BCS ground state

The fact that creation of Cooper pairs is energetically favorable (it has a positive binding energy) naturally leads to the following question what is the preferred state of the system under conditions where Cooper pairs can be formed A tempting 

We explore next the implications of the BCS ground state wavefunction. Eor properly normalized pair states we require that 

nkP is the probability that a Cooper pair of wave-vector k is present in the ground state and uk the probability that it is not. We note that from the definition of these quantities, n k = Uk and U-k = Uk- Moreover, in terms of these quantities the normal (non-superconducting) ground state is described by i k = 1, Wk = Ofor k pand Mk = 1, l kl = Ofor k /jp. which is a consequence of the fact that at zero temperature all states with wave-vectors up to the Fermi momentum p are filled. 

The first term describes the electron interactions in the single-particle picture with the ions frozen, and the second term describes the electron interactions mediated by the exchange of phonons, when the ionic motion is taken into account. In principle, we can write the first term as a sum over single-particle terms and the second as a sum over pair-wise interactions  

The single-particle wavefunctions iAk) are eigenfunctions of the first term, with eigenvalues 

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The quantity Ck is actually the excitation energy above the BCS ground state, associated with breaking the pair of wave-vector k (see Problem 2). For this reason, we refer to Ck as the energy of quasiparticles associated with excitations above the superconducting ground state. 

Show that the second solution to the BCS equation, Eq. (8.17), corresponds to an energy higher than the ground state energy, Eq. (8.22), by 2fk-Show that the energy cost of removing the pair of wave-vector k from the BCS ground state is ... 

Figure Al.6.26. Stereoscopic view of ground- and excited-state potential energy surfaces for a model collinear ABC system with the masses of HHD. The ground-state surface has a minimum, corresponding to the stable ABC molecule. This minimum is separated by saddle points from two distmct exit chaimels, one leading to AB + C the other to A + BC. The object is to use optical excitation and stimulated emission between the two surfaces to steer the wavepacket selectively out of one of the exit chaimels (reprinted from [54]).

Singlet and Triplet Configurations. Let us first consider an interaction which involves just the ground-state determinant Aq and the four singly excited deter-mJna-n+e which Can bc deiivcd from an orbital... 

At the lowest temperature where the para-H2 and ortho-H2 concentrations are in thermal equilibrium, the rotational ground state and the lowest excited state (J = 0 and 1) are about equally populated, hence the comparable line intensities at 354 and 587 cm-1 at 77 K. With increasing temperature, the J = 1 state is more highly populated, and states with J > 1 are increasingly populated as well, at the expense of the J = 0 ground state, so that the So(l) line shows up much more prominently than So(0) at the higher temperatures. Profiles obtained at temperatures T > 100 K may similarly be fitted by simple three-parameter model profiles if one accounts for the higher So(J) and Qo(J) lines, J > 1, as well. Very satisfactory fits of the laboratory data have resulted [15]. The profiles of the individual lines vary with temperature. Fairly accurate empirical spectra may be constructed, even at temperatures for which no measurements exist, when the empirical temperature dependences of the three BC parameters are known, see Chapter 5 below. 

Examples of this type of reaction, where A, B, and C are atoms, are hard to find. Clear, well understood examples are particularly rare, and one must look instead in the uncertain field of elementary steps postulated as parts of complex mechanisms. A necessary condition for the reaction to occur is for the AB bond to be much stronger than the BC bond. The chances for success are presumably increased if AB has a low lying electronically excited state. They are further increased if formation of AB in the electronic ground state is forbidden by spin conservation. Since there is little detailed knowledge of even the few processes of the above type which have been proposed, we can give only a cursory discussion. 

For simplicity we have assumed a constant transition dipole function and neglected multiplication with Ephoton. The f(n) are conventional one-dimensional Franck-Condon (FC) factors for the (bound) vibrational wavefunctions of the BC entity (Graybeal 1988 ch.l5.6). Note, that the vibrational wavefunction in the upper state is evaluated at the equilibrium Re of the electronic ground state. 

Fig. 6.9. Left-hand side Vibrational excitation function N(ro) and weighting function W(ro) versus the initial oscillator coordinate ro for three values of the coupling parameter e. The equilibrium separation of the free BC molecule is f = 0.403 A and the equilibrium value within the parent molecule is re = 0.481 A. Right-hand side Final vibrational state distributions P(n) for fixed energy E the quantum mechanical and the classical distributions are normalized to the same height at the maxima. The classical distributions are obtained with the help of (6.32). The lowest part of the figure contains also the pure Franck-Condon (FC) distribution (

Fig. 9.1. Left-hand side Representation of an elastic potential energy surface. It has the general form (6.35) with coupling strength parameter e = 0. In case (a), the equilibrium bond distance in the electronic ground state equals the equilibrium separation of the free BC fragment. The heavy arrows schematically indicate two representative trajectories starting at the respective FC points. Right-hand side The corresponding final state distributions.

The wavefunction of the parent molecule in the electronic ground state is assumed to be a product of two harmonic oscillator wavefunctions with m and n quanta of excitation along R and r, respectively. In Figure 13.2(b) only the vibrational mode of BC is excited, n = 3, while the dissociation mode is in its lowest state, m = 0. The corresponding spectrum is smooth without any reflection structures. Conversely, the wavefunction in Figure 13.2(a) shows excitation in the dissociation mode, m = 3, while the vibrational mode of BC is unexcited. The resulting spectrum displays very clear reflection structures in the same way as in the one-dimensional case. Thus, we conclude that, in general ... 

The diatomic BC has been studied by Kouba and Ohm.366 The ground state is 4S, according to calculations using full-valence-shell Cl. 54 low-lying states were investigated. 

In this paper we confirm BE s assignment of the 500nm band provide a similar assignment for the 244nm band first observed by BC, and corroborate the Pj Pq designation for the 261.4nm band. The ground state of atomic Pb is Pn(29), with the 3pj multlplet... 

All spectroscopic constants were obtained from Gordon (3 ). Theoretical calculations by Kruba ( ) predict a ground state. The recent review by Huber and Herzberg (5) did not indicate any experimental data available on the vibrational-rotational constants for BC(g). 

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