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The solutions of eigenvalue equations like the time-independent one-electron Schrödinger equation = form a a complete set of spin-orbitals , where () are normalized space orbitals and =↑ or ↓. The set can be taken orthogonal and ordered in ascending energy or in any other arbitrary way. Any one-electron state can be expanded as a linear combination of the . Moreover, we can think of a state for electrons obtained as follows. Choose in any way spin-orbitals out of the set {}, keep them in the original order but call them , ... ; now let |,...| be the state with one electron in each. Imagine labeling the indistinguishable electrons with numbers 1, 2, .... In this many-body state one has an amplitude of having electron in the one-particle state (, ). How to calculate ? A product like is in conflict with the Pauli principle because it fails to be antisymmetric in the exchange of two particles. However, the remedy is easy, because anti-symmetrized products are a basis for the antisymmetric states. To this end, let be one of the ! permutations of objects. If = 3, the set of 6 permutations comprises the rotations {(1, 2, 3), (2, 3, 1), (3, 1, 2)} and {(2, 1, 3), (3, 2, 1), (1, 3, 2)}.

If we can solve a problem with a time independent Hamiltonian , we surely meet many interesting but hard problems with a Hamiltonian where an extra term appears: sometimes the complication depends on time, but in other cases it is static. Here, () is in the Schrödinger picture, the one that comes directly from classical physics with , and so on, and sometimes we shall write () and for extra clarity. So, we have the task of solving which is notoriously difficult. We can solve formally by introducing the unitary time evolution operator such that

X-ray spectroscopy and Photoemission (see Chapter 6) show a correspondence between the electronic levels in many-electron atoms and in Hydrogen. The next Table shows the inner levels of Fe; the first column presents the spectroscopic notation, the second the corresponding Hydrogen quantum numbers and the third the measured binfing energy.

Green’s theorem (where is the outgoing normal to the surface bounding the volume ) is obtained from the divergence theorem with . Consider the one-particle Schrödinger equation defined in some volume with some boundary conditions; it is often convenient to change it into an integral equation by a method which is familiar from classical physics. One introduces a Green’s function satisfying multiplies (4.3) by , (4.4) by and subtracts; the result is (exchanging with ) The integral can be changed to a surface integral by Equation (4.1).

Often we treat excited levels as discrete, as they appear in low resolution or in some simple approximation. Sooner or later, however a closer analysis always shows that when all the degrees of freedom are considered they are in a continuum. Thus, the 2p level of H, H(2p), is coupled to a continuum of H(1s) + 1 photon, and thereby gains a width and a structure. The 2p state of H is discrete only if you accept to neglect its interaction with a continuum of photon modes that eventually take the H atom to the ground state while producing photons. In 1952, Fermi discovered a peak in the pion-proton ( − ) elastic cross section for center-of-mass kinetic energies 1.2 to 1.4 GeV; since the half width at half maximum (∼ 100 MeV) implies a lifetime ∼ 10s, which is very short, the strong interaction was implied in the formation and also in the decay. This was christened the doubly charged resonance. Such high-energy Physics contents are pertinent to this book: hundreds of resonances are familiar by now to particle physicists, but Fermi’s concept of resonances is important in quantum problems at all energy scales. Firing 500 eV electrons on Helium and measuring the loss spectrum, one observes[31] an asymmetric resonance in the ionization continuum al ∼ 60 eV which has been identified as the 22 state of He. Many more are known by now, and they are all due to the neutral, twice excited He atoms in auto-ionizing states.

Electron spectroscopies and their rich phenomenology convey information on molecules and solids and specifically on their excitations. To describe excited states we shall develop techniques based on modeling the spectra in terms of Green’s functions and also more general expectation values. ESCA is a set of spectroscopies, UPS (Ultraviolet Photoemission Spectroscopy), XPS (X-Ray Photoemission Spectroscopy)and AES (Auger Electron Spectroscopy). They can be angular resolved (ARUPS and the like) and time resolved; APECS (Auger Photoelectron Coincidence Spectroscopy) detects the photoelectron and the Auger electron most probably coming from the same atom. ESCA can make a chemical analysis with tiny samples and is able to discriminate the oxidation states and give information about the electronic structure.

Groups are central to Theoretical Physics, not only as mathematical aids to solve problems, but above all as conceptual tools. We shall develop the Group Theory that should be known by Condensed Matter theorists using a physical language and building on what the readers know already about quantum mechanical operators. In this Section, however, we need to introduce some abstract mathematical definitions.

To exemplify Group theory methods, I present LCAO calculations of molecular orbitals for simple molecules starting with a minimal basis set (only the atomic orbitals which are most directly involved in covalent bonding) neglecting overlap.

Let , where is a wave function and some operator. Then, acting with a unitary operator , one finds , where the transformed operator is . Thus, functions are transformed according to → while operators are transformed according to ; actually the two rules differ by a matter of notation. acts on everything on its right, so in the case of operators the last factor = is there just to ensure that the action of is limited to , while functions are at the extreme right and there is no need for that. We can consider (, , ) as a set of functions or as the components of an operator. As functions that transform as a basis of a representation of some symmetry Group, they transform according to the rule this is the vector representation, which is irreducible in cubic and higher Groups. If we treat them as a set of operators, we write This defines a vector operator, but the linear combinations that result are the same. A tensor is a set of operators (the components) that are mapped into linear combinations by every , that is, the multiplication Table is followed since

We used EOM several times (see Sections 4.3,4.4,5.1.2); now we extend the approach used in Equation (4.39) for the free propagator to interacting problems. Using the many-body Hamiltonian (1.63) one readily obtains with all the operators in the Heisenberg representation ( is a first-quantized one-body operator), with the notation = (, ). We multiply on the left by () and perform an interacting ground state average: