State numbers

In contrast, it is accepted practice that referring to, for example, an iron(III) complex implies that this compound contains an iron ion with a high-, intermediate-, or low-spin electron configuration. Since n for a d" configuration is, at least in principle, a measurable quantity, it has been suggested [3] that an oxidation number n, which is derived from a known d configuration, should be specified as physical or spectroscopic oxidation number (state) [4—6]. 

The essential point of the LvN method is that the number states of... 

The destruction term is defined as the sum of all irreducible transitions starting from any initial state pk p 0) with k 0 and ending with the zero wave number state (the vacuum ) ... 

The estimate of/i is based on a sample of simulated data as a result, sampling error is always associated with the estimate. The law of lar numbers states that the sample mean converges to the true mean in probability as the sample size increases ... 

The energy levels of a molecule placed in an off-resonant microwave field can be calculafed by diagonalizing fhe mafrix of fhe Floquef Hamiltonian in the basis of direct products y) ), where y) represents in the eigenstates of the molecule in the absence of the field and ) - fhe Fourier componenfs in Eq. (8.21). The states k) are equivalent to photon number states in the alternative formalism using the quantum representation of the field [11, 15, 26], The eigensfales of the Floquet Hamiltonian are the coherent superpositions... 

The -functions of phonon states after the electronic transition induced by differently ses. (a) The 2-function at the minimal AA"+, that is, maximal squeezing (the chirp parameter see marker a in Fig. 1). (b) The AX+ is in its next maximum, (c) An intermediate state maximum and minimum. As we consider lower and lower chirp parameters, the maxima and ome less prominent (4), approaching the number state with equal distribution along a circle. 

In this book, Le will denote the Lewis number (Le = h/kCp). The Lewis number states that the heat transfer coefficient is to the mass transfer coefficient as the value of the medium s specific heat serves for both heat transfer and mass transfer. 

The numbers in column (4) represent the numbers of moles of atoms of each of the components of the compound found in the 100-g sample. These three numbers state the ratio of the components of the compound— 0.703 1.406 2.812. We could write the compound as Mg0.703Al1.406O2.8i2 except, of course, the numbers used must be whole numbers. Suppose we were to divide all three amounts of moles by the smallest number such a manipulation (5) preserves the ratio since all three are divided by the same number, and the division does result in a whole-number ratio. This final ratio can be used to correctly write the empirical formula, MgAl204. 

A powerful way of achieving this goal uses the coupled-channels expansion, a method widely used in calculations of scattering cross sections [6]. In the context of quantized matter-radiation problems, the coupled-channels method amounts to expanding E, n, N ) in number states. Concentrating on the expansion in the /th mode, we write E, n, N ) as... 

Using the orthogonality of the number states ]N) along with Eqs. (12.15), (12.33), and (12.34), we transform the Schrodinger equation [Eq. (12.24)] into at set of coupled differential equations, the so-called coupled-channels equations. ... 

The coupled channels expansion can be further simplified by introducing the (number state) rotating-wave approximation (RWA), valid only when the field is jjsfif moderate intensity and the system is near resonance. As pointed out above, igtyen an initial photon number state [JVf), the components of E, n, N — 1") of. . greatest interest for a one-photon transition are (JV, , n",JV, —1 ) and y (Nj dt l[Ji, n, Nj — 1 ). If [ , ) is the ground material state, then the (Nj+m,... 

Equations (12.44) and (12.45) constitute the number state RWA for moderate field photodissociation. They are nonperturbative within the basis set adopted but are, , approximate in that they only incorporate a small number of number states and L they neglect the contribution of all modes other than that of the incident beam. 

Finally, we make a few additional remarks. First, note that a pure number state is a3j= state whose phase 0k is evenly distributed between 0 and 2n. This is a consequence of the commutation relation [3] between Nk and e,0 <. Nevertheless, dipole mafKi w elements calculated between number states are (as all quantum mechanical amplitudes) well-defined complex numbers, and as such they have well-defined phajje j S Thus, the phases of the dipole matrix elements in conjunction with the mode ph f i f/)k [Eq. (12.15)] yield well-defined matter + radiation phases that determine the outcome of the photodissociation process. As in the weak-field domain, if only gJ one incident radiation mode exists then the phase cancels out in the rate expres4<3 [Eq. (12.35)], provided that the RWA [Eqs. (12.44) and (12.45)] is adoptedf However, in complete analogy with the treatment of weak-field control, if we irradh ate the material system with two or more radiation modes then the relative pb between them may have a pronounced effect on the fully interacting state, phase control is possible. 

Extruders may be classified conveniently using a three-figure system—for example, 1-60-24—where the first number states how many screws the machine has, the second gives the diameter of the screw in millimetres, and the third the effective length of the screw as a multiple of the diameter. In a 1-60-24 there would be a single screw of diameter 60 mm and length 24 diameters. 

During isothermal annealing, the frozen-in structure of quenched glass relaxes, and F(r) can be interpreted as the probability that holes have not reached their equilibrium states. The probability of a hole in the ith wave number state having reached equilibrium in a time interval t is (nt/n) (R/l) R/Ll. Thus, we write [12]... 

The law of large numbers is fundamental to probabilistic thinking and stochastic modeling. Simply put, if a random variable with several possible outcomes is repeatedly measured, the frequency of a possible outcome approaches its probability as the number of measurements increases. The weak law of large numbers states that the average of N identically distributed independent random variables approaches the mean of their distribution. 

The initial photon state can be a number state (with a not well-defined phase) or a linear combination of number states, for instance a coherent state. We formulate the construction of coherent states in the Floquet theory and show that choosing one as the initial photon state allows us to recover the usual semiclassical time dependent Schrodinger equation, with a classical held of a well-defined phase (see Section II.C). 

In the usual Fock number state representation they are given, up to a phase factor, by... 

For a general initial condition of the photon field i (0) C jSf, we first remark that the evolution of the initial condition (that we take here at t — to —0) 4>(jc) / c( 0j can be obtained from the one of the initial condition 4>(v) 0 1 (where the constant function 1 = e relative number state of zero photons) ... 

This property is quite remarkable In the large photon number regime the coherent quantum average on a number state gives the same result as the incoherent statistical average over coherent states. 

This incoherent statistical average over the phases also gives exactly the same result as the coherent average (52) in a photon number state. 

Due to the relative character of the number operator — /S/S0, all the physical predictions of the Floquent model must be invariant with respect to a global translation of the relative photon numbers. We show that this is indeed the case for the properties discussed above. The propability P(L, t) is independent of the particular initial photon number state chosen that is, it is independent of k since... 

In adiabatic passage processes with pulsed lasers, as we will discuss in the forthcoming sections, one often encounters the following particular situation If the initial condition of the photon field were a number state, that is,... 

If one considers an initial coherent state for the photon field instead of a photon-number state, the superpositions of states have the additional optical phase, giving for (286)... 

This is confirmed by the numerical solution of the dressed Schrodinger equation (308) with a number state as the initial condition for the photon field 11 0,0) It shows that the solution dressed state vector v /(t) (the transfer state, which in the bare basis is given by / /(0,0) mainly projects on the transfer eigenvector during the process. Additional data of the dressed solution during time are shown in Fig. 21a and 21c. Figure 21a displays the probabilities of being in the bare states 1, 2, and 3 ... 

Comparing Figs. 20a and 21a, we notice that, as expected, the solution of the dressed Schrodinger equation, with a number state as initial condition for the... 

To that effect, we choose the parameters 8 = 2flo and Qmax = 4.4 f>o, corresponding to the path (c) on the surfaces in Fig. 19. As shown in Fig. 22, the solution of the semiclassical Schrodinger equation (321) leads to nearly complete population transfer from state 11) to state 3). The analysis of the surfaces shows that the state 1 0,0) connects 3 —3,3). Thus the complete population transfer from the bare state 1) to the bare state 3) must be accompanied with absorption of three pump photons and emission of three Stokes photons at the end of the process. This is confirmed by the numerical solution of the dressed Schrodinger equation (308) with the initial state as a number state for the photon field 11 0,0), shown in Fig. 23a the dressed state vector /(f) approximately projects on the transfer eigenvectors during the process. It shows... 

Now, we can see the transition from the nth occupation number state to the vacuum state, in which the oscillator will emit nha>0 energy. Indeed, from Equation (91) it follows that... 

The natural line width of the spectral line is a significant result of the dissipative quantum process which accompanies the spontaneous emission of an atom. We will treat this emission process in a dissipative two-state model. We consider the two states of the atom as the zeroth and the first occupation number state of a linearly damped oscillator. In this case, the spontaneous emission of a photon is the consequence of the transition from the first occupations number state to the equilibrium state of the damped oscillator. In this model, the spectrum density of the emitted photon follows from Equation (92)... 

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