 # 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 Hamiltonian and energy eigenstates 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 Related:  QUANTUM PHYSICS 2Physics

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 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 For the region inside the box V(x) = 0 and Equation 1 reduces to Letting the equation becomes .

Quantum reflection Quantum reflection is a physical phenomenon involving the reflection of a matter wave from an attractive potential. In classical mechanics, such a phenomenon is not possible; for instance when one magnet is pulled toward another, the observer does not expect one of the magnets to suddenly (i.e. before the magnets 'touch') turn around and retreat in the opposite direction. Definition Quantum reflection became an important branch of physics in the 21st century. In a workshop about quantum reflection, the following definition of quantum reflection was suggested: Quantum reflection is a classically counterintuitive phenomenon whereby the motion of particles is reverted "against the force" acting on them. Observation of quantum reflection has become possible thanks to recent advances in trapping and cooling atoms. Reflection of slow atoms Single-dimensional approximation and ), such that only a single coordinate (say ) is important. axis, where is the atomic mass, Fig. . , where .

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 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 . .

Fresnel diffraction In optics, the Fresnel diffraction equation for near-field diffraction, is an approximation of Kirchhoff-Fresnel diffraction that can be applied to the propagation of waves in the near field. It is used to calculate the diffraction pattern created by waves passing through an aperture or around an object, when viewed from relatively close to the object. In contrast the diffraction pattern in the far field region is given by the Fraunhofer diffraction equation. The near field can be specified by the Fresnel number, F of the optical arrangement. When the diffracted wave is considered to be in the near field. However, the validity of the Fresnel diffraction integral is deduced by the approximations derived below. where is the maximal angle described by , a and L the same as in the definition of the Fresnel number. The multiple Fresnel diffraction at nearly placed periodical ridges (ridged mirror) causes the specular reflection; this effect can be used for atomic mirrors. is the aperture, .

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 For an operator (which describes an observable is given by 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 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 The incandescent light bulb (1) emits completely random polarized photons (2) with mixed state density matrix .

Doughnut theory of the universe Bloom Toroidal Model of the Universe The doughnut theory of the universe is an informal description of the theory that the shape of the universe is a three-dimensional torus. The name comes from the shape of a doughnut, whose surface has the topology of a two-dimensional torus. The foundation for the doughnut theory started with Bell Lab’s discovery of cosmic microwave background (CMB). Supporting evidence Dr. Cosmic Background Explorer (COBE) The Cosmic Background Explorer was an explorer satellite launched in 1989 by NASA that used a Far Infrared Absolute Spectrometer (FIRAS) to measure the radiation of the universe. Led by researchers John C. Wilkinson Microwave Anisotropy Probe (WMAP) WMAP cosmic microwave background map The Wilkinson Microwave Anisotropy Probe (WMAP) was launched in 2001 as NASA’s second explorer satellite intended to map the precise distribution of CMB across the universe. Notes and references Notes References External links

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. Relying on these methods, Pauli derived the hydrogen atom spectrum in 1926, before the development of wave mechanics. Development of matrix mechanics In 1925, Werner Heisenberg, Max Born, and Pascual Jordan formulated the matrix mechanics representation of quantum mechanics. Epiphany at Helgoland 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 After Heisenberg returned to Göttingen, he showed Wolfgang Pauli his calculations, commenting at one point: In the paper, Heisenberg formulated quantum theory without sharp electron orbits. W.

Birefringence A calcite crystal laid upon a graph paper with blue lines showing the double refraction Doubly refracted image as seen through a calcite crystal, seen through a rotating polarizing filter illustrating the opposite polarization states of the two images. Explanation The simplest (and most common) type of birefringence is that of materials with uniaxial anisotropy. What's more, the extraordinary ray is an inhomogeneous wave whose power flow (given by the Poynting vector) is not exactly parallel to the wave vector. When the light propagates either along or orthogonal to the optic axis, such a lateral shift does not occur. The more general case of biaxially anisotropic materials, also known as trirefringent[citation needed] materials, is substantially more complex. Then there are three refractive indices corresponding to three principal axes of the crystal. Sources of optical birefringence Examples of uniaxial birefringent materials Fast and slow rays . and

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 Schrödinger equation and potential function 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 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 Transmission and reflection

Cotton–Mouton effect In physical optics, the Cotton–Mouton effect refers to birefringence in a liquid in the presence of a constant transverse magnetic field. It is a similar but stronger effect than the Voigt effect (in which the medium is a gas instead of a liquid). The electric analog is the Kerr effect. It was discovered in 1907 by Aimé Cotton and Henri Mouton, working in collaboration. When a linearly polarized wave propagates perpendicular to magnetic field (e.g. in a magnetized plasma), it can become elliptized. Because a linearly polarized wave is some combination of in-phase X & O modes, and because X & O waves propagate with different phase velocities, this causes elliptization of the emerging beam. Cotton effect

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. Macroscopic quantum phenomena can be observed in superfluid helium and in superconductors, 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 Fig.1 Left: only one particle; usually the small box is empty. with Ψ₀ the amplitude and

Calabi–Yau manifold A 2D slice of the 6D Calabi-Yau quintic manifold. Calabi–Yau manifolds are complex manifolds that are higher-dimensional analogues of K3 surfaces. They are sometimes defined as compact Kähler manifolds whose canonical bundle is trivial, though many other similar but inequivalent definitions are sometimes used. They were named "Calabi–Yau spaces" by Candelas et al. (1985) after E. Definitions There are many different inequivalent definitions of a Calabi–Yau manifold used by different authors. A Calabi–Yau n-fold or Calabi–Yau manifold of (complex) dimension n is sometimes defined as a compact n-dimensional Kähler manifold M satisfying one of the following equivalent conditions: These conditions imply that the first integral Chern class c1(M) of M vanishes, but the converse is not true. In particular if a compact Kähler manifold is simply connected then the weak definition above is equivalent to the stronger definition. Examples All hyper-Kähler manifolds are Calabi–Yau.

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