone]].
| |
| | |
| ===Quantum mechanics===
| |
| Eigenfunctions play an important role in many branches of physics. An important example is [[quantum mechanics]], where the [[Schrödinger equation]]
| |
| | |
| :<math> | |
| \mathcal H \psi = E \psi
| |
| </math>,
| |
| | |
| with
| |
| :<math>
| |
| \mathcal H = -\frac{\hbar^2}{2m}\nabla^2+ V(\mathbf{r},t)
| |
| </math>
| |
| | |
| has solutions of the form
| |
| | |
| :<math>
| |
| \psi(t) = \sum_k e^{-i E_k t/\hbar} \phi_k,
| |
| </math>
| |
| | |
| where <math>\phi_k</math> are eigenfunctions of the operator <math>\mathcal H</math> with eigenvalues <math>E_k</math>. The fact that only certain eigenvalues <math>E_k</math> with associated eigenfunctions <math>\phi_k</math> satisfy Schrödinger's equation leads to a natural basis for quantum mechanics and the periodic table of the elements, with each <math>E_k</math> an allowable energy state of the system. The success of this equation in explaining the spectral characteristics of hydrogen is considered one of the greatest triumphs of 20th century physics.
| |
| | |
| Due to the nature of [[Self-adjoint operator|Hermitian Operators]] such as the [[Hamiltonian (quantum mechanics)|Hamiltonian]] operator <math>\mathcal H</math>, its eigenfunctions are [[orthogonal functions]]. This is not necessarily the case for eigenfunctions of other operators (such as the example <math>A</math> mentioned above). Orthogonal functions <math>f_i</math>, <math>i=1, 2, \dots,</math> have the property that
| |
| | |
| :<math> | |
| 0 = \int f_i^{*} f_j
| |
| </math>
| |
| | |
| where <math>
| |
| f_i^{*}
| |
| </math> is the [[complex conjugate]] of <math>f_i</math>
| |
| | |
| whenever <math>i\neq j</math>, in which case the set <math>\{f_i \,|\, i \in I\}</math> is said to be orthogonal. Also, it is [[Linear independence|linearly independent]].
| |
| | |
| ==Notes==
| |
| {{reflist}}
| |
| | |
| ==References==
| |
| *''Methods of Mathematical Physics'' by R. Courant, D. Hilbert ISBN 0-471-50447-5 (Volume 1 Paperback) ISBN 0-471-50439-4 (Volume 2 Paperback) ISBN 0-471-17990-6 (Hardback)
| |
| | |
| ==See also==
| |
| * [[Eigenvalue, eigenvector and eigenspace]]
| |
| * [[Hilbert–Schmidt theorem]]
| |
| * [[Spectral theory of ordinary differential equations]]
| |
| * [[Fixed point combinator]]
| |
| * More images (non-GPL) at [https://daugerresearch.com/orbitals/index.shtml Atom in a Box]
| |
| | |
| [[Category:Functional analysis]]
| |
| | |
| [[de:Eigenfunktion]]
| |
| [[zh:本徵函數]]
| |
Nice to be here there, I am Adrianne and I totally get that name. Vermont has always been simple home and I affection every day living here. Gardening is what I do a week. I am a people manager but nevertheless , soon I'll be on my own. You can find my website here: http://prometeu.net
Look at my webpage: Clash of clans hack cydia