Sturm-Liouville theory

The map can be viewed as a linear operator mapping a function u to another function Lu. We may study this linear operator in the context of functional analysis. If we put w = 1 in equation (1), it can be written as This is precisely the eigenvalue problem; that is, we are trying to find the eigenvalues λ and eigenvectors u of the L operator. However, to be honest we must also include the boundary conditions. Let's say that we want to look at the problem over the inter ...

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Sturm-Liouville theory, Sturm-Liouville theory - Sturm-Liouville theory, Sturm-Liouville theory - Sturm-Liouville form, Sturm-Liouville theory - Examples, Sturm-Liouville theory - Sturm-Liouville differential operators, Sturm-Liouville theory - Some highly technical details, Sturm-Liouville theory - Useful consequences of the preceding technicalities, Sturm-Liouville theory - Example, Sturm-Liouville theory - Application to normal modes

Read more here: » Sturm-Liouville theory: Encyclopedia II - Sturm-Liouville theory - Sturm-Liouville differential operators

In mathematics, boundary conditions are imposed on the solutions of ordinary differential equations and partial differential equations, to fit the solutions to the actual problem. In the problems most frequently considered, only one of the infinitely many solutions of the differential equation satisfies the boundary conditions. In a physical model simulation, the boundary conditions describe the behavior ...

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• Boundary condition - Bibliography

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Consider two bodies (not affected by gravity), each of mass M, attached to three springs with stiffness K. They are attached in the following manner: where the edge points are fixed and cannot move. We'll use x1(t) to denote the displacement of the leftmost mass, and x2(t) to denote the displacement of the rightmost. If we denote the second derivative of x(t) with respect to time as x″, the equations of motion are: See also:

Normal mode, Normal mode - Example - normal modes of coupled oscillators, Normal mode - Standing waves, Normal mode - Normal modes in quantum mechanics

Read more here: » Normal mode: Encyclopedia II - Normal mode - Example - normal modes of coupled oscillators

Laplace's equation in spherical coordinates is: (see nabla in cylindrical and spherical coordinates). Separation of variables leads to solutions expressed in terms of trigonometric functions and Legendre polynomials. Note that the spherical coordinates and in this article are used in the physicist's way, as opposed to the mathematician's definition of spherical coordinates. That is, is the colatitude or polar angle, ranging from and the azimuth or longitude, ranging from . The general solution which remains finite towards infinity is a linear combination of functions of the form< ...

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Spherical harmonics, Spherical harmonics - Introduction, Spherical harmonics - First few spherical harmonics, Spherical harmonics - Generalizations

Read more here: » Spherical harmonics: Encyclopedia II - Spherical harmonics - Introduction

Suppose that f(x), a complex-valued function of a real variable, is periodic with period 2π, and is square-integrable over the interval from −π to π. Let Each Fn is called a Fourier coefficient. Then, the Fourier series representation of f(x) is given by Each term in this sum is called a Fourier mode or a harmonic. In the important special case of a real-valued function f< ...

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Fourier series, Fourier series - Definition of Fourier series, Fourier series - Example, Fourier series - Convergence of Fourier series, Fourier series - Orthogonality, Fourier series - Some positive consequences of the homomorphism properties of exp, Fourier series - Shifting property, Fourier series - Convolution theorems, Fourier series - Plancherel's and Parseval's theorem, Fourier series - General formulation

Read more here: » Fourier series: Encyclopedia II - Fourier series - Definition of Fourier series

The spherical harmonics in a certain sense capture the symmetry properties of the two-sphere. The symmetry properties of the two-sphere are given by the Lie groups SO(3) and its double-cover SU(2). The spherical harmonic transform under the integer-spin representations of these groups; they are a part of the representation theory of these groups. However, the two-sphere can also be understood to be the Riemann sphere. The complete set of symmetries of the Riemann sphere can be understood to be described by the Mobius transformation group SL( ...

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Spherical harmonics, Spherical harmonics - Introduction, Spherical harmonics - First few spherical harmonics, Spherical harmonics - Generalizations

Read more here: » Spherical harmonics: Encyclopedia II - Spherical harmonics - Generalizations

In quantum mechanics, a state of a system is described by a wavefunction of (x, t) which solves the Schrödinger equation. The square of the absolute value of ,i.e. is the probability (density) to measure the particle in place x at time t. Usually, when involving some sort of potential, the wavefunction is decomposed into a superposition of energy eigenstates, each oscill ...

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Normal mode, Normal mode - Example - normal modes of coupled oscillators, Normal mode - Standing waves, Normal mode - Normal modes in quantum mechanics

Read more here: » Normal mode: Encyclopedia II - Normal mode - Normal modes in quantum mechanics

These are the first few spherical harmonics: More spherical harmonics up to Y10 ...

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Spherical harmonics, Spherical harmonics - Introduction, Spherical harmonics - First few spherical harmonics, Spherical harmonics - Generalizations

Read more here: » Spherical harmonics: Encyclopedia II - Spherical harmonics - First few spherical harmonics

Let f(x) = x be the identity function for x from −π to π. Outside this domain, the Fourier series implicitly requires that we define the function periodically. We will compute the Fourier coefficients for this function. Notice that cos(nx) is an even function, while f and sin(nx) are odd functions. Notice that an are 0 because x and x cos(nx) are odd functions. Hence the Fourier series for f(x) = x is: See also:

Fourier series, Fourier series - Definition of Fourier series, Fourier series - Example, Fourier series - Convergence of Fourier series, Fourier series - Orthogonality, Fourier series - Some positive consequences of the homomorphism properties of exp, Fourier series - Shifting property, Fourier series - Convolution theorems, Fourier series - Plancherel's and Parseval's theorem, Fourier series - General formulation

Read more here: » Fourier series: Encyclopedia II - Fourier series - Example

A standing wave is a continuous form of normal mode. In a standing wave, all the space elements (i.e (x,y,z) coordinates) are oscillating in the same frequency and in phase (reaching the equilibrium point together), but each has a different amplitude. The general form of a standing wave is: Ψ(t) = f(x,y,z)(Acos(ωt) + Bsin(ωt)) where f(x, y, z) represents the dependence of amplitude on location and ...

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Normal mode, Normal mode - Example - normal modes of coupled oscillators, Normal mode - Standing waves, Normal mode - Normal modes in quantum mechanics

Read more here: » Normal mode: Encyclopedia II - Normal mode - Standing waves

The Fourier basis functions are orthogonal in the discrete space where δ(x) is the Dirac delta function and δT(x) is the Dirac comb function. The Fourier basis functions are orthogonal in the continuous space as well: where δnm is the Kronecker delta function. ...

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Fourier series, Fourier series - Definition of Fourier series, Fourier series - Example, Fourier series - Convergence of Fourier series, Fourier series - Orthogonality, Fourier series - Some positive consequences of the homomorphism properties of exp, Fourier series - Shifting property, Fourier series - Convolution theorems, Fourier series - Plancherel's and Parseval's theorem, Fourier series - General formulation

Read more here: » Fourier series: Encyclopedia II - Fourier series - Orthogonality

While the Fourier coefficients an and bn can be formally defined for any function for which the integrals make sense, whether the series so defined actually converges to f(x) depends on the properties of f. The simplest answer is that if f is square-integrable then (this is convergence in the norm of the space L2). There are also many known tests that ensure that the series converges at a given poin ...

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Fourier series, Fourier series - Definition of Fourier series, Fourier series - Example, Fourier series - Convergence of Fourier series, Fourier series - Orthogonality, Fourier series - Some positive consequences of the homomorphism properties of exp, Fourier series - Shifting property, Fourier series - Convolution theorems, Fourier series - Plancherel's and Parseval's theorem, Fourier series - General formulation

Read more here: » Fourier series: Encyclopedia II - Fourier series - Convergence of Fourier series

Another important property of the Fourier series is the Plancherel theorem Parseval's theorem, a special case of the Plancherel theorem, states that which can be restated for the real-valued f(x) case above, These theorems may be proven using the orthogonality relationships. ...

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Fourier series, Fourier series - Definition of Fourier series, Fourier series - Example, Fourier series - Convergence of Fourier series, Fourier series - Orthogonality, Fourier series - Some positive consequences of the homomorphism properties of exp, Fourier series - Shifting property, Fourier series - Convolution theorems, Fourier series - Plancherel's and Parseval's theorem, Fourier series - General formulation

Read more here: » Fourier series: Encyclopedia II - Fourier series - Plancherel's and Parseval's theorem

Because "basis functions" eikx are homomorphisms of the real line (more precisely, of the "circle group") we have some useful identities: Fourier series - Shifting property. If then (if G is the transform of g) Fourier series - Convolution theorems. Main article: Convolution If h( ...

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Fourier series, Fourier series - Definition of Fourier series, Fourier series - Example, Fourier series - Convergence of Fourier series, Fourier series - Orthogonality, Fourier series - Some positive consequences of the homomorphism properties of exp, Fourier series - Shifting property, Fourier series - Convolution theorems, Fourier series - Plancherel's and Parseval's theorem, Fourier series - General formulation

Read more here: » Fourier series: Encyclopedia II - Fourier series - Some positive consequences of the homomorphism properties of exp

Mathematical formulation of quantum mechanics - The old quantum theory and the need for new mathematics. Main article: Old quantum theory In the decade of 1890, Planck was able to derive the blackbody spectrum and solve the classical ultraviolet catastrophe by making the unorthodox assumption that, in the interaction of radiation with matter, energy could only be exchanged in discrete units which he called quanta. Planck postulated a direct proportionality between the frequency of radiation and the quantum of energy at that freque ...

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Mathematical formulation of quantum mechanics, Mathematical formulation of quantum mechanics - History of the formalism, Mathematical formulation of quantum mechanics - The old quantum theory and the need for new mathematics, Mathematical formulation of quantum mechanics - The new quantum theory, Mathematical formulation of quantum mechanics - Later developments, Mathematical formulation of quantum mechanics - Mathematical structure of quantum mechanics, Mathematical formulation of quantum mechanics - Postulates of quantum mechanics, Mathematical formulation of quantum mechanics - Pictures of dynamics, Mathematical formulation of quantum mechanics - Representations, Mathematical formulation of quantum mechanics - Time as an operator, Mathematical formulation of quantum mechanics - The problem of measurement, Mathematical formulation of quantum mechanics - Wavefunction collapse, Mathematical formulation of quantum mechanics - The relative state interpretation, Mathematical formulation of quantum mechanics - List of mathematical tools

Read more here: » Mathematical formulation of quantum mechanics: Encyclopedia II - Mathematical formulation of quantum mechanics - History of the formalism

The most commonly used differential operator is the action of taking the derivative itself. Common notations for this operator include: where the variable one is differentiating to is clear, and where the variable is declared explicitly. First derivatives are signified as above, but when taking higher, n-th derivatives, the following alterations are useful: The D notation's use and creation is credited to Oliver Heaviside, who considered differential operators of the form See also:

Differential operator, Differential operator - Notations, Differential operator - Adjoint of an operator, Differential operator - Properties of differential operators, Differential operator - Several variables, Differential operator - Coordinate-independent description, Differential operator - Examples

Read more here: » Differential operator: Encyclopedia II - Differential operator - Notations

A physical system is generally described by three basic ingredients: states; observables; and dynamics (or law of time evolution) or, more generally, a group of physical symmetries. A classical description can be given in a fairly direct way by a phase space model of mechanics: states are points in a symplectic phase space, observables are real-valued functions on it, time evolution is given by a one-parameter group of symplectic transformations of the phase space, and physical symmetries are realized by symplectic transformations. A quantum ...

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Mathematical formulation of quantum mechanics, Mathematical formulation of quantum mechanics - History of the formalism, Mathematical formulation of quantum mechanics - The old quantum theory and the need for new mathematics, Mathematical formulation of quantum mechanics - The new quantum theory, Mathematical formulation of quantum mechanics - Later developments, Mathematical formulation of quantum mechanics - Mathematical structure of quantum mechanics, Mathematical formulation of quantum mechanics - Postulates of quantum mechanics, Mathematical formulation of quantum mechanics - Pictures of dynamics, Mathematical formulation of quantum mechanics - Representations, Mathematical formulation of quantum mechanics - Time as an operator, Mathematical formulation of quantum mechanics - The problem of measurement, Mathematical formulation of quantum mechanics - Wavefunction collapse, Mathematical formulation of quantum mechanics - The relative state interpretation, Mathematical formulation of quantum mechanics - List of mathematical tools

Read more here: » Mathematical formulation of quantum mechanics: Encyclopedia II - Mathematical formulation of quantum mechanics - Mathematical structure of quantum mechanics

The picture given in the preceding paragraphs is sufficient for description of a completely isolated system. However, it fails to account for one of the main differences between quantum mechanics and classical mechanics, that is the effects of measurement. Mathematical formulation of quantum mechanics - Wavefunction collapse. When von Neumann proposed his mathematical postulate for quantum mechanics he included the following "measurement postulate". Carrying out a measurement of an observable ...

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Mathematical formulation of quantum mechanics, Mathematical formulation of quantum mechanics - History of the formalism, Mathematical formulation of quantum mechanics - The old quantum theory and the need for new mathematics, Mathematical formulation of quantum mechanics - The new quantum theory, Mathematical formulation of quantum mechanics - Later developments, Mathematical formulation of quantum mechanics - Mathematical structure of quantum mechanics, Mathematical formulation of quantum mechanics - Postulates of quantum mechanics, Mathematical formulation of quantum mechanics - Pictures of dynamics, Mathematical formulation of quantum mechanics - Representations, Mathematical formulation of quantum mechanics - Time as an operator, Mathematical formulation of quantum mechanics - The problem of measurement, Mathematical formulation of quantum mechanics - Wavefunction collapse, Mathematical formulation of quantum mechanics - The relative state interpretation, Mathematical formulation of quantum mechanics - List of mathematical tools

Read more here: » Mathematical formulation of quantum mechanics: Encyclopedia II - Mathematical formulation of quantum mechanics - The problem of measurement

Differentiation is linear, i.e., D(f + g) = (Df) + (Dg) D(af) = a(Df) where f and g are functions, and a is a constant. Any polynomial in D with function coefficients is also a differential operator. We may also compose differential operators by the rule (D1oDSee also:

Differential operator, Differential operator - Notations, Differential operator - Adjoint of an operator, Differential operator - Properties of differential operators, Differential operator - Several variables, Differential operator - Coordinate-independent description, Differential operator - Examples

Read more here: » Differential operator: Encyclopedia II - Differential operator - Properties of differential operators

In differential geometry and algebraic geometry it is often convenient to have a coordinate-independent description of differential operators between two vector bundles. Let E and F be two vector bundles over a manifold M. An operator is a mapping of sections, P: Γ(E) → Γ(F) which maps the stalk of the sheaf of germs of Γ(E) at a point x ∈ M to the fibre of F at x: ...

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Differential operator, Differential operator - Notations, Differential operator - Adjoint of an operator, Differential operator - Properties of differential operators, Differential operator - Several variables, Differential operator - Coordinate-independent description, Differential operator - Examples

Read more here: » Differential operator: Encyclopedia II - Differential operator - Coordinate-independent description