Studies in Composing Hydrogen Atom Wavefunctions
... devised by early organ builders to enrich the sound of individual notes. Similarly, mixing two eigenfunctions produces a spatial pattern that evolves periodically in time, with a frequency equal to the difference between the characteristic frequencies of the eigenfunctions. While celeste produces a ...
... devised by early organ builders to enrich the sound of individual notes. Similarly, mixing two eigenfunctions produces a spatial pattern that evolves periodically in time, with a frequency equal to the difference between the characteristic frequencies of the eigenfunctions. While celeste produces a ...
Creating fractional quantum Hall states with atomic clusters
... As shown in Fig. 3(c), a strong antibunching appears for ramp speeds δ̇ 0.3 2 . For the particle numbers investigated here, the physical characteristics of the Laughlin state thus appear for ramp durations comparable to those required for the adiabatic spin flip of a single particle. So far we co ...
... As shown in Fig. 3(c), a strong antibunching appears for ramp speeds δ̇ 0.3 2 . For the particle numbers investigated here, the physical characteristics of the Laughlin state thus appear for ramp durations comparable to those required for the adiabatic spin flip of a single particle. So far we co ...
... locations or nodes by means of single photons traveling qubits, which are guided through waveguides. Interestingly, this coherent interface, which is responsible for the state of the storage qubits to be mapped onto the traveling qubits or the entanglement between them, is itself a qubit system, t ...
Chapter 41. One-Dimensional Quantum Mechanics
... The quantum-mechanical solution for a particle in a finite potential well has some important properties: • The particle’s energy is quantized. • There are only a finite number of bound states. There are no stationary states with E > U0 because such a particle would not remain in the well. • The ...
... The quantum-mechanical solution for a particle in a finite potential well has some important properties: • The particle’s energy is quantized. • There are only a finite number of bound states. There are no stationary states with E > U0 because such a particle would not remain in the well. • The ...
Microscopic theory for quantum mirages in quantum corrals D. Porras, J. Ferna´ndez-Rossier,
... focus where the magnetic impurity is located but also on the empty focus, which can be as far as 110 Å away from the impurity. Remarkably, the phantom dip is not observed if neither the tip or the impurity are not at the foci. The phenomenon of the phantom dip is referred to as the quantum mirage.11 ...
... focus where the magnetic impurity is located but also on the empty focus, which can be as far as 110 Å away from the impurity. Remarkably, the phantom dip is not observed if neither the tip or the impurity are not at the foci. The phenomenon of the phantom dip is referred to as the quantum mirage.11 ...
Angular Momentum and Central Forces
... each individual components of L, however the components of L does not commute with each other. L2 is given by, L =L +L +L When a measurement is made, we can find the total angular momentum and only one other component at a time. For example, if a wave function is an eigenfunction of Lz then it is no ...
... each individual components of L, however the components of L does not commute with each other. L2 is given by, L =L +L +L When a measurement is made, we can find the total angular momentum and only one other component at a time. For example, if a wave function is an eigenfunction of Lz then it is no ...
Bohmian Trajectories of the Two-Electron Helium Atom
... the wavefunction as giving a measure of the relative probability of finding the system at a particular point in configuration space. The constancy of the norm of the wavefunction then expressed the fact that there is always a probability equal to unity of finding the system at some point in configu ...
... the wavefunction as giving a measure of the relative probability of finding the system at a particular point in configuration space. The constancy of the norm of the wavefunction then expressed the fact that there is always a probability equal to unity of finding the system at some point in configu ...
Comparisons between classical and quantum mechanical
... the Pauli exclusion principle, meaning that each quantum state can be occupied with at most one fermion, while bosons on the other hand have integer spin, and any number of bosons can populate a given quantum state. In this thesis, we will deal primarily with the latter type. The statistical propert ...
... the Pauli exclusion principle, meaning that each quantum state can be occupied with at most one fermion, while bosons on the other hand have integer spin, and any number of bosons can populate a given quantum state. In this thesis, we will deal primarily with the latter type. The statistical propert ...
Positronium: Review of symmetry, conserved quantities and decay
... density at the origin 共nucleus or positron兲 increasing the likelihood of capture.2 These orbitals have zero orbital angular momentum (l⫽0); the total angular momentum 共J兲 in this case derives from the alignment or nonalignment of the two spin one-half particles, i.e., J⫽1 or 0 which are referred to ...
... density at the origin 共nucleus or positron兲 increasing the likelihood of capture.2 These orbitals have zero orbital angular momentum (l⫽0); the total angular momentum 共J兲 in this case derives from the alignment or nonalignment of the two spin one-half particles, i.e., J⫽1 or 0 which are referred to ...
Stability Of Matter
... merged as a consequence from this theory. In the same year, Pauli discovered his ‘Esclusion Prinicle’, which will be of great importance and we shall talk about it in great detail later. This development culminated in January 1926 with Erwin Schrödinger’s discovery of the equation now named after h ...
... merged as a consequence from this theory. In the same year, Pauli discovered his ‘Esclusion Prinicle’, which will be of great importance and we shall talk about it in great detail later. This development culminated in January 1926 with Erwin Schrödinger’s discovery of the equation now named after h ...
Geometric manipulation of the quantum states of two
... laser-pulse-controlled Bloch vector rotation, U3共␦兲 on Tm3+ doped in yttrium aluminum gamet (YAG) crystal. The resulting quantum states of the ions were measured using photon echoes [15] as the analogue of spin echoes in NMR systems [16]. The rephasing process of the photon echo cancels the dynamic ...
... laser-pulse-controlled Bloch vector rotation, U3共␦兲 on Tm3+ doped in yttrium aluminum gamet (YAG) crystal. The resulting quantum states of the ions were measured using photon echoes [15] as the analogue of spin echoes in NMR systems [16]. The rephasing process of the photon echo cancels the dynamic ...
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... In Section 2, we first give a brief review of the symplectic formulation of the Gaussian wave packet dynamics by Lubich [20] and Ohsawa and Leok [26]; this formulation is more amenable to symmetry analysis because the set of ODEs as a whole is formulated as a single Hamiltonian system. Then, in Sect ...
... In Section 2, we first give a brief review of the symplectic formulation of the Gaussian wave packet dynamics by Lubich [20] and Ohsawa and Leok [26]; this formulation is more amenable to symmetry analysis because the set of ODEs as a whole is formulated as a single Hamiltonian system. Then, in Sect ...
Eikonal Approximation K. V. Shajesh
... have to an exact result in a scattering problem is the result got by the method of partial wave expansion. We shall thus find it very useful to use the results obtained from the method of partial wave expansion as a benchmark. The method of partial wave expansion breaks down the initial wavefunction ...
... have to an exact result in a scattering problem is the result got by the method of partial wave expansion. We shall thus find it very useful to use the results obtained from the method of partial wave expansion as a benchmark. The method of partial wave expansion breaks down the initial wavefunction ...
Squeezed spin states
... Spin or angular momentum systems [2] have often been regarded as squeezed if the uncertainty of one spin component, say (AS ) or (AS„ ), is smaller than 2[(S,)~ [3]. This definition implies that a coherent spin state (CSS) [4] is already squeezed if it is placed in an appropriate system of coordinat ...
... Spin or angular momentum systems [2] have often been regarded as squeezed if the uncertainty of one spin component, say (AS ) or (AS„ ), is smaller than 2[(S,)~ [3]. This definition implies that a coherent spin state (CSS) [4] is already squeezed if it is placed in an appropriate system of coordinat ...
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.