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Quantum mechanics for chemists
Kedar Nath Force free motion of rigid body - Cases of symmetric top - Heavy. You can hone your targeting by slowly exhaling, which slows time and zooms in slightly. Introduction of Electrodynamics - Griffith. A function f is in H 2 U if and only if. Thus H 2 U consists of those functions that are L 2 on the circle, and whose negative frequency Fourier coefficients vanish. The Bergman spaces are another family of Hilbert spaces of holomorphic functions.
Clearly L 2, h D is a subspace of L 2 D ; in fact, it is a closed subspace, and so a Hilbert space in its own right. This is a consequence of the estimate, valid on compact subsets K of D , that. Thus convergence of a sequence of holomorphic functions in L 2 D implies also compact convergence , and so the limit function is also holomorphic.
Another consequence of this inequality is that the linear functional that evaluates a function f at a point of D is actually continuous on L 2, h D. The Riesz representation theorem implies that the evaluation functional can be represented as an element of L 2, h D. The integrand. This integral kernel satisfies a reproducing property. For instance, in harmonic analysis the Poisson kernel is a reproducing kernel for the Hilbert space of square-integrable harmonic functions in the unit ball.
That the latter is a Hilbert space at all is a consequence of the mean value theorem for harmonic functions. Many of the applications of Hilbert spaces exploit the fact that Hilbert spaces support generalizations of simple geometric concepts like projection and change of basis from their usual finite dimensional setting. In particular, the spectral theory of continuous self-adjoint linear operators on a Hilbert space generalizes the usual spectral decomposition of a matrix , and this often plays a major role in applications of the theory to other areas of mathematics and physics.
In the theory of ordinary differential equations , spectral methods on a suitable Hilbert space are used to study the behavior of eigenvalues and eigenfunctions of differential equations. For example, the Sturm—Liouville problem arises in the study of the harmonics of waves in a violin string or a drum, and is a central problem in ordinary differential equations.
Hilbert spaces form a basic tool in the study of partial differential equations. Many weak formulations involve the class of Sobolev functions , which is a Hilbert space. A suitable weak formulation reduces to a geometrical problem the analytic problem of finding a solution or, often what is more important, showing that a solution exists and is unique for given boundary data. For linear elliptic equations, one geometrical result that ensures unique solvability for a large class of problems is the Lax—Milgram theorem.
This strategy forms the rudiment of the Galerkin method a finite element method for numerical solution of partial differential equations. The question then reduces to finding u in this space such that for all v in this space. The Lax—Milgram theorem then ensures the existence and uniqueness of solutions of this equation. Hilbert spaces allow for many elliptic partial differential equations to be formulated in a similar way, and the Lax—Milgram theorem is then a basic tool in their analysis. With suitable modifications, similar techniques can be applied to parabolic partial differential equations and certain hyperbolic partial differential equations.
The field of ergodic theory is the study of the long-term behavior of chaotic dynamical systems. The protypical case of a field that ergodic theory applies to is thermodynamics , in which—though the microscopic state of a system is extremely complicated it is impossible to understand the ensemble of individual collisions between particles of matter —the average behavior over sufficiently long time intervals is tractable.
The laws of thermodynamics are assertions about such average behavior. In particular, one formulation of the zeroth law of thermodynamics asserts that over sufficiently long timescales, the only functionally independent measurement that one can make of a thermodynamic system in equilibrium is its total energy, in the form of temperature. An ergodic dynamical system is one for which, apart from the energy—measured by the Hamiltonian —there are no other functionally independent conserved quantities on the phase space.
The von Neumann mean ergodic theorem  states the following:. That is, the long time average of an observable f is equal to its expectation value over an energy surface. One of the basic goals of Fourier analysis is to decompose a function into a possibly infinite linear combination of given basis functions: the associated Fourier series.
The classical Fourier series associated to a function f defined on the interval [0, 1] is a series of the form. The example of adding up the first few terms in a Fourier series for a sawtooth function is shown in the figure. All basis functions have nodes at the nodes of the sawtooth, but all but the fundamental have additional nodes.
The oscillation of the summed terms about the sawtooth is called the Gibbs phenomenon. A significant problem in classical Fourier series asks in what sense the Fourier series converges, if at all, to the function f. Hilbert space methods provide one possible answer to this question. Consequently, any square-integrable function can be expressed as a series.
The problem can also be studied from the abstract point of view: every Hilbert space has an orthonormal basis , and every element of the Hilbert space can be written in a unique way as a sum of multiples of these basis elements. The coefficients appearing on these basis elements are sometimes known abstractly as the Fourier coefficients of the element of the space. In many circumstances, it is desirable not to decompose a function into trigonometric functions, but rather into orthogonal polynomials or wavelets for instance,  and in higher dimensions into spherical harmonics.
For instance, if e n are any orthonormal basis functions of L 2 [0, 1] , then a given function in L 2 [0, 1] can be approximated as a finite linear combination . In various applications to physical problems, a function can be decomposed into physically meaningful eigenfunctions of a differential operator typically the Laplace operator : this forms the foundation for the spectral study of functions, in reference to the spectrum of the differential operator.
Spectral theory also underlies certain aspects of the Fourier transform of a function. Whereas Fourier analysis decomposes a function defined on a compact set into the discrete spectrum of the Laplacian which corresponds to the vibrations of a violin string or drum , the Fourier transform of a function is the decomposition of a function defined on all of Euclidean space into its components in the continuous spectrum of the Laplacian.
The Fourier transformation is also geometrical, in a sense made precise by the Plancherel theorem , that asserts that it is an isometry of one Hilbert space the "time domain" with another the "frequency domain". This isometry property of the Fourier transformation is a recurring theme in abstract harmonic analysis , as evidenced for instance by the Plancherel theorem for spherical functions occurring in noncommutative harmonic analysis. In the mathematically rigorous formulation of quantum mechanics , developed by John von Neumann ,  the possible states more precisely, the pure states of a quantum mechanical system are represented by unit vectors called state vectors residing in a complex separable Hilbert space, known as the state space , well defined up to a complex number of norm 1 the phase factor.
Broyden's method in Hilbert space | SpringerLink
In other words, the possible states are points in the projectivization of a Hilbert space, usually called the complex projective space. The exact nature of this Hilbert space is dependent on the system; for example, the position and momentum states for a single non-relativistic spin zero particle is the space of all square-integrable functions, while the states for the spin of a single proton are unit elements of the two-dimensional complex Hilbert space of spinors. Each observable is represented by a self-adjoint linear operator acting on the state space.
Each eigenstate of an observable corresponds to an eigenvector of the operator, and the associated eigenvalue corresponds to the value of the observable in that eigenstate. The inner product between two state vectors is a complex number known as a probability amplitude.
During an ideal measurement of a quantum mechanical system, the probability that a system collapses from a given initial state to a particular eigenstate is given by the square of the absolute value of the probability amplitudes between the initial and final states. The possible results of a measurement are the eigenvalues of the operator—which explains the choice of self-adjoint operators, for all the eigenvalues must be real.
The probability distribution of an observable in a given state can be found by computing the spectral decomposition of the corresponding operator. For a general system, states are typically not pure, but instead are represented as statistical mixtures of pure states, or mixed states, given by density matrices : self-adjoint operators of trace one on a Hilbert space. Moreover, for general quantum mechanical systems, the effects of a single measurement can influence other parts of a system in a manner that is described instead by a positive operator valued measure.
Thus the structure both of the states and observables in the general theory is considerably more complicated than the idealization for pure states.
Quantum Mechanics Lecture Notes Pdf
Any true physical color can be represented by a combination of pure spectral colors. As physical colors can be composed of any number of spectral colors, the space of physical colors may aptly be represented by a Hilbert space over spectral colors. Humans have three types of cone cells for color perception, so the perceivable colors can be represented by 3-dimensional Euclidean space.
The many-to-one linear mapping from the Hilbert space of physical colors to the Euclidean space of human perceivable colors explains why many distinct physical colors may be perceived by humans to be identical e. When u and v are orthogonal, one has. By induction on n , this is extended to any family u 1 , …, u n of n orthogonal vectors,. Whereas the Pythagorean identity as stated is valid in any inner product space, completeness is required for the extension of the Pythagorean identity to series.
Furthermore, the sum of a series of orthogonal vectors is independent of the order in which it is taken. By definition, every Hilbert space is also a Banach space. Furthermore, in every Hilbert space the following parallelogram identity holds:. Conversely, every Banach space in which the parallelogram identity holds is a Hilbert space, and the inner product is uniquely determined by the norm by the polarization identity.
The parallelogram law implies that any Hilbert space is a uniformly convex Banach space. This subsection employs the Hilbert projection theorem. More generally, this holds in any uniformly convex Banach space. This result is especially significant in applied mathematics , especially numerical analysis , where it forms the basis of least squares methods.
A very useful criterion is obtained by applying this observation to the closed subspace F generated by a subset S of H. It carries a natural norm, defined by. This norm satisfies the parallelogram law , and so the dual space is also an inner product space. The dual space is also complete, and so it is a Hilbert space in its own right.
The Riesz representation theorem affords a convenient description of the dual. The Riesz representation theorem states that this mapping is an antilinear isomorphism. In the real case, the antilinear isomorphism from H to its dual is actually an isomorphism, and so real Hilbert spaces are naturally isomorphic to their own duals. The Riesz representation theorem relies fundamentally not just on the presence of an inner product, but also on the completeness of the space.
In fact, the theorem implies that the topological dual of any inner product space can be identified with its completion. An immediate consequence of the Riesz representation theorem is also that a Hilbert space H is reflexive , meaning that the natural map from H into its double dual space is an isomorphism. Conversely, every bounded sequence in a Hilbert space admits weakly convergent subsequences Alaoglu's theorem. Among several variants, one simple statement is as follows: . This fact and its various generalizations are fundamental for direct methods in the calculus of variations.
Minimization results for convex functionals are also a direct consequence of the slightly more abstract fact that closed bounded convex subsets in a Hilbert space H are weakly compact , since H is reflexive.
Any general property of Banach spaces continues to hold for Hilbert spaces. The open mapping theorem states that a continuous surjective linear transformation from one Banach space to another is an open mapping meaning that it sends open sets to open sets. A corollary is the bounded inverse theorem , that a continuous and bijective linear function from one Banach space to another is an isomorphism that is, a continuous linear map whose inverse is also continuous.
This theorem is considerably simpler to prove in the case of Hilbert spaces than in general Banach spaces. The geometrical Hahn—Banach theorem asserts that a closed convex set can be separated from any point outside it by means of a hyperplane of the Hilbert space. This is an immediate consequence of the best approximation property: if y is the element of a closed convex set F closest to x , then the separating hyperplane is the plane perpendicular to the segment xy passing through its midpoint.
Conversely, if an operator is bounded, then it is continuous. The space of such bounded linear operators has a norm , the operator norm given by. The sum and the composite of two bounded linear operators is again bounded and linear. A converse is also true in the sense that, for a non-negative operator A , there exists a unique non-negative square root B such that. In a sense made precise by the spectral theorem , self-adjoint operators can usefully be thought of as operators that are "real".
Normal operators decompose into the sum of a self-adjoint operators and an imaginary multiple of a self adjoint operator. Normal operators can also usefully be thought of in terms of their real and imaginary parts. The unitary operators form a group under composition, which is the isometry group of H. An element of B H is compact if it sends bounded sets to relatively compact sets. Many integral operators are compact, and in fact define a special class of operators known as Hilbert—Schmidt operators that are especially important in the study of integral equations.
Fredholm operators differ from a compact operator by a multiple of the identity, and are equivalently characterized as operators with a finite dimensional kernel and cokernel.