what is the proof of the q-analouge of the Bohr-mollerup theorem?
952581
http://planetmath.org/?op=getobj&from=objects&id=3808
697038
This is kinda hard to draw on here...\n......_\nK--Br|\n......_\n\nThe lines around Br represent 2 electrons, and the -- represents a bond...ignore the periods
215530
I'm assuming that you mean the black-body radiation law. You need two things: the first is that light of a particular frequency can only have energy in integer multiples of Planck's constant times the frequency of the light. The second is that the ratio of the probabilities that two energy states will be populated is given by an exponential of the energy difference divided by Boltzmann's constant times the temperature. The total energy of light of a particular frequency can then be described by a geometric series with ratio the exponential of -h*nu/(kT). When this sum is evaluated, you get Planck's Law.
302578
commutator: [x, H] = (xH - Hx)\n\n\napply to Hilbert vector ψ\n\n(xH - Hx)ψ\n\n\n= x(p²/2m + V)ψ - (p²/2m + V)xψ\n\n= x*p²ψ/2m + x*V*ψ - p²(x*ψ)/2m - V*x*ψ\n\n= x*p²ψ/2m - p²(x*ψ)/2m \n\n= (x*p²ψ - p²(x*ψ))/2m\n\n\n1-dim case: p = -ih∂\n\n\n= -h²(x*∂²ψ - ∂²(x*ψ))/2m\n\n= -h²(x*∂²ψ - ∂(∂x*ψ + x*∂ψ))/2m\n\n= -h²(x*∂²ψ - (∂∂x*ψ + ∂x*∂ψ + ∂x*∂ψ + x*∂∂ψ))/2m\n\n= -h²(x*∂²ψ - 2*∂ψ - x*∂²ψ)/2m\n\n= h²(∂ψ)/m\n\n= ih(-ih∂ψ)/m\n\n= ih*pψ/m\n\n\n=> [x, H] = -ihp/m
145018
It does not just have to do with Bohr's quantum number (n). It is also to do with the angular momentum quantum number(L) and electron spin (+1/2 or -1/2). Therefore, if n=1, L can only = 0 and the result is 2 electons. One with a spin of +1/2 and the other with a spin of -1/2. \nIf n=3, L can be either 0,1, or 2. \nMeaning, \nif n=3 and L=0 this results in 2 electrons on the sub-shell. \nIf n=3 and L=1 this results in 6 electrons on the sub-shell. \nIf n=3 and L=2 this results in 10 electrons on the sub-shell.\n\nThis results in a maximum of 10+6+2 = 18 electrons in the shell.\n\nThis is because if L=2, the maximum number of possible electron states on a sub-shell of n=3 are as follows:\n\nGiven, m = -L, -(L-1)..0..(L-1), L\n\nm(magnetic quantum number) can have a value of -2, -1, 0, 1, 2 with the electrons having a spin of +1/2 or -1/2 resulting in a maximum of 10 electons in that sub-shell.\n\nThis is all based on the Pauli Exclusion Principle which states no two fermions (electrons) can exist on the same sub-shell with the same spin and magnetic quantum number(m).
798748
Bohr's theory worked great to explain the spectrum of hydrogen, but it doesn't work for anything with more than a single electron.  The previous answer is correct, also...
1105659
D'un point de vue mathématique, calculer la vitesse de la balle revient à résoudre un système de n équations différentielles avec conditions initiales. C'est un peu laborieux, donc dans la pratique on utilise effectivement un radar à effet Doppler, le plus souvent sous forme de canon.\n\n Ce radar est pourvu d'un émetteur à radio-fréquences (dans la gamme des micro-ondes) dont les ondes sont réfléchies par la balle en déplacement. Le délai de retour de l'onde est proportionnel à la vitesse de la balle. Ce délai est amplifié, filtré et analysé par une puce DSP qui pratique une correction d'erreurs et envoie les données analogiques à l'afficheur. \n\nA noter que les joueurs amateurs peuvent eux aussi chronométrer leur coup droit, grâce à un module de type Speed Trac qui s'accroche sur le filet. Ludique et pas trop cher.
829980
No picture to be found.  \n\nAre M, N, O, P all on the same line?  \n\nIf so, NOQ is isoceles, which makes MNQ and QOP congruent angles.  Since two sides and the included angle uniquely define the third side, so MQ and QP are congruent. and MQP is isoceles.
456169
There are many proofs but with fallacies. Please visit:\nhttp://www.math.toronto.edu/mathnet/falseProofs/first1eq2.html\n\nhttp://www.math.toronto.edu/mathnet/falseProofs/second1eq2.html
261371
It's not that they are at two places at the same time, but there is an equal probability of finding the proton in both places. When you get to the level of quantum mechanics, due to Heisenberg's uncertainty principle we rely on probability curves to get an approximation of the physical world, the funny thing is that the areas with the most probability, are those predicted by Bohr without using quantum mechanics.
875711
Try the following:\n\nhttp://www.iop.org/EJ/article/0305-4470/34/12/101/a112l1.html\n\nhttp://arxiv.org/PS_cache/hep-th/pdf/9302/9302097.pdf\n\nhttp://arxiv.org/PS_cache/quant-ph/pdf/0004/0004090.pdf\n\nhttp://fisteo12.ific.uv.es/~santamar/qapplet/metro.html
1141031
True, at least assuming the axiom of choice. A Hamel basis of R or C over Q has cardinality of the continuum. Without the axiom of choice, I doubt that it can be proved.
144611
You might find this paper useful\nhttp://arxiv.org/abs/math-ph/0106027
335873
In electromagnetic theory, the continuity equation is derived from two of Maxwell's equations. It states that the divergence of the current density is equal to the negative rate of change of the charge density,\n\n    &#92;nabla &#92;cdot &#92;mathbf{J} = - {&#92;partial &#92;rho &#92;over &#92;partial t}\n\n[edit]\n\nDerivation\n\nOne of Maxwell's equations states that\n\n    &#92;nabla &#92;times &#92;mathbf{H} = &#92;mathbf{J} + {&#92;partial &#92;mathbf{D} &#92;over &#92;partial t}.\n\nTaking the divergence of both sides results in\n\n    &#92;nabla &#92;cdot &#92;nabla &#92;times &#92;mathbf{H} = &#92;nabla &#92;cdot &#92;mathbf{J} + {&#92;partial &#92;nabla &#92;cdot &#92;mathbf{D} &#92;over &#92;partial t},\n\nbut the divergence of a curl is zero, so that\n\n    &#92;nabla &#92;cdot &#92;mathbf{J} + {&#92;partial &#92;nabla &#92;cdot &#92;mathbf{D} &#92;over &#92;partial t} = 0. &#92;qquad &#92;qquad (1)\n\nAnother one of Maxwell's equations states that\n\n    &#92;nabla &#92;cdot &#92;mathbf{D} = &#92;rho.&#92;,\n\nSubstitute this into equation (1) to obtain\n\n    &#92;nabla &#92;cdot &#92;mathbf{J} + {&#92;partial &#92;rho &#92;over &#92;partial t} = 0,&#92;,\n\nwhich is the continuity equation.\n[edit]\n\nInterpretation\n\nCurrent density is the movement of charge density. The continuity equation says that if charge is moving out of a differential volume (i.e. divergence of current density is positive) then the amount of charge within that volume is going to decrease, so the rate of change of charge density is negative. Therefore the continuity equation amounts to a conservation of charge.\n[edit]\n\nFluid dynamics\n\nIn fluid dynamics, a continuity equation is an equation of conservation of mass. Its differential form is\n\n    {&#92;partial &#92;rho &#92;over &#92;partial t} + &#92;nabla &#92;cdot (&#92;rho &#92;mathbf{u}) = 0\n\nwhere ρ is density, t is time, and u is fluid velocity.\n[edit]\n\nQuantum mechanics\n\nIn quantum mechanics, the conservation of probability also yields a continuity equation. Let P(x, t) be a probability density and write\n\n    &#92;nabla &#92;cdot &#92;mathbf{j} = -{ &#92;partial &#92;over &#92;partial t} P(x,t)\n\nwhere J is probability flux.
6117
quantum theory\nn.\nA theory in physics based on the principle that matter and energy have the properties of both particles and waves, created to explain the radiation of energy from a blackbody, the photoelectric effect, and the Bohr theory, and now used to account for a wide range of physical phenomena, including the existence of discrete packets of energy and matter, the uncertainty principle, and the exclusion principle. \nAny of various specific applications of this theory.\n\n\nquantum theory, modern physical theory concerned with the emission and absorption of energy by matter and with the motion of material particles; the quantum theory and the theory of relativity together form the theoretical basis of modern physics. Just as the theory of relativity assumes importance in the special situation where very large speeds are involved, so the quantum theory is necessary for the special situation where very small quantities are involved, i.e., on the scale of molecules, atoms, and elementary particles. Aspects of the quantum theory have provoked vigorous philosophical debates concerning, for example, the uncertainty principle and the statistical nature of all the predictions of the theory.\nRelationship of Energy and Matter\n\nAccording to the older theories of classical physics, energy is treated solely as a continuous phenomenon, while matter is assumed to occupy a very specific region of space and to move in a continuous manner. According to the quantum theory, energy is held to be emitted and absorbed in tiny, discrete amounts. An individual bundle or packet of energy, called a quantum (pl. quanta), thus behaves in some situations much like particles of matter; particles are found to exhibit certain wavelike properties when in motion and are no longer viewed as localized in a given region but rather as spread out to some degree.\n\nFor example, the light or other radiation given off or absorbed by an atom has only certain frequencies (or wavelengths), as can be seen from the line spectrum associated with the chemical element represented by that atom. The quantum theory shows that those frequencies correspond to definite energies of the light quanta, or photons, and result from the fact that the electrons of the atom can have only certain allowed energy values, or levels; when an electron changes from one allowed level to another, a quantum of energy is emitted or absorbed whose frequency is directly proportional to the energy difference between the two levels.\n\nDual Nature of Waves and Particles\n\nThe restriction of the energy levels of the electrons is explained in terms of the wavelike properties of their motions: electrons occupy only those orbits for which their associated wave is a standing wave (i.e., the circumference of the orbit is exactly equal to a whole number of wavelengths) and thus can have only those energies that correspond to such orbits. Moreover, the electrons are no longer thought of as being at a particular point in the orbit but rather as being spread out over the entire orbit. Just as the results of relativity approximate those of Newtonian physics when ordinary speeds are involved, the results of the quantum theory agree with those of classical physics when very large “quantum numbers” are involved, i.e., on the ordinary large scale of events; this agreement in the classical limit is required by the correspondence principle of Niels Bohr. The quantum theory thus proposes a dual nature for both waves and particles, one aspect predominating in some situations, the other predominating in other situations.\n\nEvolution of Quantum Theory\n\nEarly Developments\n\nWhile the theory of relativity was largely the work of one man, Albert Einstein, the quantum theory was developed principally over a period of thirty years through the efforts of many scientists. The first contribution was the explanation of black body radiation in 1900 by Max Planck, who proposed that the energies of any harmonic osci
834583
You can use the Rydberg formula, which relates the reciprocal of the wavelength of the photon absorbed or emitted to the squares of the reciprocals of the principal quantum numbers.\n\n1/wavelength = R(1/n_l^2 - 1/n_u^2)\n\nSo, for the listed values of n, \n\nA.  1/w = R(1/ 2^2 - 1/4^2) = 0.1875R\nB.  1/w = -0.75R\nC.  1/w = 0.21R\nD.  1/w = -0.04861R\n\nSince energy and frequency are proportional to 1/wavelength,\nlooks like the order is D, A, C, B.
887107
B.the electron subshell