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We study the time-resolved fluorescence spectrum in two-level systems interacting with an incident coherent field, both in the weak and intermediate coupling regimes. For a single two-level system in the intermediate coupling case, as time flows, the spectrum develops distinct features, that are not captured by a semi-classical treatment of the incident field. Specifically, for a field on resonance with the atomic transition energy, the usual Mollow spectrum is replaced by a four peak structure, and for a frequency that is half of the atomic transition energy, the time-dependent spectrum develops a second harmonic peak with a superimposed Mollow triplet. In the long-time limit, our description recovers results previously found in the literature. After analyzing why a different behavior is observed in the quantum and classical dynamics, the reason for the occurrence of a second harmonic signal in a two-level system is explained via a symmetry analysis of the total (electron and photon) system, and in terms of a three level system operating in limiting regimes. We find an increased second harmonic signal in an array of two-level systems, suggesting a superradiance-like enhancement for multiple two-level systems in cavity setups. Finally, initial explorative results are presented for two-level model atoms entering and exiting a cavity, which hint at an interesting interplay between cavity-photon screening and atomic dynamics effects., 19 pages, 11 figures. To be published in Phys. Rev. A

In general, the process that we use to prepare a system of atoms, molecules or whatever we want for a measurement produces many copies, but not all in the same quantum state. A pure state in which all the molecules, say, are in the same state, is a limiting case. In general, the system will be in a mixed state. One reason for that is that thermal excitations are unavoidable. Let the possible states be vectors of a Hilbert space with a basis \(\{\langle \psi _{n}|\}.\) The expectation value of an operator \(\hat{A}\) is $$ \langle \hat{A}\rangle =\sum _{n}P_{n}\langle \psi _{n}|\hat{A}|\psi _{n}\rangle ,$$ where \(P_{n}\) is the (classical) probability of finding the system in \(\langle \psi _{n}|\).

Even in flat Euclidean space it may be useful to use curvilinear coordinates; for instance, in 3d problems having central symmetry, we obtain an important simplification when the line element \(ds^2= dx^{2}+dy^{2}+dz^{2}\) is replaced by \(ds^{2}=dr^{2}+r^{2}d\theta ^{2}+r^{2}\sin ^{2}(\theta ) d\phi ^{2}.\) In a curved space we have no other choice, because Cartesian coordinates may exist only locally, that is, in an infinitesimal neighborhood. One example is the surface of a sphere of radius R. The spherical coordinates with r set equal to the radius R of the sphere do the job. In the latter case, we deal with a 2d curved subspace embedded in a 3d Euclidean space. The curved coordinates are intrinsic to the surface, and one can ignore the existence of a radial dimension.

When we dealt with the H atom in Chap. 17, we started from the classical formulation in terms of an effective particle having a reduced mass (see Sect. 2.5.1). The same argument works in Quantum Mechanics, with the momenta replaced with the quantum operators, starting from the Hamiltonian $$ H=-\frac{\hbar ^{2}}{2 m_{e}}\nabla _{e}^{2}-\frac{\hbar ^{2}}{2 m_{p}}\nabla _{p}^{2}+V(\rho ); $$ here, \(\nabla _{e}\) acts on \(r_{e}\), \(\nabla _{p}\) acts on \(r_{p}\), and \(\overrightarrow{\rho }=\overrightarrow{r_{e}}-\overrightarrow{r_{p}}\). The wave function \(\psi =\psi (\overrightarrow{r_{e}},\overrightarrow{r_{p}})\) depends on the coordinates and spins of both particles, however we omit spins, since H does not depend on them in the non-relativistic limit.

We need to extend the quantum mechanical theory to the case of N particles; separately, the particles would be described by a Schrodinger equation or the Pauli equation, depending on their spins. As in the classical case, the Hamiltonian of the system will be the sum of those of the particles plus an interaction term (possibly). For independent particles (no interaction), the Hamiltonian is, in obvious notation $$ \hat{H}(1,2,\ldots , N)= \sum _{n}^N \hat{h}(n), $$ where h(n) describes particle n. The wave function \(\varPsi (1,2,\ldots , N)\) depends on all the orbital and spin degrees of freedom of the particles, so it is a spinor in the spin space of each particle.

The 3-dimensional plane wave is the product of one-dimensional plane waves and the kinetic energy is the sum of the contributions of motions along x, y, z. More generally, the problem is separable into Cartesian coordinates if the potential energy is of the form $$\begin{aligned} V(x,y, z)=U_{x}(x)+U_{y}(y)+U_{z}(z) \end{aligned}$$ where \(U_{x}(x), U_{y}(y)\) e \(U_{z}(z)\) are arbitrary functions.

The usual laws of the circuits (Ohm’s law, Kirchoff’s law, and so on) are valid in the macroscopic world. However, when the linear dimensions are less than an electron mean free path, which may be on the order of 10 nm (22 in Al and 55 in Cu), the motion of the electrons is ballistic, that is, the circuit obeys quantum laws.

Classically, a point charge that circulates on a ring of radius r produces a current \(i = \frac{ev}{2 \pi r}\), which causes a magnetic dipole moment \(\overrightarrow{\mu }= \frac{i}{c} S \overrightarrow{n}\), \(S= \pi r^{2}\), in obvious notation; \(\overrightarrow{\mu }= \frac{e v}{2c} r \overrightarrow{n}= \frac{e}{2c} \overrightarrow{r} \wedge \overrightarrow{v}= \frac{e}{2 m c} \overrightarrow{L}.\).

Following the method of Galileo, Theoretical Physics uses Mathematics as natural and essential language to describe reality. But even Galileo would probably be surprised by the degree of success of the mathematical description of the world. Simple elegant laws like the Planck radiation formula agree with reality accurately over all the electromagnetic spectrum; on the other hand, the need for abstract mathematical concepts stemming from complex analysis, such as topology, Group Theory, and infinite-dimensional spaces shows that the fabric of the Universe is highly nontrivial. Is God a mathematician? is the title of an interesting book by Mario Livio. The Nobel Laureate in Physics Eugene Wigner in 1960 spoke about the unreasonable effectiveness of Mathematics in the understanding of Reality.

The Schrodinger–Pauli theory is characterized by the fact that every system must have a ground state, whose energy is a lower bound to the energies of all states. This is not true in Classical Mechanics, in which, for instance, a H atom could have any energy.

Among the fundamental constants of Physics, Planck’s constant h is an action [h = energy \(\times \) time= angular momentum]. It is a tiny action, \(h \sim 6,626196 10^{-34}\) J s. Most often, one uses the notation \(\hbar =\frac{h}{2 \pi }=1.054 10^{-34}\) J s. All the phenomena in which h is non-negligible are quantum phenomena, while sometimes phenomena in which the characteristic actions are \(\gg h\) may have classical aspects, which allow for a simpler description; however all the microscopic mechanisms are fundamentally quantum, and many phenomena such as magnetism, superfluidity and superconductivity are just macroscopic quantum phenomena; their quantum mechanical description is far from trivial. Historically, the breakdown of classical Physics (after many triumphs) was urged by the clear inability of the latter to explain a few quantum effects that where known by 1900. This breakdown was shocking, because well-understood laws of Physics that had been validated by experiment in many ways were seen to lead to wrong conclusions in a class of phenomena quite unexpectedly.

The Indian physicist S. Pancharatnam, working in quantum Optics in 1956, introduced the novel concept of a geometrical phase. Let \(H(\xi )\) denote a Hamiltonian that depends on some parameters \(\xi \), with ground state \(|\psi (\xi )\rangle \). One can define a phase difference \(\varDelta \varphi _{12}\) between two ground states \(|\psi (\xi _{1})\rangle \) and \(|\psi (\xi _{2})\rangle \): $$ \langle \psi (\xi _{1})|\psi (\xi _{2})\rangle =|\langle \psi (\xi _{1})|\psi (\xi _{2})\rangle | e^{-i \varDelta \varphi _{12} }. $$

Let us start with the Heavyside (Oliver Heaviside (1850–1925) was probably the first to use the \(\delta \) before Dirac, and the work of George Green also implies the concept. Often the names are not historically fair) \(\theta \) discontinuous function, also known as the step function, defined by $$\begin{aligned} \theta (x) = \left\{ \begin{array}{ll} 1 &{} \text {se } x>0, \\ {1 \over 2} &{} \text {if } x=0, \\ 0 &{} \text {se } x

The operators that represent variables of classical dynamics are built by analogy with classical analogues. (Since the classical description, as we know, can be changed by canonical transformation, this statement implies that in Quantum Mechanics, we must enjoy the same freedom. We shall see later how this arises) Sometimes, there are complications due to the fact that some operators fail to commute; such cases will be noted below. Therefore, for the simple case of one particle in a potential, $$\begin{aligned} \hat{H} = - {\hbar ^{2} \over 2m} \nabla ^{2} +V(\hat{x}) \end{aligned}$$ is the Hamiltonian operator. By the same criterion, one can write down the Hamiltonian operator for systems of particles.

The oscillator Hamiltonian in the coordinate representation is: $$\hat{H} = {p^{2} \over 2m} +\frac{1}{2} m \omega ^{2} x^{2}.$$

The system referred to above could be a particle, an atom, or even a macroscopic superconductor (then, x stands for a very large set of coordinates), so the statement is quite strong and general. In any case, all the information that is available from all possible experiments is in \(\varPsi _{a}(x, t) \). The wave function must be taken to be normalized. In the case of a single degree of freedom \(\varPsi _{a}(x, t), \) the normalization condition reads as \(\int |\varPsi (x, t)|^{2} dx=1\), while in general, one must integrate the square modulus over all the variables. The function is complex, therefore \( \varPsi (x,t)=| \varPsi (x,t)| e^{i\phi (x, t)}\), where \(\phi (x, t)\) is the phase. One can change \(\phi (x, t)\) by a constant phase factor (for instance, multiplying \( \varPsi (x, t)\) by i) but the physical state remains the same; nevertheless, the phase difference between two wave functions does matter a lot, since the wave functions do interfere. We shall see that the phase can be changed in several ways (e.g. rotations, Galileo transformations, gauge changes).

The (classical) electromagnetic fields in vacuo that satisfy given boundary conditions can be calculated through Maxwell’s equations. In the Gauss system they read as: $$\begin{aligned} \begin{array}{l} \overrightarrow{\nabla }\cdot \overrightarrow{E} = 4 \pi \rho , \\ \overrightarrow{\nabla }\cdot \overrightarrow{B} = 0, \\ \overrightarrow{\nabla }\wedge \overrightarrow{E} = -\frac{1}{c}\frac{\partial \overrightarrow{B}}{\partial t}, \\ \overrightarrow{\nabla }\wedge \overrightarrow{B} = \frac{1}{c}\frac{\partial \overrightarrow{E}}{\partial t}+\frac{4 \pi }{c}\overrightarrow{j}, \end{array} \end{aligned}$$ where \(\overrightarrow{j}\) and \(\rho \) are current density and charge density. In all, they are 4 functions of space and time. The fields can be computed and measured, however, it amazing that the Maxwell equations succeed in giving us 6 measurable quantities (3 components of \(\overrightarrow{E}\) and 3 of \(\overrightarrow{B}\)) having only 4 quantities in input. This is a most remarkable property of the electromagnetic field. Moreover, we can obtain the same field more easily by working out 4 quantities, namely the scalar potential \(\phi \) and the vector potential \(\overrightarrow{A},\) such that $$\begin{aligned} \begin{array}{l} B=\overrightarrow{\nabla } \wedge \overrightarrow{A}, \\[3mm] \overrightarrow{E}=-\overrightarrow{\nabla } \phi -\frac{1}{c}\frac{\partial \overrightarrow{A}}{\partial t}. \end{array} \end{aligned}$$

We study theoretically the memory effects due to different kinds of initial conditions in the transport properties of one-dimensional systems described by the Tomonaga-Luttinger model. We show that in presence of electron–electron interactions the sudden switching of a weak link between two initially uncontacted reservoirs induces a qualitative change in the transient current with respect to the contacted case. In particular the different switching process produces a change in the powerlaw temporal relaxation of the current towards the steady-state as well as a significant suppression of the transient oscillations. Even more dramatic is the response to a sudden interaction quench, which remarkably leads to a current–voltage characteristic differing from the one displayed if the system was initially interacting.

In Fisica, la conoscenza di ogni argomento e sempre operativa, cioe comporta l'abilita di risolvere problemi; ma di solito non basta mettere i dati in qualche formula risolutiva. Lo studente deve riconoscere nel problema una fattispecie della teoria generale, e questo puo essere arduo. Pero un problema abbastanza impegnativo da richiedere una comprensione approfondita della materia non implica necessariamente una soluzione lunga ed elaborata. In questo volume gli Autori si sono proposti di presentare esercizi, ciascuno dei quali puo essere svolto con pochi calcoli, tipicamente in un'ora. Tuttavia per lo studente essi rappresentano una significativa verifica della sua preparazione. La formulazione, che in alcuni problemi e modellistica, ha il pregio di evidenziare gli aspetti concettuali. I procedimenti proposti spesso non sono unici e il lettore potrebbe escogitarne altri ugualmente validi."

We review the Cini-Sawatzky approach to the line shape analysis of Auger CVV transitions, in its basic and original formulation. Then, several extensions of the theory are reviewed, namely the inclusion spin-orbit coupling, dynamical/plasmon screening, overlap effects, off-site interactions, the treatment of disorder and the formulation for partially filled bands. We conclude by highlighting what we consider to be future directions in the field., 15 pages, 7 figures, to appear in J. Electron Spectroscopy and Rel. Phen

We introduce a {\em local} formalism, in terms of eigenstates of number operators, having well defined point symmetry, to solve the Hubbard model at weak coupling on a $N\times N$ square lattice (for even $N$). The key concept is that of W=0 states, that are the many-body eigenstates of the kinetic energy with vanishing Hubbard repulsion. At half filling, the wave function demonstrates an antiferromagnetic order, a lattice step translation being equivalent to a spin flip. Further, we state a general theorem which allows to find all the W=0 pairs (two-body W=0 singlet states). We show that, in special cases, this assigns the ground state symmetries at least in the weak coupling regime. The N=4 case is discussed in detail. To study the doped half filled system, we enhance the Group Theory analysis of the $4 \times 4$ Hubbard model introducing an Optimal Group which explains all the degeneracies in the one-body and many-body spectra. We use the Optimal Group to predict the possible ground state symmetries of the $4 \times 4$ doped antiferromagnet by means of our general theorem and the results are in agreement with exact diagonalization data. Then we create W=0 {\it electron} pairs over the antiferromagnetic state. We show analitycally that the effective interaction between the electrons of the pairs is attractive and forms bound states. Computing the corresponding binding energy we are able to definitely predict the exact ground state symmetry., 18 pages, 2 embedded figures

We study the Cu_{5}O_{4} cluster by exact diagonalization of a three-band Hubbard model and show that bound electron or hole pairs are obtained at appropriate fillings, and produce superconducting flux quantisation. The results extend earlier cluster studies and illustrate a canonical transformation approach to pairing that we have developed recently for the full plane. The quasiparticles that in the many-body problem behave like Cooper pairs are W=0 pairs, that is, two-hole eigenstates of the Hubbard Hamiltonian with vanishing on-site repulsion. The cluster allows W=0 pairs of d symmetry, due to a spin fluctuation, and s symmetry, due to a charge fluctuation. Flux quantisation is shown to be a manifestation of symmetry properties that hold for clusters of arbitrary size.

The Cu-O plane and the clusters that possess the same C4v symmetry around a Cu ion have 2-hole eigenstates of the kinetic energy with vanishing on-site repulsion (W=0 pairs). Cluster calculations by exact diagonalisation show that these are the quasiparticles that lead to a paired ground state, and have superconducting flux-quantisation properties. Here, we extend the theory to the full plane, and show that the W=0 quasiparticles are again the natural explanation of superconducting flux-quantisation. Moreover, by a new approach which is exact in principle, we calculate the effective interaction Weff between two holes added to the ground state of the repulsive three-band Hubbard model. To explain how a noninteracting electron gas becomes a superconductor when switching the local Coulomb interaction, we obtain a closed-form analytic expression including the effects of all virtual transitions to 4-body intermediate states (exchange of an electron-hole pair). Our scheme is ready to include other interactions which are not considered in the Hubbard model but may be important. In the plane, the W=0 pairs have 1 B 2 and 1 A 2 symmetry. The effective interaction in these channels is attractive and leads to a Cooper-like instability of the Fermi liquid, while it is repulsive for triplet pairs. From Weff, we derive an integral equation for the pair eigenfunction; the binding energy |∆| of the pairs is in the range of tens of meV. However, our symmetry-based method is far more general than the model.

In the three-band Hubbard model, the fully symmetric Cu-O clusters with a central Cu and appropriate hole numbers have many-body ground states in which holes are paired. Here we investigate the mechanism of pairing, focusing on clusters with ≤ 21 atoms and 4 holes. We show that spin-flip diagrams yield the effective interaction. The analysis agrees with the results of exact diagonalization and shows that singlet-triplet splitting measures the strength of the pairing interaction. Moreover, the quantization at half-integer values of a magnetic text flux is consistent with supercon-ducting pairing.

We propose a quantum-mechanical effect specific of the symmetry of the Cu-O plane of cuprate superconductors. The mechanism considers hole pairs in single-particle degenerate states which, because of the configuration interaction, form singlet states with zero on-site repulsion (W = 0 pairs). In the many-body problem the effective interaction due to the spin exchange with the holes ofb1 symmetry is attractive. The exact diagonalization of the extended Hubbard Hamiltonian with 4 holes using well-established parameters for several clusters shows that the ground state has1B2(xy) symmetry and that binding energies are of the order of 10 meV. Unlike other electronic mechanisms proposed so far, this effect is due to the symmetry of the system and does not rely on off-site interactions; when the symmetry is broken the repulsion wins. Besides the correct symmetry of the pair, we obtain good agreement with experiment for the binding energies and the singlet-triplet energy separation of the quasiparticles.

We show that a tight-binding model device consisting of a laterally connected ring at half filling in a tangent time-dependent magnetic field can in principle be designed to pump a purely spin current. The process exploits the spin-orbit interaction in the ring. This behavior is understood analytically and found to be robust with respect to temperature and small deviations from half filling., 4 figures, 1 typo corrected

Magnetic moments excited by currents in nanoscopic circuits containing loops are dominated by quantum effects and depend nonlinearly on exciting bias, at variance from classical expectations. The creation of a magnetic dipole by a bias-induced current can be reversed: i.e. connected rings excited by a time-dependent internal flux produce ballistic currents in the external wires even in the absence of an external bias. By studying the real-time quantum evolution of tight-binding models in different geometries, several kinds of crucial experimental tests of these ideas can be envisaged, resulting in potentially useful devices. As a direct consequence of the above mentioned nonlinearity, one can achieve, by employing suitable flux protocols, single-parameter nonadiabatic pumping. Next, we present several results that depend in an essential way on the spin magnetic interaction and study the conditions for a maximally spin polarized current. The effects of the spin-orbit interactions on the above thought experiments are explored.

We consider a quantum ring laterally connected to open one-dimensional leads described within the Luttinger liquid (LL) model and subject to a time-dependent driving magnetic field o(t ). The closed loop is obtained by connecting two points of the LL by a weak link, which is treated exactly to lowest order in perturbation theory. Analytical results for the current across the weak link, as well as in the external circuit and inside the loop, are obtained. We pay special attention to the case of smooth magnetic pulses such that the response of the system is approximately adiabatic. We find that the response to the flux o is linear only if o is a small fraction of a flux quantum o0. By inserting whole quanta, a current goes in the external wires as two bunches, separated by a time proportional to the length of the loop, and a net pumped current in the external wires is produced. This kind of pumping is consistent with the predictions of the corresponding tight-binding model in the uncorrelated limit, but is not destroyed by electron-electron interactions.

We study the zero-temperature Green function for the local polaron model that describes the interaction of an electron in an empty band with a boson continuum; the coupling takes place at one site in the lattice. To this end, we propose a new exact approach which consists in two steps: first, the problem is reduced to the solution of an infinite linear algebraic system of recursion equations by the method of excitation amplitudes; then, a systematic expansion of the solution is generated by Feenberg's formalism. Previously, the problem had been solved exactly in the narrow-band limit; otherwise, the solution was achieved for one or a few boson modes. Here we consider a boson continuum and obtain an infinite expansion of the self-energy in terms of the dressed Green function. In this case, the recursive, continued-fraction technique is equivalent to self-consistent perturbation theory. The results are illustrated by selected numerical examples, and show that the satellites are much more sensitive to the boson dispersion than the main line.

We propose a configuration interaction mechanism that leads to hole pairs with zero repulsion energy which are exact eigenstates of the two-band Hubbard Hamiltonian. The two-hole pairs have symmetry and arise from degenerate states at the Fermi level; we show that they must exist independently of the band filling. To discuss the possible relevance of these states to the problem of pairing in high- superconductors we solve five- and nine-site model clusters with four holes. The quasiparticles become dressed by the interaction with the background holes and get paired with a set of valence band parameters well established for high- cuprates. We predict a number of important experimental facts such as binding energies of the correct order of magnitude, the symmetry of the order parameter and the singlet - triplet energy separation of the Cooper pairs.

A quantum-mechanical effect specific of the symmetry of the Cu-O plane in cuprate superconductors is presented. We show that pairs of holes in degenerate states are such that their mutual on-site Coulomb interaction W vanishes identically. A procedure for obtaining such W = O two-hole states in clusters of any size is outlined. Small clusters also display the effect provided they have the full C-4v symmetry and a Cu atom at the centre. In the many-body problem that arises when degenerate states are at the Fermi level, these holes interact indirectly through their screening clouds. This interaction is analysed by the exact diagonalization of an extended Hubbard Hamiltonian in doped CuO4 clusters and is shown to be attractive with the literature values of all parameters. Bound states of B-1(2) symmetry and binding energies of the order of 10 meV are obtained. The hole-hole interaction readily returns strongly repulsive if impurity potentials lower the symmetry. Off-site interactions fit well into the scheme. If the current estimates of the parameters are correct, this correlation effect alone can explain pairing.

A new formalism, based on the method of Excitation Amplitudes, is presented in order to set up a theory of the second harmonic generation (SHG) and other Inelastic Light Scattering phenomena, like the dynamical Start effect, when many-photon effects are considered. We develop the quantum theory and its semiclassical limit on equal footing, and show that the maximum intensity of the Second Harmonic fields does not generally correspond exactly to twice the incident frequency. The most interesting case occurs in highly polarizable systems when the exciting field is strong enough to produce appreciable many-photon effects, yet there are still observable quantum fluctuations. The inclusion of electron-photon coupling to all orders then leads to new effects, including a contribution of the Dynamical Stark Effect to the SHG response, as shown by model calculations.

We develop a simple variational approximation for the ground state of a conduction band in the presence of a localized core-hole potential, which fulfils the Anderson orthogonality theorem. Using this as the initial state of the Auger decay, we derive an expression for the line shape, including correlation effects in the Ladder Approximation. The computed spectrum is a combination of one-hole and two-hole-one-electron contributions; it is similar to the experimental spectra of low -electron-density systems that show the apparent negative U behaviour. These results show that in such systems the conduction electrons effectively screen the core hole before it decays. Also, we present evidence for a transition from complete relaxation to no relaxation with increasing band filling; the “unrelaxed” contribution to the spectra reflects a two-holes density of states.

Characterize the fluctuations of the energy in the grand canonical ensemble and prove that $$\left\langle {{{\left( {\Delta E} \right)}^2}} \right\rangle = {\left\langle {{{\left( {\Delta E} \right)}^2}} \right\rangle _{can}} + \left\langle {{{\left( {\Delta N} \right)}^2}} \right\rangle \left( {\frac{{\partial U}}{{\partial N}}} \right)_{T,V}^2$$ where U=〈E〉 is the average energy, N the average number of particles, 〈(ΔN)2〉 its fluctuations, and 〈(ΔE)2〉 can the energy fluctuations as obtained from the canonical ensemble.

A plane rigid rotator has the following Hamiltonian $${\hat H_0} = \frac{{\hat L_z^2}}{{2I}}$$ where I is the momentum of inertia and \({\hat L_z} = - i\hbar \tfrac{d}{{d\phi }}\) is the component of the angular momentum in the z direction (ϕ is the azimuthal angle). A small perturbation $${\hat H_1} = \lambda \cos (2\hat \phi ) \lambda \ll 1$$ is applied to the rotator. Determine the average value of \({\hat H_1}\) on each unperturbed eigenstate. Then, determine the off-diagonal elements of the perturbation matrix between the degenerate states. Finally, find the first order correction to the energy of the ground state and the splitting induced on the enery of the first excited state.

Consider a three dimensional gas of bosons with spin 0 and single particle energy given by $$\varepsilon = \frac{{{p^2}}}{{2m}}$$ where p is the absolute value of the momentum and m the mass of the particle. Write down the equation determining the critical temperature (T c ) for the Bose-Einstein condensation in the ground state.

Let Â=† be an observable operator with a complete set of eigenstates |ϕ n 〉 with eigenvalues α n (n=0, 1, 2,...). A generic state is given by $$\left. {|\psi } \right\rangle = \left. {N(3|{\psi _1}} \right\rangle \left. { - 4i|{\psi _2}} \right\rangle )$$ where |ψ1〉,|ψ2〉 are orthonormal. Find N and the probability P 4 that a measurement of  yields α4. What does it happen in case of degeneracy? Specialize these calculations to the case |ϕ n 〉=|ψ n 〉.

Let us consider a particle subject to the following three dimensional harmonic potential $$U\left( r \right) = \frac{{M{\omega ^2}{r^2}}}{2}$$ with r the radial distance in spherical polar coordinates. Discuss the properties of the associated Schrodinger equation for r≈0 and r→+∞.

Determine the uncertainty relations between the orbital angular momentum \(\hat L = \left( {{{\hat L}_x},{{\hat L}_y},{{\hat L}_z}} \right)\) and the components of the position and of the momentum operators \(\hat r = \left( {\hat x,\hat y,\hat z} \right),\hat p = \left( {{{\hat p}_x},{{\hat p}_y},{{\hat p}_z}} \right)\). Then, find the operator \({\hat L_z}\) in spherical polar coordinates and explain why the operators \(\hat \phi\) (azimuthal angle) and \({\hat L_z}\) can be measured simultaneously. What are the functions of \(\hat \phi\) whose commutator with \({\hat L_z}\) has a physical sense?

In Quantum Mechanics, the state of a particle in one dimension and in presence of a potential U(x,t), is entirely described by a complex wave function ψ(x,t) obeying the time dependent Schrodinger equation $$i\hbar \frac{{\partial \psi \left( {x,t} \right)}}{{\partial t}} = - \frac{{{\hbar ^2}}}{{2m}}\frac{{{\partial ^2}\psi \left( {x,t} \right)}}{{\partial {x^2}}} + U\left( {x,t} \right)\psi \left( {x,t} \right)$$ where m is the mass of the particle and ħ is the Planck constant, h, divided by 2π.

An atomic nucleus of Helium consists of a gas of 0.18 nucleons in a volume of 1 fm3 (1fm = 10−13cm). In this system, we can find two kinds of nucleons (protons And neutrons) with spin S=1/2 and their masses, m n ≈ 1GeV, may be considered equal. Compute the Fermi energy for the system and comment about its nature at the ambient temperature. Assume that the particles are independent with a single particle energy $$\varepsilon = \frac{{{p^2}}}{{2{m_n}}}$$ where p is the absolute value of the momentum. Also, make use of: h=7×10−24GeVS, 10−14cm≈7×10−25s, k=8.61×10−5eV/K

The effects of long-range interactions in quantum transport are still largely unexplored, mainly due to the difficulty of devising efficient embedding schemes. In this work we present a substantial progress in the interacting resonant level model by reducing the problem to the solution of Kadanoff-Baym-like equations with a correlated embedding self-energy. The method allows us to deal with short- and long-range interactions and is applicable from the transient to the steady-state regime. Furthermore, memory effects are consistently incorporated and the results are not plagued by negative densities or nonconservation of the electric charge. We employ the method to calculate densities and currents with long-range interactions appropriate to low-dimensional leads, and show the occurrence of a jamming effect, which drastically reduces the screening time and suppresses the zero-bias conductance. None of these effects are captured by short-range model interactions.

We know that the free energy F(T,V,N) of a thermodynamic system is extensive. Show that $$N{\left( {\frac{{\partial F}}{{\partial N}}} \right)_{T,V}} + V{\left( {\frac{{\partial F}}{{\partial V}}} \right)_{T,N}} = Nf = F$$ with f the free energy density expressed in suitable variables. Given this result, from the differential properties of F(T,V,N), show that $$\Phi = N\mu$$ with Φ the Gibbs potential defined as Φ=F+PV. In the above expression, μ is the chemical potential properly defined in terms of F(T,V,N).

A gas is in contact with a surface. On the surface we find N 0 localized and distinguishable sites adsorbing N (N≤N 0) molecules of the gas (each site can adsorb zero or one molecule of the gas). Find the grand canonical partition function of the system, and determine the chemical potential as a function of the average number of particles 〈N〉 which are adsorbed by the surface. You can think that the canonical partition function of an adsorbed molecule is a function only of the temperature, Q(T), and that all the adsorbed molecules are non interacting.

A method for computing the intensity of electric dipole second-harmonic generation from semiconductor surfaces and interfaces is presented. It uses third-order perturbation theory and a Green's function technique to embody Fresnel's formulas. Calculations for As/Si(111) are compared to experimental data, and show fair agreement for the spectral and angular dependence of the second-harmonic signal; the calculated absolute intensity, however, is too high. To investigate the origin of the discrepancy, we generalize the theory to all orders in the electron-photon coupling. The new scheme, based on the method of excitation amplitudes, is outlined. It allows to set up a theory of the second-harmonic generation response and other inelastic light scattering phenomena, like the dynamical Stark effect. Many-photon effects can modify the response in a qualitative way, and the maximum intensity of the second-harmonic fields does not generally correspond exactly to twice the incident frequency. This formalism also leads to new exact sum rules for nonlinear optics, that complement those obtained recently by Bassani and Scandolo.

The Cini-Sawatzky theory of CVV Auger transitions does not give simultaneous agreement with the position and line shape of the ${\mathit{M}}_{45}$${\mathit{N}}_{45}$${\mathit{N}}_{45}$ Auger spectrum of Ag. In accord with recent results for Au, this failure is attributed to the neglect of off-site interactions between the final-state holes. The magnitude of the discrepancy between theory and experiment for Ag is twice that found for Au and this is understood in terms of the difference in the ratio of off-site interaction to d-band width in the two metals.

Nanoscopic rings pierced by external magnetic fields and asymmetrically connected to wires behave in sharp contrast with classical expectations. By studying the real-time evolution of tight-binding models in different geometries, we show that the creation of a magnetic dipole by a bias-induced current is a process that can be reversed: connected rings excited by an internal ac flux produce ballistic currents in the external wires. In particular we point out that, by employing suitable flux protocols, single-parameter nonadiabatic pumping can be achieved, and an arbitrary amount of charge can be transferred from one side to the other. We also propose a set up that could serve a memory device, in which both the operations of {\it writing} and {\it erasing} can be efficiently performed., 5 pages, 6 figures, accepted by Phys. Rev. B