One of the most important quantum-mechanical properties is the property of exchange. The ideas behind this arise from the fact that elementary quanta are indistinguishable in the real word, but they are completely distinguishable in our theory, where we use different labels for the different particles. The procedure by which we make our theory indistinguishable as well leads to a new “force,” called quantum-mechanical exchange. In this chapter, we develop just what exchange is, we introduce the Heisenberg spin-exchange model and prove the Marshall sign rule, and we solve the one-dimensional version of the model using the so-called Bethe ansatz. You will find it amazing to learn the quantum origin of magnetism and details about how we work with quantum systems that have many particles. Always remember that permanent magnets are one of the best examples of macroscopic quantum mechanics.
One of the most important quantum-mechanical properties is the property of exchange. The ideas behind this arise from the fact that elementary quanta are indistinguishable in the real word, but they are completely distinguishable in our theory, where we use different labels for the different particles. The procedure by which we make our theory indistinguishable as well leads to a new “force,” called quantum-mechanical exchange. In this chapter, we develop just what exchange is, we introduce the Heisenberg spin-exchange model and prove the Marshall sign rule, and we solve the one-dimensional version of the model using the so-called Bethe ansatz. You will find it amazing to learn the quantum origin of magnetism and details about how we work with quantum systems that have many particles. Always remember that permanent magnets are one of the best examples of macroscopic quantum mechanics.
Paramagnetism could be explained by assuming that all spins are independent. To explain the phenomenon of permanent magnetization or ferromagnetism therefore, one must take into account the interaction between spins. In ferromagnets there must therefore be another force that tends to align the elementary magnets or spins. The mechanism for this alignment force was discovered by Heisenberg in 1928. It is quantum mechanical in origin and is called the exchange force. It is a subtle interplay between the Pauli principle and the Coulomb potential. The one-dimensional Ising model behaves therefore as a paramagnet! When Ising first derived this result, he conjectured that the higher-dimensional analogues would probably behave in the same way. The Ising model becomes equivalent to a contour model. The lines on the dual lattice separate into connected pieces which are either closed curves or curves which begin and end at the boundary of the lattice. These curves are called contours.
Abstract This chapter introduces the Heisenberg model, a fully quantum mechanical model that describes the magnetism of localized magnetic moments. The one-dimensional version of the Heisenberg model, the Heisenberg quantum spin chain, provides a good picture of magnetic materials that belong to a class of insulating magnetic materials where the interaction of the magnetic moments in one particular direction is much larger than in the perpendicular directions, and which can be described with high accuracy as quasi- one-dimensional magnets. A detailed description of the Heisenberg quantum spin chain is followed by a discussion of its various special cases, in particular the special case of the anisotropic Heisenberg quantum spin chain, the so-called XXZ quantum spin chain. It considers the solution of eigenvalue problem of this quantum spin and leads to Bethe’s conjecture for the wave function.
Abstract One of the outstanding differences between classical theory and quantum mechanics is the importance which the latter attaches to the identity of particles. Although perfectly well realised before the introduction of modern quantum theory, the fact that two electrons are experimentally indistinguishable did not introduce any novel features into the dynamics of many electron systems on Bohr’s theory. However, Heisenberg showed that these identity relations are of fundamental importance in many body problems, and was able to explain the formerly very puzzling characteristic of the spectrum of neutral helium, its division into two non-combining term systems. Owing to the identity of the particles, the wave function representing such a system must have certain symmetry properties; for the particles known in Nature this symmetry characteristic is that the function be either symmetric or anti-symmetric in the co-ordinates of the particles. For particles with a spin such as the electron and proton, only antisymmetry is allowed (Pauli’s principle), while for α-particles the wave function must be symmetrical. Since Heisenberg’s explanation of the helium spectrum the theory of anti-symmetric electron wave functions has been extended to describe the general features of the spectra of all atoms, while it appears from the work of Heitler and London that the theory is of fundamental importance in the explanation of chemical combination between atoms. Again, in furnishing a new statistics it is an essential feature of the theory of metals. As the effect of the identity relations depends on the fact that exchanging the electrons does not alter the system in any way, these effects are known as exchange phenomena.
Exchange forces lie at the base of some of the most striking explanatory successes of modern physics.An exchange interaction, for instance, is held to be responsible in some key sense for the chemical bond.An exchange force is the origin of ferromagnetism, providing the force that lines up individual spins into an overall pattern.In nuclear physics an exchange force is characteristic for the description of nucleon-nucleon forces.And, in perhaps the most far-reaching example, an exchange interaction is taken as definitional for the quantum field-theoretic description of nature, in which not only the fundamental forces (gravity, electromagnetism, the weak and strong nuclear forces), but also such non-fundamental ones as that operative between pairs of superconducting electrons in a lattice, are traced to the exchange of field quanta.In all these cases, moreover, exchange forces are a peculiarly quantum mechanical phenomenon, something characteristic of the new era that began in 1925 and continues into the present.And as a peculiarly quantum mechanical phenomenon, they are central to the new understanding of matter and force that was ushered in by the theory.For such a central concept, the historical origins of exchange forces are distinctly unclear.What was the path by which these diverse manifestations of exchange-in atomic physics, in condensed matter, in nuclear physics, in quantum field theory-came to go by the same name?Indeed, what is the common thread running through all these uses?The very notion of exchange is nebulous.Consider the following definition from a technical dictionary: exchange [QUANT MECH] (1) Operation of exchanging the space and spin coordinates in a Schrodinger--Pauli wave function representing two identical particles; this operation must leave the wave function unchanged, except possibly for a sign.(2) Process of exchanging a real or virtual particle between two other particles.'Here two distinct meanings of the term 'exchange' are on offer.The first notion
The behavior of classical macroscopic particles – from baseballs to planets – is governed by Newtonian mechanics as we have described in the first chapters of this book. The behavior of very small particles, on the other hand, is governed by an entirely different collection of equations and rules that go under the name of “quantum mechanics” and these particles are referred to as quantum particles. These rules have analogies to classical mechanics, but are fundamentally different and somewhat mysterious as we discussed in Chapter 8. The fundamental principles of quantum mechanics cannot be proven to be correct. They were developed based on the results of experimental observation, by analogy with classical mechanics, and through the consideration of logical implications of the formalisms used. There is no way to derive the fundamental elements of quantum mechanics, any more than one could predict from first principles that Newtonian forces should add as vectors. On the other hand, quantum mechanics is among the most rigorously tested theories in science, and the predictions of quantum mechanics have, so far, always been in agreement with experiment. In this chapter, we will present quantum mechanics as a series of postulates and examine their implications in application to simple one-dimensional systems. In the next chapter we will examine quantum effects in real systems including atoms, molecules, crystals, and artificially engineered nano-structures. Postulates of quantum mechanics QM Postulate 1: A quantum particle is described by a wave function Ψ( x , y , z , t ) which is a function of spatial coordinates and time. All physically determinable quantities can be derived from the wave function Ψ( x , y , z , t ) ; quantities which cannot be found from the wave function are not physically meaningful. Note that some things that one could ask about a quantum particle, for example the exact position and simultaneous exact momentum of a particle, are not derivable from the wave function and are therefore considered indeterminate (or uncertain ). These are things which cannot ever be known about a particle, and, as such, they are meaningless to contemplate.
This thesis is divided into 4 chapters. The first chapter gives a brief discussion on the
need for the study of Quantum spin systems, and an introduction to Heisenberg Model. In the
second chapter, we start the discussion with Quantum entanglement and then go on to show
the importance of anti ferromagnetic ground state from the view of entanglement. In the third
chapter, we discuss the ways to obtain the ground states of Heisenberg model, both ferromagnet
and antiferromagnet. Then we discuss the effect of magnetic field on the antiferromagnetic
ground states, for a good understanding of magnetic properties in Chapter(4). In the fourth and
the final chapter, we discuss how Bonner Fisher model uses the Bethe ansatz to obtain all the
properties of finite Heisenberg chains. Then we analyze the magnetic properties, like Magnetic
moment and Susceptibility, of Heisenberg antiferromagnets for finite as well as infinite chains.