Blochs theorem when i started to think about it, i felt that the. Various interactions in solidstate nuclear magnetic. Write down hamiltons equations for the following hamiltonian h 1 2m. The scheme is lagrangian and hamiltonian mechanics. Hamiltonian structure of the algebraic riccati equation. Pdf fractional lagrangian and hamiltonian formulations. C timedependent states electron wavepacket and group velocity. Frequently attributed to albert einstein actual quote. However, it is more convenient to use quantizedstate wave functions explicitly in expanding the field operator. In this part of the paper, we summarized the various forms of hamiltonians subject to the secular approximation of highfield truncation.
In qm we learn that the ground state must have the symmetry of the hamiltonian so there cant be a dipole moment interactions between ions and electrons have no preferred direction in space. As an alternative to lagranges equations, a hamiltonian formulation of rigid body dy. Among the various states of condensed matter that solid state theory seeks to describe are. Hamiltonjacobi equations, viscosity solutions and asymptotics of hamiltonian systems by diogo aguiar gomes doctor of philosophy in mathematics university of california at berkeley professor lawrence c. In fact, in many systems where the above approximations are invalid e.
This lecture does not replace a course on solid state physics, which covers the. A defining the base states and the hamiltonian matrix 9. The hamiltonian 1,3 describes the known highenergy physics on the energy scale of hartree, and one aims at describing the lowenergy properties using reduced e ective, phenomenological theories. Thus, the problem of pairing correlations has been simplified considerably. Hamiltonian structure of the algebraic riccati equation and its in nitesimal vstability nanaz fathpour edmond a. Evans, chair the objective of this dissertation is to understand the.
Hamiltonian mechanics is a theory developed as a reformulation of classical mechanics and predicts the same outcomes as nonhamiltonian classical mechanics. The hamiltonian is the legendre transform of the lagrangian when holding q and t fixed and defining p as the dual variable, and thus both approaches give the same equations for the same generalized momentum. Piyapong yuantong hamiltonian matrices and the algebraic riccati equation theorem consider the hamiltonian matrix h with no eigenvalue on the imaginary axis and the invariant subspace. Fractional lagrangian and hamiltonian formulations in field theory. Pdf lecture notes for the undergraduate course on solid state physics. Application of those results to the equations of mechanical pendulum, to josephson systems of solid state physics and to questions from celestial mechanics are given.
Various interactions in solidstate nuclear magnetic resonance. Inspired by, but distinct from, the hamiltonian of classical mechanics, the hamiltonian of optimal control theory was developed by lev pontryagin as. Solid state physics mcgill physics mcgill university. Hamiltons equations are often a useful alternative to lagranges equations, which take the form of secondorder differential equations. This is my first lecture on solid state physics, and the script should therefore be considered. The rst is naturally associated with con guration space, extended by time, while the latter is the natural description for working in phase space. This proves that the hamiltons principle holds good even for nonconservative systems. The hamiltonian method ilarities between the hamiltonian and the energy, and then in section 15. Pdf treating some solid state problems with the dirac equation. Now we come to a very important concept of solid state physics which allows one to. Historically, it was an important reformulation of classical mechanics, which later contributed to the formulation of statistical mechanics. The interactions of spin systems in nmr have been extensively described by levitt 29, veshtort and griffin 30, and bak et al 31.
Individual chapters and problem sheets are available below. Review of energy dispersion relations in solids mit. Chapter 6 interaction of light and matter atomic or molecular gases in low concentration show sharp energy eigen spectra. Hamiltonian function a realvalued function hx,y is considered to be a conserved quantity for a system of ordinary di. State hamiltons principle for nonconservative system and hence derive lagranges equations of motion for nonconservative holonomic systems.
The above equations can be applied to exciton states, regardless of the dimension of materials. With a nonzero hamiltonian, the dynamics itself through the conserved hamiltonian showed that the appropriate parameter is path length. Pdf the ambiguity involved in the definition of effectivemass hamiltonians for nonrelativistic models is resolved using the dirac equation. From the hamiltonian equations to the schrodinger equation the case of an electron propagating in a crystal lattice 9. Hamiltonian matrices and the algebraic riccati equation. The hamiltonian formulation of mechanics describes a system in terms of generalised co motion of the system. To analyze the dynamics one should use classical hamilton equations. The total hamiltonian can then be written as the sum between. This is a second course in classical mechanics, given to final year undergraduates. Simon oxford university everything should be made as simple as possible, but no simpler. Problems forsolid state physics 3rdyearcourse6 hilary term2011 professor steven h. Treating some solid state problems with the dirac equation. Alagrangian function for which the legendre transformation is applicable is said to be. The aim of the book is to introduce a reader familiar to more classical techniques of ordinary differential equations to the powerful approach of modern critical point theory.
Pdf nonlinear quantum langevin equations for bosonic. In other words, if xt,yt is a solution of the system then hxt,yt is constant for all time which also. For the love of physics walter lewin may 16, 2011 duration. We consider a hamiltonian h of electrons invariant under a discrete set of. This equation specifies the manyparticle hamiltonian of our standard model of. The lagrange equations 3 are therefore equivalent to 6. Complex variables in classical hamiltonian mechanics. Hamiltonian mechanics december 5, 2012 1 phase space phase space is a dynamical arena for classical mechanics in which the number of independent dynamical. A proof that the optical field is inserted into the hamiltonian in the form p p. Critical point theory and hamiltonian systems springerlink. Hamiltonian mechanics is an equivalent but more abstract reformulation of classical mechanic theory. Now we come to a very important concept of solid state physics which allows one to treat electrons in a. This is the standard complete introduction to solid state physics. This proves that the hamilton s principle holds good even for nonconservative systems.
The hamiltonian is a function used to solve a problem of optimal control for a dynamical system. Complex variables in classical hamiltonian mechanics in the classical hamiltonian formalism, the state of a mechanical system is given by a set of pairs of conjugated variables, of variables fqj. Solid state physics part ii optical properties of solids mit. The rst is naturally associated with con guration space, extended by time, while the latter is. This is di erent than the rst example, since here we are picking coordinates rather than. Problems forsolid state physics 3rdyearcourse6 hilary. Hamiltonian systems table of contents 1 derivation from lagranges equation 1 2 energy conservation and. Its original prescription rested on two principles. First that we should try to express the state of the mechanical system using the minimum representation possible and which re ects the fact that the physics of the problem is coordinateinvariant. David tong cambridge lecture notes on classical dynamics. Having established that, i am bound to say that i have not been able to think of a problem in classical mechanics that i can solve more easily by hamiltonian methods than by newtonian or lagrangian methods. The main motivation to use hamiltonian mechanics instead of lagrangian mechanics comes from the symplectic structure of hamiltonian systems. The full set of lecture notes, weighing in at around pages, can be downloaded here.
It can be understood as an instantaneous increment of the lagrangian expression of the problem that is to be optimized over a certain time horizon. Historically, it contributed to the formulation of statistical mechanics and quantum mechanics hamiltonian mechanics was first formulated by william rowan hamilton in 1833, starting from lagrangian mechanics, a previous reformulation of classical mechanics introduced by joseph louis lagrange. Properties of bloch functions outline momentum and crystal momentum k. It can scarcely be denied that the supreme goal of all theory is to make the irreducible basic.
Lagrange equations consist of a set of k secondorder differential equations describing the variables qk being the time derivatives of the other k variables qk. Chapter 2 lagranges and hamiltons equations in this chapter, we consider two reformulations of newtonian mechanics, the lagrangian and the hamiltonian formalism. The crystal orbital hamilton population cohp method as a. Lecture notes for solid state physics rudolf peierls centre.
The kinetic and potential energies of the system are written and, where is the displacement, the mass, and. This is done in the following equations for both quantum wells and quantum dots. It uses a different mathematical formalism, providing a more abstract understanding of the theory. Inparticular, we want to know the con ditions under which the legendre transformation can be used. Then, the manybody hamiltonian in a solid crystal contains the kinetic terms of.
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