Quantum Mechanical Model
... A function of the coordinates (x, y, and z) of the electron’s position in 3-D space ...
... A function of the coordinates (x, y, and z) of the electron’s position in 3-D space ...
photon particle - wave duality
... given by Planck based on a picture in which electromagnetic radiation is carried in particle-like bunches called “photons.” 2. The failure of classical physics to adequately describe the photoelectric effect, and the subsequent successful treatment given by Einstein based on the photon picture. ...
... given by Planck based on a picture in which electromagnetic radiation is carried in particle-like bunches called “photons.” 2. The failure of classical physics to adequately describe the photoelectric effect, and the subsequent successful treatment given by Einstein based on the photon picture. ...
The Pauli exclusion principle states that no two fermions
... the way atoms share electrons. It explains the variety of chemical elements and their ...
... the way atoms share electrons. It explains the variety of chemical elements and their ...
Quantum Physics
... wavelengths but predicts that the intensity should continue to increase as the wavelength decreases. Planck assumed that an atom oscillating at frequency f can only have certain energies given by E = nhf. He quantized the energy of an atom, and in so doing, also quantized the emitted electromagnetic ...
... wavelengths but predicts that the intensity should continue to increase as the wavelength decreases. Planck assumed that an atom oscillating at frequency f can only have certain energies given by E = nhf. He quantized the energy of an atom, and in so doing, also quantized the emitted electromagnetic ...
A paradox in quantum measurement theory - Philsci
... Hence, the detection rate at A can be altered, depending on this choice. But if the detection rate at A alters depending on this choice, then it seems that the rate of detection at B must also change, because the total detection rate must be constant. The problem is that we can separate the screens ...
... Hence, the detection rate at A can be altered, depending on this choice. But if the detection rate at A alters depending on this choice, then it seems that the rate of detection at B must also change, because the total detection rate must be constant. The problem is that we can separate the screens ...
Wave Mechanics
... around the mean value. As is the case in the analysis of statistical data, this is done in terms of the usual statistical quantity, the standard deviation, written ∆x. This quantity is determined in terms of the average value of the squares of the distance of each value x from the average value of t ...
... around the mean value. As is the case in the analysis of statistical data, this is done in terms of the usual statistical quantity, the standard deviation, written ∆x. This quantity is determined in terms of the average value of the squares of the distance of each value x from the average value of t ...
Lecture 3: The Wave Function
... wavelength λ has a momentum p = hλ−1 . This means that ψ3 and ψ4 , being sinusoidal waves, have well-defined wavelengths and therefore well-defined momenta with small uncertainties in their respective momenta, with ψ4 having a smaller wavelength and therefore a larger momentum than ψ3 . On the other h ...
... wavelength λ has a momentum p = hλ−1 . This means that ψ3 and ψ4 , being sinusoidal waves, have well-defined wavelengths and therefore well-defined momenta with small uncertainties in their respective momenta, with ψ4 having a smaller wavelength and therefore a larger momentum than ψ3 . On the other h ...
powerpoint
... Owing to the continuousness condition, we must demand that Ψ(x=0) = 0 and Ψ(x=L) = 0. ekx cannot satisfy these simultaneously (ek0 = 1). cos x is not promising (cos 0 = 1), either. (Other than this, all four functions are single-valued and finite). ...
... Owing to the continuousness condition, we must demand that Ψ(x=0) = 0 and Ψ(x=L) = 0. ekx cannot satisfy these simultaneously (ek0 = 1). cos x is not promising (cos 0 = 1), either. (Other than this, all four functions are single-valued and finite). ...
Waves and Particles: Basic Concepts of Quantum
... A wave function which fulfills this condition (namely, for which the integral on the left hand side of equation 34 is finite), is called square integrable. When the right hand side is equal to 1, it is also normalized (though often we will absorb the term “normalized” into the ...
... A wave function which fulfills this condition (namely, for which the integral on the left hand side of equation 34 is finite), is called square integrable. When the right hand side is equal to 1, it is also normalized (though often we will absorb the term “normalized” into the ...
Part II. Statistical mechanics Chapter 9. Classical and quantum
... In such a case, A(t) A, the equilibrium value of the observable. The goal of equilibrium statistical mechanics is to find the values A for a ρ subject to certain imposed constraints; e.g. ρeq (qi , pi ,U,V ) is the set of all possible system trajectories such that U and V are constant. The more ge ...
... In such a case, A(t) A, the equilibrium value of the observable. The goal of equilibrium statistical mechanics is to find the values A for a ρ subject to certain imposed constraints; e.g. ρeq (qi , pi ,U,V ) is the set of all possible system trajectories such that U and V are constant. The more ge ...
A PRIMER ON THE ANGULAR MOMENTUM AND PARITY
... twice, you should end up with the same thing. Trying that: P̂ ψ = pψ where p is the ”quantum number” for parity. And again: P̂ P̂ ψ = P̂ pψ = p2 ψ. ...
... twice, you should end up with the same thing. Trying that: P̂ ψ = pψ where p is the ”quantum number” for parity. And again: P̂ P̂ ψ = P̂ pψ = p2 ψ. ...
$doc.title
... Ψ is specified by the quantum numbers n & m There are as many states as there are possible n,m combinations (N.B. n & m are positive) Two distinct wave functions are DEGENERATE if they have the same energy. e.g. the states 1,3 and 3,1 are degenerate if a = b If a/b is irrational there are no deg ...
... Ψ is specified by the quantum numbers n & m There are as many states as there are possible n,m combinations (N.B. n & m are positive) Two distinct wave functions are DEGENERATE if they have the same energy. e.g. the states 1,3 and 3,1 are degenerate if a = b If a/b is irrational there are no deg ...
Quantum Effects in Spin Ice 1. Thermodynamic properties of
... quantum systems which exhibit exotic properties such as emergent hidden or topological order, quantum entanglement or fractionalized excitations has made the study of quantum spin liquids and the so-called “spin ice” materials an area of considerable activity over the last decade. This has also coin ...
... quantum systems which exhibit exotic properties such as emergent hidden or topological order, quantum entanglement or fractionalized excitations has made the study of quantum spin liquids and the so-called “spin ice” materials an area of considerable activity over the last decade. This has also coin ...
Quantum Solutions For A Harmonic Oscillator
... 1. The motion of a fictitious particle of mass (m1+m2) positioned at the center of mass, and subject only to the external forces acting on the system. 2. The motion of a fictitious particle of reduced mass μ, subject only to the internal forces. The center of mass of this system does not move. Note ...
... 1. The motion of a fictitious particle of mass (m1+m2) positioned at the center of mass, and subject only to the external forces acting on the system. 2. The motion of a fictitious particle of reduced mass μ, subject only to the internal forces. The center of mass of this system does not move. Note ...
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.