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.
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[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]
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.[1] Macroscopic quantum phenomena can be observed in superfluid helium and in superconductors,[2] but also in dilute quantum gases and in laser light. Although these media are very different, their behavior is very similar as they all show macroscopic quantum behavior. Consequences of the macroscopic occupation[edit] Fig.1 Left: only one particle; usually the small box is empty. with Ψ₀ the amplitude and the phase. 1. 2. 3. and
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. Likewise, it can never have zero energy, meaning that the particle can never "sit still". 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] where is the length of the box and is time. and
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[edit] Main quantum mechanics interpreters Nature of interpretation[edit] Two qualities vary among interpretations:
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[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, at x = -L/2 and x = L/2. 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! But time passed, and science moved on. 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:
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[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 and Proof:
EPR paradox Albert Einstein The EPR paradox is an early and influential critique leveled against the Copenhagen interpretation of quantum mechanics. Albert Einstein and his colleagues Boris Podolsky and Nathan Rosen (known collectively as EPR) designed a thought experiment which revealed that the accepted formulation of quantum mechanics had a consequence which had not previously been noticed, but which looked unreasonable at the time. The scenario described involved the phenomenon that is now known as quantum entanglement. According to quantum mechanics, under some conditions, a pair of quantum systems may be described by a single wave function, which encodes the probabilities of the outcomes of experiments that may be performed on the two systems, whether jointly or individually. The routine explanation of this effect was, at that time, provided by Heisenberg's uncertainty principle. The EPR paper, written in 1935, was intended to illustrate that this explanation is inadequate. EPR paper[edit]
DNA molecules can 'teleport', Nobel Prize winner claims A Nobel Prize winning biologist has ignited controversy after publishing details of an experiment in which a fragment of DNA appeared to ‘teleport’ or imprint itself between test tubes. According to a team headed by Luc Montagnier, previously known for his work on HIV and AIDS, two test tubes, one of which contained a tiny piece of bacterial DNA, the other pure water, were surrounded by a weak electromagnetic field of 7Hz. Eighteen hours later, after DNA amplification using a polymerase chain reaction, as if by magic the DNA was detectable in the test tube containing pure water. Oddly, the original DNA sample had to be diluted many times over for the experiment to work, which might explain why the phenomenon has not been detected before, assuming that this is what has happened. The phenomenon might be very loosely described as 'teleportation' except that the bases project or imprint themselves across space rather than simply moving from one place to another. What does all of this mean?