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 . . .
Matrix mechanicsMatrix 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.
EinselectionIn quantum mechanics, einselection, short for environment - induced superselection, is a name coined by Wojciech H. Zurek for a process which explains the phenomenon of wavefunction collapse and the emergence of classical descriptions of reality from quantum descriptions. Classicality is an emergent property induced in open quantum systems by their environments. Since only quasi-local, essentially classical states survive the decoherence process, einselection can in many ways explain the emergence of a (seemingly) classical reality in a fundamentally quantum universe (at least to local observers). Definition corresponding to different pointer states become orthogonal: Details To study einselection, an operational definition of pointer states has been introduced. This is the "predictability sieve" criterion, based on an intuitive idea: Pointer states can be defined as the ones which become minimally entangled with the environment in the course of their evolution. ). .
Free particleHow Cosmic Inflation Creates an Infinity of Universes [Video]Give a Gift & Get a Gift - Free!ApplicationsFrom Eternity to Book Club: Chapter Eleven | Cosmic VarianceWelcome to this week’s installment of the From Eternity to Here book club. Part Three of the book concludes with Chapter Eleven, “Quantum Time.” Excerpt: This distinction between “incomplete knowledge” and “intrinsic quantum indeterminacy” is worth dwelling on. Title notwithstanding, the point of the chapter is not that there’s some “quantum” version of time that we have to understand. However, we still need to talk about quantum mechanics for the purposes of this book, for one very good reason: we’ve been making a big deal about how the fundamental laws of physics are reversible, but wave function collapse (under the textbook Copenhagen interpretation) is an apparent counterexample. Along the way, I get to give my own perspective on what quantum mechanics really means. So I present a number of colorful examples of two-state systems involving cats and dogs.
Solution of Schrödinger equation for a step potentialIn 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
PearltreesMacroscopic quantum phenomenaQuantum 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. Although these media are very different, their behavior is very similar as they all show macroscopic quantum behavior. Consequences of the macroscopic occupation Fig.1 Left: only one particle; usually the small box is empty. with Ψ₀ the amplitude and the phase. 1. 2. 3. and
From Eternity to Book Club: Chapter Twelve | Cosmic VarianceWelcome to this week’s installment of the From Eternity to Here book club. Part Four opens with Chapter Twelve, “Black Holes: The Ends of Time.” Excerpt: Unlike boxes full of atoms, we can’t make black holes with the same size but different masses. The size of a black hole is characterized by the “Schwarzschild radius,” which is precisely proportional to its mass. If you know the mass, you know the size; contrariwise, if you have a box of fixed size, there is a maximum mass black hole you can possibly fit into it. It’s not surprising to find a chapter about black holes in a book that talks about relativity and cosmology and all that. Black holes are important to our story for a couple of reasons. The other reason black holes are important, of course, is that the answer that Bekenstein and Hawking derive is somewhat surprising, and ultimately game-changing.
Particle in a boxIn 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 where is the length of the box and is time. and
Interpretations of quantum mechanicsAn 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: