Landau Problem Revisited
نویسندگان
چکیده
The motion of an electron in a constant magnetic field is an interesting physical problem. Essentially this is a two-dimensional problem. Landau [l] solved the Schrodinger equation in a particular gauge, the now well-known Landau gauge. His famous solution is a plane wave in one direction and a harmonic oscillator wavefunction in the other direction. Fifty odd years later Laughlin [2] solved the same Schrodinger equation in the symmetrical gauge. This paved the way for the study of the fractional Quantum Hall effects [3]. In fact these two solutions are two different representations of the same problem. Laughlinís solution respects more the rotational symmetry of the problem. However, the symmetry involved is a little bit subtle since it is a symmetry in a magnetic field. The correct symmetry generators for translations as well as rotation are the sum of a mechanical part plus a field part which depends on the magnetic field B. They are thus constants of motion in any gauge. In [4] we are able to show that the hamiltonian can be written entirely in terms of these constants of motion in a very simple way. In this gauge invariant formulation we can employ the algebraic techniques of raising and lowering operators to solve the Landau problem completely. In this formulation the presence of degeneracy of the problem is quite obvious. In this paper we are going to solve the wavefunctions in our formulation [4]. To obtain the wavefunctions we have to choose a gauge. In the symmetric gauge we obtained results similar to those obtained by Laughlin [2]. However, our results are indeed a refinement of those presented by Laughlin. The wavefunctions for the higher Landau levels are better represented. We also worked out the coherent states for all the Landau levels. We find that the Landau wavefunctions are a linear combination of the coherent state wavefunctions in our formulation. The interrelations are fully worked out.
منابع مشابه
Investigation of Temperature Effect in Landau-Zener Avoided Crossing
Considering a temperature dependent two-level quantum system, we have numerically solved the Landau-Zener transition problem. The method includes the incorporation of temperature effect as a thermal noise added Schrödinger equation for the construction of the Hamiltonian. Here, the obtained results which describe the changes in the system including the quantum states and the transition pro...
متن کاملSOLVING FUZZY LINEAR PROGRAMMING PROBLEMS WITH LINEAR MEMBERSHIP FUNCTIONS-REVISITED
Recently, Gasimov and Yenilmez proposed an approach for solving two kinds of fuzzy linear programming (FLP) problems. Through the approach, each FLP problem is first defuzzified into an equivalent crisp problem which is non-linear and even non-convex. Then, the crisp problem is solved by the use of the modified subgradient method. In this paper we will have another look at the earlier defuzzifi...
متن کاملWeakly Nonlocal Irreversible Thermodynamics
Weakly nonlocal thermodynamic theories are critically revisited. The irreversible thermodynamic theory of nonlocal phenomena is given, based on a modified form of the entropy current. Several classical equations are derived , including Guyer-Krumhansl, Ginzburg-Landau and Cahn-Hilliard type equations.
متن کاملWeakly Nonlocal Irreversible Thermodynamics- the Ginzburg-landau Equation
The variational approach to weakly nonlocal thermodynamic theories is critically revisited in the light of modern nonequilibrium thermody-namics. The example of Ginzburg-Landau equation is investigated in detail.
متن کاملExact solutions of the 2D Ginzburg-Landau equation by the first integral method
The first integral method is an efficient method for obtaining exact solutions of some nonlinear partial differential equations. This method can be applied to non integrable equations as well as to integrable ones. In this paper, the first integral method is used to construct exact solutions of the 2D Ginzburg-Landau equation.
متن کامل