 # 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.

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. The density operator for this system is 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 An example of pure and mixed states is light polarization. . and . . .

Free particle Applications 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

Interpretations of quantum mechanics An interpretation of quantum mechanics is a set of statements which attempt to explain how quantum mechanics informs our understanding of nature. Although quantum mechanics has held up to rigorous and thorough experimental testing, many of these experiments are open to different interpretations. There exist a number of contending schools of thought, differing over whether quantum mechanics can be understood to be deterministic, which elements of quantum mechanics can be considered "real", and other matters. This question is of special interest to philosophers of physics, as physicists continue to show a strong interest in the subject. They usually consider an interpretation of quantum mechanics as an interpretation of the mathematical formalism of quantum mechanics, specifying the physical meaning of the mathematical entities of the theory. History of interpretations Main quantum mechanics interpreters Nature of interpretation Two qualities vary among interpretations:

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. However, when the well becomes very narrow (on the scale of a few nanometers), quantum effects become important. The particle may only occupy certain positive energy levels. 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 simplest form of the particle in a box model considers a one-dimensional system. where is the length of the box and is time. and . . .

Quantum Physics Revealed As Non-Mysterious This is one of several shortened indices into the Quantum Physics Sequence. Hello! You may have been directed to this page because you said something along the lines of "Quantum physics shows that reality doesn't exist apart from our observation of it," or "Science has disproved the idea of an objective reality," or even just "Quantum physics is one of the great mysteries of modern science; no one understands how it works." There was a time, roughly the first half-century after quantum physics was invented, when this was more or less true. Certainly, when quantum physics was just being discovered, scientists were very confused indeed! The series of posts indexed below will show you - not just tell you - what's really going on down there. Some optional preliminaries you might want to read: Reductionism: We build models of the universe that have many different levels of description. And here's the main sequence:

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. The quantum mechanical interpretation is unlike the classical interpretation, where if the total energy of the particle is less than potential energy barrier of the walls it cannot be found outside the box. In the quantum interpretation, there is a non-zero probability of the particle being outside the box even when the energy of the particle is less than the potential energy barrier of the walls (cf quantum tunnelling). 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, at x = -L/2 and x = L/2. and

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. Since 2000 there has been extensive experimental work on quantum gases, particularly Bose–Einstein Condensates. 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). with Ψ₀ the amplitude and

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. C,D,E,F, but not G,H, are energy eigenstates. H is a coherent state, a quantum state which approximates the classical trajectory. 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 and Proof:

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