A conformal field theory approach to the fractional quantum Hall
... One of the main subjects that condensed matter theory is concerned with is the classification of different states (or phases) of matter when many particles are brought together. Until a few decades ago, it was thought that all quantum mechanical phases can be described using Landau’s symmetry-breaki ...
... One of the main subjects that condensed matter theory is concerned with is the classification of different states (or phases) of matter when many particles are brought together. Until a few decades ago, it was thought that all quantum mechanical phases can be described using Landau’s symmetry-breaki ...
Lecture notes, Chapter 4. Energy Levels
... The simplest system to be analyzed is a particle in a box: classically, in 3D, the particle is stuck inside the box and can never leave. Another classical analogy would be a ball at the bottom of a well so deep that no matter how much kinetic energy the ball possess, it will never be able to exit th ...
... The simplest system to be analyzed is a particle in a box: classically, in 3D, the particle is stuck inside the box and can never leave. Another classical analogy would be a ball at the bottom of a well so deep that no matter how much kinetic energy the ball possess, it will never be able to exit th ...
Quantum graphs and the integer quantum Hall effect
... energy spectra of independent electron’s model, a result which led Osadchy and Avron [11] to draw the phase diagram for Hofstadter’s model [12]. This phase diagram is fractal and depicts infinitely many phases, each one characterized by the integer value of the Hall conductance. Such fractal phase d ...
... energy spectra of independent electron’s model, a result which led Osadchy and Avron [11] to draw the phase diagram for Hofstadter’s model [12]. This phase diagram is fractal and depicts infinitely many phases, each one characterized by the integer value of the Hall conductance. Such fractal phase d ...
Spin-charge separation in ultra
... and interesting systems. In spite of their apparent conceptual simplicity, both the ground state and the excitations exhibit strong correlation effects and posses a number of exotic properties, ranging from spin-charge separation to fractional statistics (see [6, 7, 8] and ref. therein). Progress in ...
... and interesting systems. In spite of their apparent conceptual simplicity, both the ground state and the excitations exhibit strong correlation effects and posses a number of exotic properties, ranging from spin-charge separation to fractional statistics (see [6, 7, 8] and ref. therein). Progress in ...
qm1 - Michael Nielsen
... Postulate 3: If we measure in an orthonormal basis e1 ,..., ed , then we obtain the result j with probability P ( j ) ej ...
... Postulate 3: If we measure in an orthonormal basis e1 ,..., ed , then we obtain the result j with probability P ( j ) ej ...
Second Quantization
... useful. In the language of the states introduced above, we consider a Hilbert space which includes the states |α1 i (one particle), |α1 , α2 i (two particles), |α1 , α2 , α3 i (three particles), and so on. The basic object of second quantization is the creation operator a†α . Acting on some state in ...
... useful. In the language of the states introduced above, we consider a Hilbert space which includes the states |α1 i (one particle), |α1 , α2 i (two particles), |α1 , α2 , α3 i (three particles), and so on. The basic object of second quantization is the creation operator a†α . Acting on some state in ...
The Universal Uncertainty Principle v1
... properties: temperature and entropy. The surprising result of that research was that the equation for the black hole entropy was included into the equation for the temperature of the black hole. This meant that the black hole entropy equation emerged naturally as there was no need to introduce any a ...
... properties: temperature and entropy. The surprising result of that research was that the equation for the black hole entropy was included into the equation for the temperature of the black hole. This meant that the black hole entropy equation emerged naturally as there was no need to introduce any a ...
Lecture 7: Stationary Perturbation Theory In most practical
... The characteristic equation is a quartic equation in the 1st order energy correction which must be computed from a 4 × 4 determinant. This seems a rather tedious task. However, with a few very general considerations one can see that there is only one non-zero matrix element: V01 = −3eEaBohr , and t ...
... The characteristic equation is a quartic equation in the 1st order energy correction which must be computed from a 4 × 4 determinant. This seems a rather tedious task. However, with a few very general considerations one can see that there is only one non-zero matrix element: V01 = −3eEaBohr , and t ...
Theory of Open Quantum Systems - ITP Lecture Archive
... remain valid. In explicit there are are operators Kα such that X trE (U ρS̄ U ∗ ) = Kα ρKα∗ . α ...
... remain valid. In explicit there are are operators Kα such that X trE (U ρS̄ U ∗ ) = Kα ρKα∗ . α ...
Angular Momentum
... is known then the other two components remain undetermined. This situation is represented by the following diagram: ...
... is known then the other two components remain undetermined. This situation is represented by the following diagram: ...
Dark energy and normalization of cosmological wave function in
... with λ a dimensionless parameter which cannot be zero because of self-consistency. Deriving WD equation from this action had been tactfully carried out in [9]. There the WD equation is ...
... with λ a dimensionless parameter which cannot be zero because of self-consistency. Deriving WD equation from this action had been tactfully carried out in [9]. There the WD equation is ...
Wave function
A wave function in quantum mechanics describes the quantum state of an isolated system of one or more particles. There is one wave function containing all the information about the entire system, not a separate wave function for each particle in the system. Its interpretation is that of a probability amplitude. Quantities associated with measurements, such as the average momentum of a particle, can be derived from the wave function. It is a central entity in quantum mechanics and is important in all modern theories, like quantum field theory incorporating quantum mechanics, while its interpretation may differ. The most common symbols for a wave function are the Greek letters ψ or Ψ (lower-case and capital psi).For a given system, once a representation corresponding to a maximal set of commuting observables and a suitable coordinate system is chosen, the wave function is a complex-valued function of the system's degrees of freedom corresponding to the chosen representation and coordinate system, continuous as well as discrete. Such a set of observables, by a postulate of quantum mechanics, are Hermitian linear operators on the space of states representing a set of physical observables, like position, momentum and spin that can, in principle, be simultaneously measured with arbitrary precision. Wave functions can be added together and multiplied by complex numbers to form new wave functions, and hence are elements of a vector space. This is the superposition principle of quantum mechanics. This vector space is endowed with an inner product such that it is a complete metric topological space with respect to the metric induced by the inner product. In this way the set of wave functions for a system form a function space that is a Hilbert space. The inner product is a measure of the overlap between physical states and is used in the foundational probabilistic interpretation of quantum mechanics, the Born rule, relating transition probabilities to inner products. The actual space depends on the system's degrees of freedom (hence on the chosen representation and coordinate system) and the exact form of the Hamiltonian entering the equation governing the dynamical behavior. In the non-relativistic case, disregarding spin, this is the Schrödinger equation.The Schrödinger equation determines the allowed wave functions for the system and how they evolve over time. A wave function behaves qualitatively like other waves, such as water waves or waves on a string, because the Schrödinger equation is mathematically a type of wave equation. This explains the name ""wave function"", and gives rise to wave–particle duality. The wave of the wave function, however, is not a wave in physical space; it is a wave in an abstract mathematical ""space"", and in this respect it differs fundamentally from water waves or waves on a string.For a given system, the choice of which relevant degrees of freedom to use are not unique, and correspondingly the domain of the wave function is not unique. It may be taken to be a function of all the position coordinates of the particles over position space, or the momenta of all the particles over momentum space, the two are related by a Fourier transform. These descriptions are the most important, but they are not the only possibilities. Just like in classical mechanics, canonical transformations may be used in the description of a quantum system. Some particles, like electrons and photons, have nonzero spin, and the wave function must include this fundamental property as an intrinsic discrete degree of freedom. In general, for a particle with half-integer spin the wave function is a spinor, for a particle with integer spin the wave function is a tensor. Particles with spin zero are called scalar particles, those with spin 1 vector particles, and more generally for higher integer spin, tensor particles. The terminology derives from how the wave functions transform under a rotation of the coordinate system. No elementary particle with spin 3⁄2 or higher is known, except for the hypothesized spin 2 graviton. Other discrete variables can be included, such as isospin. When a system has internal degrees of freedom, the wave function at each point in the continuous degrees of freedom (e.g. a point in space) assigns a complex number for each possible value of the discrete degrees of freedom (e.g. z-component of spin). These values are often displayed in a column matrix (e.g. a 2 × 1 column vector for a non-relativistic electron with spin 1⁄2).In the Copenhagen interpretation, an interpretation of quantum mechanics, the squared modulus of the wave function, |ψ|2, is a real number interpreted as the probability density of measuring a particle as being at a given place at a given time or having a definite momentum, and possibly having definite values for discrete degrees of freedom. The integral of this quantity, over all the system's degrees of freedom, must be 1 in accordance with the probability interpretation, this general requirement a wave function must satisfy is called the normalization condition. Since the wave function is complex valued, only its relative phase and relative magnitude can be measured. Its value does not in isolation tell anything about the magnitudes or directions of measurable observables; one has to apply quantum operators, whose eigenvalues correspond to sets of possible results of measurements, to the wave function ψ and calculate the statistical distributions for measurable quantities.The unit of measurement for ψ depends on the system, and can be found by dimensional analysis of the normalization condition for the system. For one particle in three dimensions, its units are [length]−3/2, because an integral of |ψ|2 over a region of three-dimensional space is a dimensionless probability.