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Quantum harmonic oscillator

Quantum harmonic oscillator
Some trajectories of a harmonic oscillator according to Newton's laws of classical mechanics (A-B), and according to the Schrödinger equation of quantum mechanics (C-H). In A-B, the particle (represented as a ball attached to a spring) oscillates back and forth. In C-H, some solutions to the Schrödinger Equation are shown, where the horizontal axis is position, and the vertical axis is the real part (blue) or imaginary part (red) of the wavefunction. The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. One-dimensional harmonic oscillator[edit] Hamiltonian and energy eigenstates[edit] Wavefunction representations for the first eight bound eigenstates, n = 0 to 7. Corresponding probability densities. where m is the particle's mass, ω is the angular frequency of the oscillator, is the position operator, and is the momentum operator, given by One may write the time-independent Schrödinger equation, The functions Hn are the Hermite polynomials, and

Finite potential well The finite potential well (also known as the finite square well) is a concept from quantum mechanics. It is an extension of the infinite potential well, in which a particle is confined to a box, but one which has finite potential walls. Unlike the infinite potential well, there is a probability associated with the particle being found outside the box. Particle in a 1-dimensional box[edit] For the 1-dimensional case on the x-axis, the time-independent Schrödinger equation can be written as: where is Planck's constant, is the mass of the particle, is the (complex valued) wavefunction that we want to find, is a function describing the potential energy at each point x, and is the energy, a real number, sometimes called eigenenergy. For the case of the particle in a 1-dimensional box of length L, the potential is zero inside the box, but rises abruptly to a value at x = -L/2 and x = L/2. Inside the box[edit] For the region inside the box V(x) = 0 and Equation 1 reduces to Letting the equation becomes .

Mehler kernel In mathematics and physics, the Mehler kernel is the fundamental solution, or non-relativistic propagator of the Hamiltonian for the quantum harmonic oscillator. Mehler (1866) gave an explicit formula for it, called Mehler's formula. The Mehler kernel provides the fundamental solution for the most general solution φ(x, y; t) to Specifically, Mehler's kernel is By a simple transformation, this is, apart from a multiplying factor, the bivariate Gaussian probability density given by When t=0 variables x and y coincide resulting in the formula The bivariate probability density can be written as an infinite series involving the one-dimensional probability densities and Hermite polynomials of x and y.

Particle in a box In quantum mechanics, the particle in a box model (also known as the infinite potential well or the infinite square well) describes a particle free to move in a small space surrounded by impenetrable barriers. The model is mainly used as a hypothetical example to illustrate the differences between classical and quantum systems. In classical systems, for example a ball trapped inside a large box, the particle can move at any speed within the box and it is no more likely to be found at one position than another. The particle in a box model provides one of the very few problems in quantum mechanics which can be solved analytically, without approximations. One-dimensional solution[edit] The barriers outside a one-dimensional box have infinitely large potential, while the interior of the box has a constant, zero potential. The simplest form of the particle in a box model considers a one-dimensional system. where is the length of the box and is the position of the particle within the box. and . .

Mock modular form In mathematics, a mock modular form is the holomorphic part of a harmonic weak Maass form, and a mock theta function is essentially a mock modular form of weight 1/2. The first examples of mock theta functions were described by Srinivasa Ramanujan in his last 1920 letter to G. H. History[edit] "Suppose there is a function in the Eulerian form and suppose that all or an infinity of points are exponential singularities, and also suppose that at these points the asymptotic form closes as neatly as in the cases of (A) and (B). Ramanujan's original definition of a mock theta function, from (Ramanujan 2000, Appendix II) Ramanujan's 12 January 1920 letter to Hardy, reprinted in (Ramanujan 2000, Appendix II), listed 17 examples of functions that he called mock theta functions, and his lost notebook (Ramanujan 1988) contained several more examples. Ramanujan associated an order to his mock theta functions, which was not clearly defined. Definition[edit] Fix a weight k, usually with 2k integral.

Macroscopic quantum phenomena Quantum mechanics is most often used to describe matter on the scale of molecules, atoms, or elementary particles. However some phenomena, particularly at low temperatures, show quantum behavior on a macroscopic scale. The best-known examples of macroscopic quantum phenomena are superfluidity and superconductivity; another example is the quantum Hall effect. Between 1996 to 2003 four Nobel prizes were given for work related to macroscopic quantum phenomena.[1] Macroscopic quantum phenomena can be observed in superfluid helium and in superconductors,[2] but also in dilute quantum gases and in laser light. Quantum phenomena are generally classified as macroscopic when the quantum states are occupied by a large number of particles (typically Avogadro's number) or the quantum states involved are macroscopic in size (up to km size in superconducting wires). Consequences of the macroscopic occupation[edit] Fig.1 Left: only one particle; usually the small box is empty. with Ψ₀ the amplitude and

Harmonic series (mathematics) In mathematics, the harmonic series is the divergent infinite series: The fact that the harmonic series diverges was first proven in the 14th century by Nicole Oresme,[1] but this achievement fell into obscurity. Proofs were given in the 17th century by Pietro Mengoli,[2] Johann Bernoulli,[3] and Jacob Bernoulli.[4] The harmonic series is counterintuitive to students first encountering it, because it is a divergent series though the limit of the nth term as n goes to infinity is zero. Because the series gets arbitrarily large as n becomes larger, eventually this ratio must exceed 1, which implies that the worm reaches the end of the rubber band. Another example is: given a collection of identical dominoes, it is clearly possible to stack them at the edge of a table so that they hang over the edge of the table without falling. There are several well-known proofs of the divergence of the harmonic series. for every positive integer k. The harmonic series diverges very slowly. where The series

Solution of Schrödinger equation for a step potential In quantum mechanics and scattering theory, the one-dimensional step potential is an idealized system used to model incident, reflected and transmitted matter waves. The problem consists of solving the time-independent Schrödinger equation for a particle with a step-like potential in one dimension. Typically, the potential is modelled as a Heaviside step function. Calculation[edit] Schrödinger equation and potential function[edit] Scattering at a finite potential step of height V0, shown in green. The time-independent Schrödinger equation for the wave function is The barrier is positioned at x = 0, though any position x0 may be chosen without changing the results, simply by shifting position of the step by −x0. The first term in the Hamiltonian, is the kinetic energy of the particle. Solution[edit] The step divides space in two parts: x < 0 and x > 0. both of which have the same form as the De Broglie relation (in one dimension) Boundary conditions[edit] Transmission and reflection[edit]

Zonal spherical harmonics In the mathematical study of rotational symmetry, the zonal spherical harmonics are special spherical harmonics that are invariant under the rotation through a particular fixed axis. The zonal spherical functions are a broad extension of the notion of zonal spherical harmonics to allow for a more general symmetry group. On the two-dimensional sphere, the unique zonal spherical harmonic of degree ℓ invariant under rotations fixing the north pole is represented in spherical coordinates by where Pℓ is a Legendre polynomial of degree ℓ. , where x is a point on the sphere representing the fixed axis, and y is the variable of the function. In n-dimensional Euclidean space, zonal spherical harmonics are defined as follows. to be the dual representation of the linear functional in the finite-dimensional Hilbert space Hℓ of spherical harmonics of degree ℓ. for all Y ∈ Hℓ. Relationship with harmonic potentials[edit] where is the surface area of the (n-1)-dimensional sphere. Properties[edit] If Y1,...

Free particle Density matrix Explicitly, suppose a quantum system may be found in state with probability p1, or it may be found in state with probability p2, or it may be found in state with probability p3, and so on. By choosing a basis (which need not be orthogonal), one may resolve the density operator into the density matrix, whose elements are[1] For an operator (which describes an observable is given by[1] In words, the expectation value of A for the mixed state is the sum of the expectation values of A for each of the pure states Mixed states arise in situations where the experimenter does not know which particular states are being manipulated. Pure and mixed states[edit] In quantum mechanics, a quantum system is represented by a state vector (or ket) . is called a pure state. and a 50% chance that the state vector is . A mixed state is different from a quantum superposition. Example: Light polarization[edit] The incandescent light bulb (1) emits completely random polarized photons (2) with mixed state density matrix .

Matrix mechanics Matrix mechanics is a formulation of quantum mechanics created by Werner Heisenberg, Max Born, and Pascual Jordan in 1925. In some contrast to the wave formulation, it produces spectra of energy operators by purely algebraic, ladder operator, methods.[1] Relying on these methods, Pauli derived the hydrogen atom spectrum in 1926,[2] before the development of wave mechanics. Development of matrix mechanics[edit] In 1925, Werner Heisenberg, Max Born, and Pascual Jordan formulated the matrix mechanics representation of quantum mechanics. Epiphany at Helgoland[edit] In 1925 Werner Heisenberg was working in Göttingen on the problem of calculating the spectral lines of hydrogen. "It was about three o' clock at night when the final result of the calculation lay before me. The Three Fundamental Papers[edit] After Heisenberg returned to Göttingen, he showed Wolfgang Pauli his calculations, commenting at one point:[4] In the paper, Heisenberg formulated quantum theory without sharp electron orbits. W.

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