bound states

In mathematics and physics, continuous spectrum is, roughly speaking, a non-countable set of eigenvalues of an operator. An operator acting on a Hilbert space is said to have a continuous spectrum if its eigenvalues can be changed continuously. If the spectrum of an operator is not continuous, we say that it is has discrete spectrum. Some of the basic questions in spectral theory are to characterise the discrete spectrum and purely continuous spectrum, just as a measure, such as a probability measure, can typically ...

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There are two categories of perturbation theory: time-independent and time-dependent. In this section, we discuss time-independent perturbation theory, in which the perturbation Hamiltonian is static (i.e., possesses no time dependence.) Time-independent perturbation theory was invented by Erwin Schrödinger in 1926, shortly after he invented wave mechanics. We begin with an unperturbed Hamiltonian H0, which is also assumed to have no time dependence. It has known energy levels ...

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Perturbation theory quantum mechanics, Perturbation theory quantum mechanics - Applications of perturbation theory, Perturbation theory quantum mechanics - Time-independent perturbation theory, Perturbation theory quantum mechanics - Effects of degeneracy, Perturbation theory quantum mechanics - Time-dependent perturbation theory

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Let us now consider the potential V(r) = V0 for r < r0, i.e., inside a sphere of radius r0 and zero outside. We first consider bound states, i.e., states which display the particle mostly inside the box (confined states). Those have an energy E less than the potential outside the sphere, i.e., they have negative energy, and we shall see that there are a discrete num ...

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Particle in a spherically symmetric potential, Particle in a spherically symmetric potential - General considerations, Particle in a spherically symmetric potential - Vacuum case, Particle in a spherically symmetric potential - Spherical square well, Particle in a spherically symmetric potential - Infinite spherical square well

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The most ubiquitous, and perhaps simplest, example of a hydrogen bond is found between water molecules. In a discrete water molecule, water has two hydrogen atoms and one oxygen atom. Two molecules of water can form a hydrogen bond between them. The oxygen of one water molecule has two lone pairs of electrons, each of which can form a hydrogen bond with hydrogens on two other water molecules. This can repeat so that every water molecule is H-bonded with four other molecules (two through its two lone pairs, and two through its two hydrogen atoms. ...

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Hydrogen bond, Hydrogen bond - Hydrogen bond in water, Hydrogen bond - Hydrogen bond in proteins and DNA, Hydrogen bond - Symmetric hydrogen bond, Hydrogen bond - Dihydrogen bond, Hydrogen bond - Advanced theory of the hydrogen bond

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Hilbert spaces were named after David Hilbert, who studied them in the context of integral equations. The origin of the designation "der abstrakte Hilbertsche Raum" is John von Neumann in his famous work on unbounded Hermitian operators published in 1929. Von Neumann was perhaps the mathematician who most clearly recognized their importance as a result of his seminal work on the foundations of quantum mechanics begun with Hilbert and Lothar (Wolfgang) Nordheim and continued with Eugene Wigner. The name "Hilbert space" was soon adopted by oth ...

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Hilbert space, Hilbert space - Introduction, Hilbert space - Definition, Hilbert space - Examples, Hilbert space - Euclidean spaces, Hilbert space - Sequence spaces, Hilbert space - Lebesgue spaces, Hilbert space - Sobolev spaces, Hilbert space - Operations on Hilbert spaces, Hilbert space - Bases, Hilbert space - Orthogonal complements and projections, Hilbert space - Reflexivity, Hilbert space - Bounded operators, Hilbert space - Unbounded operators

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The task is to find an expression for the S-Matrix element using the reduction formula. Before starting to accomplish this, it is useful to show the following trick: We will use this in the following calculation: This operation is called particle extraction. This is true because p is not equal to k. See also:

S matrix, S matrix - S-matrix and evolution operator U, S matrix - L.S.Z. Lehman Symanzik Zimmermann reduction formula, S matrix - Wick's theorem, S matrix - Bibliography

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Eigenvalue eigenvector and eigenspace - Computing eigenvalues and eigenvectors of matrices. Suppose that we want to compute the eigenvalues of a given matrix. If the matrix is small, we can compute them symbolically using the characteristic polynomial. However, this is often impossible for larger matrices, in which case we must use a numerical method. For more details on this topic, ...

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Eigenvalue eigenvector and eigenspace, Eigenvalue eigenvector and eigenspace - Definitions, Eigenvalue eigenvector and eigenspace - Examples, Eigenvalue eigenvector and eigenspace - Eigenvalue equation, Eigenvalue eigenvector and eigenspace - Spectral theorem, Eigenvalue eigenvector and eigenspace - Eigenvalues and eigenvectors of matrices, Eigenvalue eigenvector and eigenspace - Computing eigenvalues and eigenvectors of matrices, Eigenvalue eigenvector and eigenspace - Properties, Eigenvalue eigenvector and eigenspace - Conjugate eigenvector, Eigenvalue eigenvector and eigenspace - Generalized eigenvalue problem, Eigenvalue eigenvector and eigenspace - Entries from a ring, Eigenvalue eigenvector and eigenspace - Infinite-dimensional spaces, Eigenvalue eigenvector and eigenspace - Applications, Eigenvalue eigenvector and eigenspace - Notes

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The time independent solution of 3D Schrödinger equation with hamiltonian p2 / 2m0 + V(r) where m0 is the particle's mass, can be separated in the variables r, θ and φ so that the wavefunction ψ reads: are the usual Spherical harmonics, while RSee also:

Particle in a spherically symmetric potential, Particle in a spherically symmetric potential - General considerations, Particle in a spherically symmetric potential - Vacuum case, Particle in a spherically symmetric potential - Spherical square well, Particle in a spherically symmetric potential - Infinite spherical square well

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Hydrogen bonding also plays an important role in determining the three-dimensional structures adopted by proteins and nucleic acids. In these macromolecules, bonding between parts of the same macromolecule cause it to fold into a specific shape, which helps determine the molecule's physiological or biochemical role. The double helical structure of DNA, for example, is due largely to hydrogen bonding between the base pairs, which ...

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Hydrogen bond, Hydrogen bond - Hydrogen bond in water, Hydrogen bond - Hydrogen bond in proteins and DNA, Hydrogen bond - Symmetric hydrogen bond, Hydrogen bond - Dihydrogen bond, Hydrogen bond - Advanced theory of the hydrogen bond

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Let us now consider V(r)=0 (if V0, replace everywhere E with E − V0). Introducing the dimensionless variable the equation becomes a Bessel equation for J defined by (whence the notational choice of J): which regular solutions for positive energies are given by so-called Bessel functions of the first kind Jl + 1 / 2(ρ) so that the soluti ...

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Particle in a spherically symmetric potential, Particle in a spherically symmetric potential - General considerations, Particle in a spherically symmetric potential - Vacuum case, Particle in a spherically symmetric potential - Spherical square well, Particle in a spherically symmetric potential - Infinite spherical square well

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Perturbation theory is an extremely important tool for describing real quantum systems, as it turns out to be very difficult to find exact solutions to the Schrödinger equation for Hamiltonians of even moderate complexity. The Hamiltonians to which we know exact solutions, such as the hydrogen atom, the quantum harmonic oscillator and the particle in a box, are too idealized to adequately describe most systems. Using perturbation theory, we can use the known solutions of these simple Hamiltonians to generate solutions for a range of more co ...

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Perturbation theory quantum mechanics, Perturbation theory quantum mechanics - Applications of perturbation theory, Perturbation theory quantum mechanics - Time-independent perturbation theory, Perturbation theory quantum mechanics - Effects of degeneracy, Perturbation theory quantum mechanics - Time-dependent perturbation theory

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The spectral theorem depicts the importance of the eigenvalues and eigenvectors for characterizing a linear transformation in a unique way. In its simplest version, the spectral theorem states that, under precise conditions, a linear transformation of a vector can be expressed as the linear combination of the eigenvectors with coefficients equal to the eigenvalues times the scalar prod ...

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Eigenvalue eigenvector and eigenspace, Eigenvalue eigenvector and eigenspace - Definitions, Eigenvalue eigenvector and eigenspace - Examples, Eigenvalue eigenvector and eigenspace - Eigenvalue equation, Eigenvalue eigenvector and eigenspace - Spectral theorem, Eigenvalue eigenvector and eigenspace - Eigenvalues and eigenvectors of matrices, Eigenvalue eigenvector and eigenspace - Computing eigenvalues and eigenvectors of matrices, Eigenvalue eigenvector and eigenspace - Properties, Eigenvalue eigenvector and eigenspace - Conjugate eigenvector, Eigenvalue eigenvector and eigenspace - Generalized eigenvalue problem, Eigenvalue eigenvector and eigenspace - Entries from a ring, Eigenvalue eigenvector and eigenspace - Infinite-dimensional spaces, Eigenvalue eigenvector and eigenspace - Applications, Eigenvalue eigenvector and eigenspace - Notes

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So we have where because Substituting the explicit expression for U we obtain: You can see that this formula is not explicitly covariant. ...

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S matrix, S matrix - S-matrix and evolution operator U, S matrix - L.S.Z. Lehman Symanzik Zimmermann reduction formula, S matrix - Wick's theorem, S matrix - Bibliography

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Definition of contraction: Which means that In the end, we approach at Wick's theorem: T Wick's theorem The T-product of a time-ordered free fields string can be expressed in the following manner: Applying this theorem to S-matrix elements, we discover that normal-ordered terms acting on void state give a null contribute to the sum. We conclude that m is even and only completely contracted terms remain ...

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S matrix, S matrix - S-matrix and evolution operator U, S matrix - L.S.Z. Lehman Symanzik Zimmermann reduction formula, S matrix - Wick's theorem, S matrix - Bibliography

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For a Hilbert space H, the continuous linear operators A : H → H are of particular interest. Such a continuous operator is bounded in the sense that it maps bounded sets to bounded sets. This allows to define its norm as The sum and the composition of two continuous linear operators is again continuous and linear. For y in H, the map that sends x to <y, Ax> is linear and continuous, and according to the Riesz representation theorem can theref ...

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Hilbert space, Hilbert space - Introduction, Hilbert space - Definition, Hilbert space - Examples, Hilbert space - Euclidean spaces, Hilbert space - Sequence spaces, Hilbert space - Lebesgue spaces, Hilbert space - Sobolev spaces, Hilbert space - Operations on Hilbert spaces, Hilbert space - Bases, Hilbert space - Orthogonal complements and projections, Hilbert space - Reflexivity, Hilbert space - Bounded operators, Hilbert space - Unbounded operators

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If the vector space is infinite dimensional, it may be advantageous to define the concept of spectral values. The spectral values are the set of scalars λ for which the Green's operator, , associated to the transformation is not defined, that is such that is not invertible (i.e., the inverse transformation to does not exist). If λ is an eigenvalue of , λ is also a spectral value of it. However, the reverse relation is not true: any spectral value is not an eigenvalue. There are operators on Hilbert or Banach ...

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Eigenvalue eigenvector and eigenspace, Eigenvalue eigenvector and eigenspace - Definitions, Eigenvalue eigenvector and eigenspace - Examples, Eigenvalue eigenvector and eigenspace - Eigenvalue equation, Eigenvalue eigenvector and eigenspace - Spectral theorem, Eigenvalue eigenvector and eigenspace - Eigenvalues and eigenvectors of matrices, Eigenvalue eigenvector and eigenspace - Computing eigenvalues and eigenvectors of matrices, Eigenvalue eigenvector and eigenspace - Properties, Eigenvalue eigenvector and eigenspace - Conjugate eigenvector, Eigenvalue eigenvector and eigenspace - Generalized eigenvalue problem, Eigenvalue eigenvector and eigenspace - Entries from a ring, Eigenvalue eigenvector and eigenspace - Infinite-dimensional spaces, Eigenvalue eigenvector and eigenspace - Applications, Eigenvalue eigenvector and eigenspace - Notes

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An example of an eigenvalue equation where the transformation is represented in terms of a differential operator is the time-independent Schrödinger equation in quantum mechanics HΨE = EΨE where H, the Hamiltonian, is a second-order differential operator and ΨE, the ...

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Eigenvalue eigenvector and eigenspace, Eigenvalue eigenvector and eigenspace - Definitions, Eigenvalue eigenvector and eigenspace - Examples, Eigenvalue eigenvector and eigenspace - Eigenvalue equation, Eigenvalue eigenvector and eigenspace - Spectral theorem, Eigenvalue eigenvector and eigenspace - Eigenvalues and eigenvectors of matrices, Eigenvalue eigenvector and eigenspace - Computing eigenvalues and eigenvectors of matrices, Eigenvalue eigenvector and eigenspace - Properties, Eigenvalue eigenvector and eigenspace - Conjugate eigenvector, Eigenvalue eigenvector and eigenspace - Generalized eigenvalue problem, Eigenvalue eigenvector and eigenspace - Entries from a ring, Eigenvalue eigenvector and eigenspace - Infinite-dimensional spaces, Eigenvalue eigenvector and eigenspace - Applications, Eigenvalue eigenvector and eigenspace - Notes

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The spectral theorem depicts the whole importance of the eigenvalues and eigenvectors for characterizing a linear transformation in a unique way. In its simplest version, the spectral theorem states that, under precise conditions, a linear transformation of a vector can be expressed as the linear combination of the eigenvectors with coefficients equal to the eigenvalues times the scalar prod ...

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Eigenvalue eigenvector and eigenspace, Eigenvalue eigenvector and eigenspace - Definitions, Eigenvalue eigenvector and eigenspace - Examples, Eigenvalue eigenvector and eigenspace - Eigenvalue equation, Eigenvalue eigenvector and eigenspace - Spectral theorem, Eigenvalue eigenvector and eigenspace - Eigenvalues and eigenvectors of matrices, Eigenvalue eigenvector and eigenspace - Computing eigenvalues and eigenvectors of matrices, Eigenvalue eigenvector and eigenspace - Properties, Eigenvalue eigenvector and eigenspace - Conjugate eigenvector, Eigenvalue eigenvector and eigenspace - Generalized eigenvalue problem, Eigenvalue eigenvector and eigenspace - Entries from a ring, Eigenvalue eigenvector and eigenspace - Infinite-dimensional spaces, Eigenvalue eigenvector and eigenspace - Applications, Eigenvalue eigenvector and eigenspace - Notes

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Mathematically, vλ is an eigenvector and λ the corresponding eigenvalue of a transformation if the equation: is true, where is the vector obtained when applying the transformation to vλ. Suppose is a linear transformation (which means that for all scalars a, b, and vectors v, w). Consider a basis in that vector space. Then, and vλ can be represented relative to that basis by a matrix T< ...

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Eigenvalue eigenvector and eigenspace, Eigenvalue eigenvector and eigenspace - Definitions, Eigenvalue eigenvector and eigenspace - Examples, Eigenvalue eigenvector and eigenspace - Eigenvalue equation, Eigenvalue eigenvector and eigenspace - Spectral theorem, Eigenvalue eigenvector and eigenspace - Eigenvalues and eigenvectors of matrices, Eigenvalue eigenvector and eigenspace - Computing eigenvalues and eigenvectors of matrices, Eigenvalue eigenvector and eigenspace - Properties, Eigenvalue eigenvector and eigenspace - Conjugate eigenvector, Eigenvalue eigenvector and eigenspace - Generalized eigenvalue problem, Eigenvalue eigenvector and eigenspace - Entries from a ring, Eigenvalue eigenvector and eigenspace - Infinite-dimensional spaces, Eigenvalue eigenvector and eigenspace - Applications, Eigenvalue eigenvector and eigenspace - Notes

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As the Earth rotates, every arrow pointing outward from the center of the Earth also rotates, except those arrows that lie on the axis of rotation. Consider the transformation of the Earth after one hour of rotation: An arrow from the center of the Earth to the Geographic South Pole would be an eigenvector of this transformation, but an arrow from the center of the Earth to anywhere on the equator would not be an eigenvector. Since the arrow pointing at the pole i ...

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Eigenvalue eigenvector and eigenspace, Eigenvalue eigenvector and eigenspace - Definitions, Eigenvalue eigenvector and eigenspace - Examples, Eigenvalue eigenvector and eigenspace - Eigenvalue equation, Eigenvalue eigenvector and eigenspace - Spectral theorem, Eigenvalue eigenvector and eigenspace - Eigenvalues and eigenvectors of matrices, Eigenvalue eigenvector and eigenspace - Computing eigenvalues and eigenvectors of matrices, Eigenvalue eigenvector and eigenspace - Properties, Eigenvalue eigenvector and eigenspace - Conjugate eigenvector, Eigenvalue eigenvector and eigenspace - Generalized eigenvalue problem, Eigenvalue eigenvector and eigenspace - Entries from a ring, Eigenvalue eigenvector and eigenspace - Infinite-dimensional spaces, Eigenvalue eigenvector and eigenspace - Applications, Eigenvalue eigenvector and eigenspace - Notes

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