Coherent-state analysis of the quantum bouncing ball
... properties. For instance, 具⌬x2典 and 具⌬p2典 both approach constant values 共equally well for their product兲, and additionally, the low-energy uncertainty implies that these states have long lifetimes 共discussed further in Sec. VI兲. Also worthy of further mention are the states evaluated at  = xp. The ...
... properties. For instance, 具⌬x2典 and 具⌬p2典 both approach constant values 共equally well for their product兲, and additionally, the low-energy uncertainty implies that these states have long lifetimes 共discussed further in Sec. VI兲. Also worthy of further mention are the states evaluated at  = xp. The ...
quantum physics ii
... The probing tip of a Scanning Tunnelling Microscope is placed at a height x above a metal surface and the potential difference between the tip and surface remains constant. The probability that an electron will tunnel through the gap between the tip and the surface is 8.5 1015. What is the probab ...
... The probing tip of a Scanning Tunnelling Microscope is placed at a height x above a metal surface and the potential difference between the tip and surface remains constant. The probability that an electron will tunnel through the gap between the tip and the surface is 8.5 1015. What is the probab ...
A Primer on Quantum Mechanics and Orbitals
... Actually, I lied a little. There is one physical situation that is reminiscent of the kind of 'double valued' wavefunction that I've so far said is forbidden by the rules of quantum mechanics. This situation is the case of electron (or proton, or neutron) spin. Spin is a kind of 'intrinsic' angular ...
... Actually, I lied a little. There is one physical situation that is reminiscent of the kind of 'double valued' wavefunction that I've so far said is forbidden by the rules of quantum mechanics. This situation is the case of electron (or proton, or neutron) spin. Spin is a kind of 'intrinsic' angular ...
Formal Theory of Green Functions
... which describes one-particle (or one-hole) propagations in a medium. To describe two or more particle propagations, we would necessitate the twoor many-particle Green function which will be defined in the next section. In interacting systems the Green functions obey the very complicated equations, i ...
... which describes one-particle (or one-hole) propagations in a medium. To describe two or more particle propagations, we would necessitate the twoor many-particle Green function which will be defined in the next section. In interacting systems the Green functions obey the very complicated equations, i ...
Department of Physics, Chemistry and Biology Master’s Thesis Cavities
... between these areas, most meassurements are done at macroscopic level. The ...
... between these areas, most meassurements are done at macroscopic level. The ...
TITLE: Molecules star in quantum movie STANDFIRST: The
... no less bizarre, is wave-particle duality: under some experimental conditions particles such as electrons and atoms behave as waves, whereas light – perhaps the mostly widely studied wave phenomenon in physics – often behaves as a stream of particles called photons. For decades the interference patt ...
... no less bizarre, is wave-particle duality: under some experimental conditions particles such as electrons and atoms behave as waves, whereas light – perhaps the mostly widely studied wave phenomenon in physics – often behaves as a stream of particles called photons. For decades the interference patt ...
to be completed. LECTURE NOTES 1
... The problem now is how we can generalize the Weyl correspondence to algebraic symplectic manifold. Let (M 2n , ω) be an algebraic symplectic manifold of dimension 2n. We are seeking a manifold X of dimension n, such that there is a correspondence between a subspace of smooth functions Q∞ (M 2n ) and ...
... The problem now is how we can generalize the Weyl correspondence to algebraic symplectic manifold. Let (M 2n , ω) be an algebraic symplectic manifold of dimension 2n. We are seeking a manifold X of dimension n, such that there is a correspondence between a subspace of smooth functions Q∞ (M 2n ) and ...
pages 851-900 - Light and Matter
... numbers of photons: four photons in figure i/3, for example. A wrong interpretation: photons interfering with each other One possible interpretation of wave-particle duality that occurred to physicists early in the game was that perhaps the interference effects came from photons interacting with eac ...
... numbers of photons: four photons in figure i/3, for example. A wrong interpretation: photons interfering with each other One possible interpretation of wave-particle duality that occurred to physicists early in the game was that perhaps the interference effects came from photons interacting with eac ...
Why physics does not preclude free will
... 7. It is inconsistent with the relevant direct-action theories (such as Wheeler-Feynman (1945, 1949) and Davies (1971, 1972)) to combine the retarded state |y> and advanced state.
The advanced and retarded solutions are fundamentally distinct entities,
even t ...
... 7. It is inconsistent with the relevant direct-action theories (such as Wheeler-Feynman (1945, 1949) and Davies (1971, 1972)) to combine the retarded state |y> and advanced state
Optical probing of the spin state of a single magnetic impurity in a
... + cmh共2x + 2y 兲 − Ee−h / 2. Since the coefficients ⌬z0,1 are typically negative, the ground states are ⌽±1,⫿1/2 and correspond to 兩Jz 兩 = 1 and Jtot,z = Jz + sz = ± 1 / 2 [see Fig. 2(b)]. We now calculate the interband optical matrix elements responsible for the photoluminescence (PL) process, ass ...
... + cmh共2x + 2y 兲 − Ee−h / 2. Since the coefficients ⌬z0,1 are typically negative, the ground states are ⌽±1,⫿1/2 and correspond to 兩Jz 兩 = 1 and Jtot,z = Jz + sz = ± 1 / 2 [see Fig. 2(b)]. We now calculate the interband optical matrix elements responsible for the photoluminescence (PL) process, ass ...
Isometric and unitary phase operators: explaining the Villain transform
... requires (16b), otherwise S− |−S = 0 is violated. As long as the states |+S (in (34a)) and |−S (in (34b)) are accessible, albeit with small probability for low temperatures, neglecting (15b) (respectively (16b)) will lead to errors, because the operators Ul (respectivelyŨl ) are not followed by ...
... requires (16b), otherwise S− |−S = 0 is violated. As long as the states |+S (in (34a)) and |−S (in (34b)) are accessible, albeit with small probability for low temperatures, neglecting (15b) (respectively (16b)) will lead to errors, because the operators Ul (respectivelyŨl ) are not followed by ...
Can Mind Affect Matter Via Active Information?
... ensures that, if we start with the quantum probability distribution, we will end up with the same probability distribution as in standard quantum mechanics. The quantum potential energy does not behave like an additional energy of classical type. It has no external source, but is some form of intern ...
... ensures that, if we start with the quantum probability distribution, we will end up with the same probability distribution as in standard quantum mechanics. The quantum potential energy does not behave like an additional energy of classical type. It has no external source, but is some form of intern ...
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.