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ε · μ) is the refractive index 2 . 1, which gives rise to the well-known phenomenon of refraction described by Snell’s law:. The refractive index, n, is thus related both to the (relative) dielectric constant of the material, ε r , and also to the (relative) permeability μ r 3 . The above relation for dielectric and/or permeable matter...
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On the one hand, the vector E should be perpendicular to the tangential plane of the index ellipsoid viz by the Poinsot construction. on the other hand it should belong to the plane defined by D and k . As one can show, these two conditions can only be satisfied if the direction of D is a principal direction of...
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21.5 Holography 201 It is clear that not all possibilities offered by holography have been ex- ploited systematically as yet (see, for example, the short article, in German, in the Physik Journal p. cer- tainly in other journals similar articles in a different language exist). Some of the present-day and future applications of holography (without claiming completeness) are: color holography,...
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214 24 Quantum Mechanics: Foundations. of (24.3) for ψ( r , t) does not necessarily agree pointwise with the r.h.s. in the following sense (so-called strong topology):. (24.4) In these equations the following orthonormalisation is assumed:. T (i.e., for all arbitrarily often differentiable complex functions f (λ), which decay for | λ. (24.6) This implies the following expression (also an...
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Then we have the following solutions of the Schr¨ odinger equation. (25.5) a) Firstly, we recognise that the coefficents B + (1) and B (2. x), since then all solutions can be divided into two different classes:. even parity, i.e., u. (25.7) These equations can be solved graphically (this is a typical exercise), by plotting all branches of tan(k ·...
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236 27 The Hydrogen Atom according to Schr¨ odinger’s Wave Mechanics The characteristic values for the orbital angular momentum, l . quantization of the orbital angular momentum;. where 3 the square of the orbital angular momentum can be written as. (27.5) For the radial function R(r) one obtains (using an auxiliary function w(r. r · R(r)) the following one-dimensional differential...
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The transitions to the other representations, (ii) the Heisenberg or (iii) the Dirac. interaction) representations, are described in the following. The essential point is that one uses the unitary transformations corresponding to the so-called time- displacement operators U ˆ (t, t 0. In addition we shall use indices S, H and I, which stand for “Schr¨ odinger”, “Heisenberg” and “interaction”,...
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Furthermore, M J must be one of the 2J + 1 integral or half-integral values. Unfortunately there are many different conventions in use concerning the formulation of the coefficients.). 30.2 Fine Structure of the p-Levels. Without the spin-orbit interaction the p-levels (l = 1) of the outermost elec- tron of an alkali atom are sixfold, since including spin one has...
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For f = i the Schr¨ odinger equation yields the following result:. which is extremely long near a resonance of the denominator.. of (almost) the same frequency ω α ≈ ω, e.g., V ˆ ω e − iωt. 33.2 Selection Rules 269 In the above proof we have used the identity. ‘bar’ in (33.5).. Equation (33.5) describes transitions from a...
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This means that below this temperature there is a component of the liquid which flows without energy dissipation.. Thus, even though there is a strong interaction between the bo- son particles, they can condense without pairing, and the transition to superfluidity consists in the formation of such a condensate.. The case of He 3 is fundamentally different. As a consequence,...
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Austria (see e.g., the paper in the journal “Spektrum der Wissenschaften”. Starting with a laser beam propagating in the x-direction, the photons of this beam are linearly polarized in the vertical. a beam splitter then follows which produces two coherent beams propagating, e.g., in the x-direction and the y-direction, respectively (i.e., along two different paths W 1 and W 2...
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This is the last course in our compendium on theoretical physics. the number of atoms N in the system is usually 1 or of the order of magnitude of 1), in thermodynamics and statistical physics N is typically ≈ 10 23 . Microscopic properties are however mostly unimportant with regard to the collective behavior of the system, and for a...
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and postpone the evaluation of the real part to the very end (i.e., in effect we omit it). The heat diffusion equation then leads for a given (real) frequency ω 1 or ω 2 = 365ω 1 to the following formula for calculating the wavenumbers:. one obtains for the real part k (1) j and imaginary part k (2) j...
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We now come to the Joule-Thomson effect, which deals with the station- ary flow of a fluid through a pipe of uniform cross-sectional area (S) 1 on the input side of a so-called throttle valve, V thr . On the output side there is also a pipe of uniform cross-section (S) 2 , where (S ) 2 = (S) 1...
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Molecular field theory (and thus the van der Waals expressions) are no longer applicable in the neighborhood of the critical point, because thermal fluc- tuations become increasingly important, such that they may no longer be neglected. As a consequence of thermal fluctuations, instead of the Arrott equation, for T ≥ T 0 the following critical equation of state holds 1....
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The observables of the system are represented by Hermitian 1 operators ˆ A, i.e., with real eigenvalues. For example, the spatial representation of the operator p ˆ is given by the differential operator. and the space operator r ˆ by the corresponding multiplication operator. which are the averages of the results of an extremely comprehensive series of measurements of ˆ...
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to the corresponding intensive quantity, e.g.,. one obtains, instead of the Helmholtz free energy F (T , V, m H , N), the Gibbs free energy. The second law is now. is appropriate for Hamiltonians with a Zeeman term, e.g., the Ising Hamiltonian (42.5), because the magnetic field is already taken into account in the energy values. 46.2 On the...
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The entropy constant therefore only depends on the type of gas via the log- arithm of the particle mass m.. The above result is reasonable because, with the factor N, it explicitly expresses the additivity for the entropy of an ideal gas, which (as already mentioned) depends crucially on the permutation factor N 1 ! for identical par- ticles. Only...
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In this case the third law only plays a part at even lower temperatures where quantization of the translational energy would become noticeable, i.e., at temperatures. where R describes the size scale of the sample and M eff may be the mass of a Na atom in the normal state above the Bose condensation (or in the condensed phase: an...
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For free electrons with spin g-factor 2 we have:. In the result for a classical ideal gas,. essentially by the reciprocal of the third power of the separation of two electrons at the Fermi energy). The zero-point pressure of an electron gas (also referred to as degeneracy pressure) is the phenomenon preventing the negatively charged electrons in metals from bonding...