Greens functions and onebody quantum problems Source The

Green’s functions and one-body quantum problems Source The delta not yet invented at Green’s time 1

This is the sought integral equation: next, we convert volume integral -> surface integral 2

convert volume -> surface integral using the divergence theorem. 3

Simple use of Green’s functions for Schroedinger’s equation operator identities Example: H 1=d. V(r): then, in the coordinate representation 4

Lippmann-Schwinger equation The Lippmann-Schwinger equation is most convenient when the perturbation is localized. Typical examples are impurity problems, in metals. The alternative is embedding (Inglesfield 1981) 5

Suppose we can solve except inside the surface S. We can also solve S and wave functions outside are related to g by Green’s theorem It is assumed that G outside S is not modified by the presence of the localized perturbation. 6

S 7

S 8

John E. Inglesfield, Emeritus Professor , Cardiff 9

Inglesfield embedding The Lippmann-Schwinger equation is less convenient when the perturbation is extended. Typical examples are surface problems, when one wants to treat the effects of surface creation, reconstruction, or contamination. In such cases one can resort to slab models, but there are serious drawbacks in any attempt to represent a bulk by a few atomic layers, with quantized normal momenta. The only practical alternative is the method of embedding. 1) in an extended system, there is a localized perturbing potential. A surface S divides the perturbed region I from the far region II where the potential is negligible. 2) the problem in I alone can be solved 3) the extended unperturbed system could be treated easily because it is highly symmetric 4) the wave functions match those of the extended system on S. II S I 10

II S I d. V 11

II S I d. V 12

d. V By the divergence theorem so everything is written in terms of f and one obtains a localized variational problem in region I equivalent to a Schrödinger equation with a potential d at S. 13

Absence of Magnetism in classical physics Ampere Postulated “molecular-ring” currents to explain the phenomenon of magnetism Bohr-Van Leeuwen Theorem (1911) Partition function for a Classical electron in canonical ensemble Classical electron in magnetic field A change of variable removes the field. 14 magnetism cannot exist (catastrophe of classical physics)

Langevin paramagnetism Paul Langevin (1872 -1946) 15

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Can we explain ferromagnetism as due to dipole-dipole interactions? Orders of magnitude estimate

Heisenberg chain: a solvable 1 d model of magnetism 18

Hans Bethe solved the linear chain Heisenberg model Born July 2, 1906 Strassburg, Germany Died March 6, 2005 Ithaca, NY, USA Residence USA Nationality American Field Physicist Institution University of Tübingen Cornell University Alma mater University of Frankfurt University of Munich Academic advisor Arnold Sommerfeld Zur Theorie der Metalle. I. Eigenwerte und Eigenfunktionen der linearen Atomkette Zeitschrift für Physik A, Vol. 71, pp. 205 -226 (1931) 19

N spins ½ on a ring with a configuration Symmetries 20

Ground state for J>0: all spins up, no spin can be raised, and so no shift can occur 21

1 reversed spin in the chain Consider for the moment no hopping, only H 0 A configuration with r=1 is conveniently denoted by |n>, n=position of down spin. 22 They are all degenerate.

Now introduce the hopping term. 23

One- Magnon solution: spin wave We must insert the periodic boundary condition for the N-spin chain 24

One- Magnon solution: the spin wave is a boson (spin 1) This magnon dispersion law looks like a free quasi-particle dispersion for small k Two- Magnon solution: scattering, bound states 25

Several reversed spins

Two Magnons a configuration is denoted by |n 1, n 2>, n 1<n 2 reversed spins provided that adiacent sites are up spins: if the reversed spins are close the energy is different and this is an effective interaction. 27

Two independent magnons? NO! New k 1 and k 2 must be found since PBC do not separately apply to n 1 and n 2 (you cannot translate one magnon across the other). However there is 28