Gaussian initial state

Here we will discuss two scenarios for the proto-neutron star cooling which we denote by A and B, where A stands for cooling of a star configuration with SC whereas B is a scenario without SC. The initial states for both scenarios are chosen to have the same mass Mi(A) = A(l>) for a given initial temperature of T = 60 MeV. The final states at T = 0, however, have different masses Mf(A) / Mf(B) while the total baryon number is conserved in the cooling evolution. The resulting mass differences are AM (A) = 0.06 M , A M(B) = 0.09 M and AM (A) = 0.05 Me, A M(B) = 0.07 M for the Gaussian and Lorentzian models, respectively. 

The most recent calculations, however, of the photoemission final state multiplet intensity for the 5 f initial state show also an intensity distribution different from the measured one. This may be partially corrected by accounting for the spectrometer transmission and the varying energy resolution of 0.12, 0.17, 0.17 and 1,3 eV for 21.2, 40.8, 48.4, and 1253.6 eV excitation. However, the UPS spectra are additionally distorted by a much stronger contribution of secondary electrons and the 5 f emission is superimposed upon the (6d7s) conduction electron density of states, background intensity of which was not considered in the calculated spectrum In the calculations, furthermore, in order to account for the excitation of electron-hole pairs, and in order to simulate instrumental resolution, the multiplet lines were broadened by a convolution with Doniach-Sunjic line shapes (for the first effect) and Gaussian profiles (for the second effect). The same parameters as in the case of the calculations for lanthanide metals were used for the asymmetry and the halfwidths ... 

This effectively states that the probability of the final state (left-hand side) is equal to that of all the initial states transforming to the final state (with probability P). Chandrasekhar expanded out the infinitesimal velocity and time changes of these quantities as Taylor series and used the Langevin equation to relate 5u and 5f. He showed that if the probability of changing velocity and position is given by a Gaussian distribution, then the probability, W(u, r, t) that a Brownian particle has a velocity u at a position r and at time t is... 

Equation (4.173) displays clearly how the cross-section is determined from the scattering dynamics via the time evolution of the initial channel state U(t — to) 4>n) and a subsequent projection onto the final channel state. In practice, the plane wave of the initial state in Eq. (4.169) can be replaced by a Gaussian wave packet, as illustrated in Fig. 1.1.1. When this wave packet is sufficiently broad, it will be localized sharply in momentum space. 

Thus the oscillator strengths for the transitions from these three levels were calculated and the theoretical absorption spectra were obtained by convolution with a Gaussian function with 0.16 eV FWHM (fig. 28). As shown in the figure, the shape of the spectrum strongly depends on the initial states. Therefore, the experimental spectrum taken at room temperature is inappropriate for the analysis of energy level structure in 4f25d configuration. Thus we compare the theoretical spectra with the excitation spectrum measured at 6 K (Reid et al., 2000). 

The two intermediate target states are set to be symmetric Gaussians with the same width parameters as those of the initial and target states and zero central momenta. The central coordinates of the wavepackets are set to be Ritsi = (2.51,1.81)t and RitS2 = (1.81,2.51)T (see Fig. 6.7). The initial guess held used to control the overall process is constructed by joining the optimal helds found for three intermediate steps the control from the initial state to the hrst intermediate target state, the control from the hrst intermediate... 

The initial state of the system is described by a normalized Gaussian wave packet of outgoing scattering states ... 

We consider the following model for computational purposes an initial state composed of an unpolarized racemic mixture of states is irradiated with pulses having Gaussian envelopes,... 

Once we fix the initial state ip,) and the final state ipj), the optimal field E(t) is obtained by some numerical procedures for appropriate values of the target time T and the penalty factor a. Though there should be many situations corresponding to the choice of ip,) and ipj), we only consider the case where they are Gaussian random vectors. It is defined by... 

Let s consider the wavefunction of the initial state, before absorption, which will be approximated using a Gaussian [17] ... 

The treatments presented in Sections 2 and 3 assume the initial state to be a ID Gaussian for diatomics and 3D for triatomics. This is correct at 0 K but, at a given temperature T, contributions from hot band(s) should be considered. In ID the contribution of hot bands can be calculated using the wavefunction for V = and the harmonic approximation ... 

It is important to spell out the limitations on the derivation of the distribution (2.2S) of fluctuations. Consider the most general initial state, which is XPinitial state is pure if the rank of the matrix p is unity. Otherwise, it is a mixture. The transition intensity to the final state / is y = Y,ijx Pijxj where x = < T />. y = x px is then a quadratic form where the amplitudes x have a gaussian probability density... 

Figure 1 shows four snapshots from the (numerically calculated) time-evolution -0(t) of the initial function -tpoix) = iVexp(—x /2)(l,l). This is a spinor with Gaussian initial functions in both components. More precisely, the pictures show (according to our interpretation) the position probability density 4> x)f + 1 2 x) p. We see that the shape of the wave packet at later times shows strange distortions, similar to distortions caused by interference phenomena. Moreover, consider the expectation value of the position x in the state which is (ac-... 

It is instructive to compute the time correlation function in the simple case that the ground and excited state potentials are harmonic but differ in their equilibrium position and frequency. This is particularly simple if the initial vibrational state is the ground state (or, in general, a coherent state (52)) so that its wave function is a Gaussian. We shall also use the Condon approximation where the transition dipole is taken to be a constant, independent of the nuclear separation, but explicit analytical results are possible even without this approximation. A quick derivation which uses the properties of coherent states (52) is as follows. The initial state on the upper approximation is, in the Condon approximation, a coherent state, i /,(0)) = a). The value of the parameter a is determined by the initial conditions which, if we start from a stationary state, are that there is no mean momentum and that the mean displacement (x) is the difference in the equilibrium position of the two potentials. In general, using m and o> to denote the mass and the vibrational frequency... 

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