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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.

One dimensional problems illustrate in simple fashion many aspects of the theory, and for this reason, every beginner in Quantum Mechanics is traditionally exposed to them. Some interesting one-particle problems in higher dimensions can be solved by separation of variables, thereby reducing them to 1d subproblems, and we shall encounter several examples in other chapters. However, in recent times there have been important technological developments in nano-devices and low-dimensional objects (like nano-wires and nano-tubes) that have made these problems much more directly relevant. In classical physics, a wire is one-dimensional in the limited sense that its length is many times its diameter. However, in Quantum Mechanics, the low dimensionality of thin films and 1d objects has a much stronger meaning. The next problem (deep potential well) shows that the confinement implies quantized levels divided by gaps that become wider the narrower the well. This implies that at low enough temperatures, the electrons behave as if the transverse dimensions were effectively frozen out. In this way, low d problems are no longer academic.

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 $$\begin{aligned} \begin{array}{l} B=\overrightarrow{\nabla } \wedge \overrightarrow{A}, \\[3mm] \overrightarrow{E}=-\overrightarrow{\nabla } \phi -\frac{1}{c}\frac{\partial \overrightarrow{A}}{\partial t}. \end{array} \end{aligned}$$

Angular resolved Auger-Photoelectron Coincidence Spectra (APECS) show a dichroic effect called DEAR-APECS in the antiferromagnetic phase of transition metal oxides and reveal well-defined structures in the Auger line shapes which are not seen otherwise in such partially filled bands. We propose a theory of the XPS and M23M45M45 APECS line shapes. Experimental photoemission and Auger spectra of CoO and NiO are compared with computed profiles. The method takes strong correlation into account, based on the exact diagonalization of octahedral cluster models with different electron numbers, including the on-site repulsion on the central ion expressed in terms of Racah parameters. The hopping parameters are written as combinations of Slater–Coster ones. We compute the relevant Auger matrix elements. The Racah A parameter and the hopping parameters which are known from the literature are refined by fitting the XPS lines; this allows for a parameter-free evaluation of the Auger lines. The Auger line shapes are explained in detail for both compounds, labeling the final states by their total spin. In the case of Co the different spin components peak at separated spectral regions and one sees a transfer of intensity to the high-spin region by varying the photoelectron-Auger electron geometry. In the NiO case the peaks overlap much more but a similar transfer from singlet to triplet final states is nevertheless unambiguously observed, and the theory accounts well for the observed resonance energies. Finally, we highlight the main questions about the dichroic effect that are still unanswered and will require a considerable extension of the theory.

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.

I vettori di traslazione primitiva del reticolo di Bravais fcc sono: \( {\mathbf{a}} = \frac{a}{2}(0,1,1),{\mathbf{b}} = \frac{a}{2}(1,0,1),{\mathbf{c}} = \frac{a}{2}(1,1,0), \) dove a e il lato del cubo.

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 propose a simple model of core–valence–valence Auger and APECS intensities from open-band solids to account for the alleged negative- U behavior of the spectra of early transition metals. In these systems the maximum of the line shape is shifted by the interaction to lower binding energy, which is the contrary of what happens in closed band materials where the two-hole Green’s function G ω (2) allows to understand the phenomenology of the spectra in terms of the U / W ratio of the on-site repulsion U to the band width W. Actually, for open-band solids, only part of the intensity comes from the decay of unscreened core-holes and is obtained by the two-body Green’s function G ω (2) , as in the case of filled bands. The rest of the intensity which arises from screened core-holes here is derived using a variational description of the relaxed ground state; this involves the two-holes-one-electron propagator G ω , that also contains one-hole contributions. We propose a practical scheme to calculate the three-body Green function by a summation of the perturbation series to all orders. We achieve that by formally rewriting the problem in terms of a fictitious three-body interaction. Our method grants non-negative densities of states, explains the apparent negative- U behavior and interpolates well between weak and strong coupling, as we demonstrate by test model calculations.

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 discuss the unique electronic properties of small clusters having a degenerate ground state in the non-interacting limit. We show that doping holes in a Cu 5 O 4 planar cluster may attract themselves in the presence of purely repulsive on-site interactions and discuss analogies and differences with the prototypical CuO 4 case. 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; moreover hole pairing involves a change in the symmetry of the ground state.

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