Postulates of QM, Qubits, Measurements - EECS: www
... Consider a system with k distinguishable (classical) states. For example, the electron in a hydrogen atom is only allowed to be in one of a discrete set of energy levels, starting with the ground state, the first excited state, the second excited state, and so on. If we assume a suitable upper bound ...
... Consider a system with k distinguishable (classical) states. For example, the electron in a hydrogen atom is only allowed to be in one of a discrete set of energy levels, starting with the ground state, the first excited state, the second excited state, and so on. If we assume a suitable upper bound ...
The Iterative Unitary Matrix Multiply Method and Its Application to
... FIG. 1 and FIG. 2 show the same calculation results. The distribution of QKR wave function is clearer in FIG. 2. Before N = 20 (the time is 220 τ = 1.05 × 106 τ ), QKR does not delocalize at all. To a resonant case, this is unexpected. At N = 20, peaks (local maxima) and valleys (local minima) appea ...
... FIG. 1 and FIG. 2 show the same calculation results. The distribution of QKR wave function is clearer in FIG. 2. Before N = 20 (the time is 220 τ = 1.05 × 106 τ ), QKR does not delocalize at all. To a resonant case, this is unexpected. At N = 20, peaks (local maxima) and valleys (local minima) appea ...
The Strong Free Will Theorem
... triple experiment of measuring the squared spin component of particle a in three orthogonal directions x, y, z, while experimenter B measures the twinned particle b in one direction, w. Then if w happens to be in the same direction as one of x, y, z, experimenter B’s measurement will necessarily yie ...
... triple experiment of measuring the squared spin component of particle a in three orthogonal directions x, y, z, while experimenter B measures the twinned particle b in one direction, w. Then if w happens to be in the same direction as one of x, y, z, experimenter B’s measurement will necessarily yie ...
Electron-electron interactions in a one-dimensional quantum
... be modulated by a gate voltage, which controls the asymmetry of the potential well that confines the electrons. Since this proposal, many spintronic devices based on the Rashba effect have been proposed, based on a single electron picture.6–9 At the same time, one-dimensional wires are now available ...
... be modulated by a gate voltage, which controls the asymmetry of the potential well that confines the electrons. Since this proposal, many spintronic devices based on the Rashba effect have been proposed, based on a single electron picture.6–9 At the same time, one-dimensional wires are now available ...
to the fiftieth anniversary of starting up the first linear acc
... free (i.e., non-bounded in atoms) ultrarelativistic electrons in storage rings is described in many papers (e.g., [1, 2]). The time duration of this process is about 104 - 105 sec. On the other hand, for the non-relativistic electron the characteristic time of the spontaneous spin flip is very long: ...
... free (i.e., non-bounded in atoms) ultrarelativistic electrons in storage rings is described in many papers (e.g., [1, 2]). The time duration of this process is about 104 - 105 sec. On the other hand, for the non-relativistic electron the characteristic time of the spontaneous spin flip is very long: ...
Lecture 2: Bogoliubov theory of a dilute Bose gas Abstract
... Given a many-body system made of identical bosons, say atoms carrying an integervalued total angular momentum, how does one construct the ground state? The simplest answer to this question occurs when bosons are noninteracting. In this case, the ground state is simply obtained by putting all bosons ...
... Given a many-body system made of identical bosons, say atoms carrying an integervalued total angular momentum, how does one construct the ground state? The simplest answer to this question occurs when bosons are noninteracting. In this case, the ground state is simply obtained by putting all bosons ...
Lecture 14 Thermodynamic Properties
... We now turn to the problem of finding the intramolecular partition function, i.e., the partition function for internal degrees of freedom, for an ideal gas. 3. Energy level spacings. We identify five energy states: nuclear-spin states, electronic states, vibrational states, rotational states, and tr ...
... We now turn to the problem of finding the intramolecular partition function, i.e., the partition function for internal degrees of freedom, for an ideal gas. 3. Energy level spacings. We identify five energy states: nuclear-spin states, electronic states, vibrational states, rotational states, and tr ...
Abstracts - Departamento de Matemáticas
... finite temperature systems are mapped to classical models on 2d lattices. The partition function is obtained from just the largest eigenvalue of the column-tocolumn transfer matrix, also called the ‘quantum transfer matrix’ which acts in an infinite dimensional space. A hierarchy of transfer matrice ...
... finite temperature systems are mapped to classical models on 2d lattices. The partition function is obtained from just the largest eigenvalue of the column-tocolumn transfer matrix, also called the ‘quantum transfer matrix’ which acts in an infinite dimensional space. A hierarchy of transfer matrice ...
Generalized variational principle for excited states using nodes of
... variational principle is because the 2s state has the same symmetry as the ground state. Since our trial function has a spherical node with radius r = 1/a, it is easy to analytically compute the two nodal expectation values, corresponding to the two nodal regions: 0 < r < 1/a and r > 1/a. By our the ...
... variational principle is because the 2s state has the same symmetry as the ground state. Since our trial function has a spherical node with radius r = 1/a, it is easy to analytically compute the two nodal expectation values, corresponding to the two nodal regions: 0 < r < 1/a and r > 1/a. By our the ...
Quantum vortices in a glass of Bose
... in the case of a dilute gas by the nonlinear Schrödinger equation (Nls), also called the Gross-Pitaevskii equation. It can be shown that the dynamics of the Nls is very close to shallow water dynamics with surface tension. One of the striking features of Becs is the possibility of point vortices to ...
... in the case of a dilute gas by the nonlinear Schrödinger equation (Nls), also called the Gross-Pitaevskii equation. It can be shown that the dynamics of the Nls is very close to shallow water dynamics with surface tension. One of the striking features of Becs is the possibility of point vortices to ...
From the Photon to Maxwell Equation. Ponderations on the Concept
... Until Section 6 the analysis is almost classical. So, in Section 7 we investigate the quantum Schrödinger equation for the photon which follows from quantum field theory, concentrating analysis on the solution, one given in [3] and the other in [69]. In [3] the author claims to produce a photon loc ...
... Until Section 6 the analysis is almost classical. So, in Section 7 we investigate the quantum Schrödinger equation for the photon which follows from quantum field theory, concentrating analysis on the solution, one given in [3] and the other in [69]. In [3] the author claims to produce a photon loc ...
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.