Two-dimensional oscillator in a magnetic field
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TwoDimensional Oscillator in a Magnetic Field T. K. Rebane Fock Research Institute of Physics, St. Petersburg University, St. Petersburg, 198504 Russia email: [email protected] Received May 31, 2011
Abstract—The energy and eigenstate spectrum of a charged particle in the electric field of a 2D anisotropic oscillator and in a uniform magnetic field is considered. The exact analytic solution to the problem is obtained for an arbitrary magnetic field strength. The characteristic features of variation of the energy spec trum depending on the magnetic field strength are analyzed. The results of this study are of interest for the quantummechanical theory of magnetism and can be used to simulate the magnetic properties of atoms and molecules. DOI: 10.1134/S1063776112010153
1. INTRODUCTION
consider its motion in the xy plane. The 2D motion of the particle is determined by the energy operator
The model of a harmonic oscillator, in which par ticles interact via elastic forces, has found wide appli cation in various quantum mechanics problems (from the theory of oscillations of molecules to the theory of artificial atoms and molecules of the “harmonium” type [1]. This is due to the fact that the exact solution to the Schrödinger equation can be easily obtained for a harmonic potential, which is also observed for sys tems in a uniform electric field owing to the possibility of separating variables. Such a separation of variables in a magnetic field is possible only in the presence of axial symmetry, and instead of an exact solution to the Schrödinger equation, one has to be content with approximate solutions. This paper will show that an anisotropic harmonic oscillator is an exception to this rule, because the problem of its energy spectrum and eigenfunctions in a uniform magnetic field of any strength has a simple and exact analytic solution despite the absence of axial symmetry. Our approach is based on employment of shift operators of energy eigenvalues and on the method of varying the vector potential [2].
2
2
2 2
2 2
H = m [ ( v x + v y + ω 1 x + ω 2 y ) ]/2,
(1)
where vx and vy are the components of the particle velocity operator in the magnetic field: iប ∂ – qA , v x = – x m ∂x cm iប ∂ q v y = – – A y . m ∂y cm
(2)
Vector potential A of the uniform magnetic field satis fies the condition curl A = Ᏼez. We choose the vector potential in the form ω 1 xe y – ω 2 ye x A ( x, y ) = Ᏼ . 2 ( ω1 + ω2 )
(3)
The lines of vector A(x, y) (3) are similar ellipses because vector A(x, y) at any point (x, y) is directed along the tangent to the ellipse: 2
2
ω 1 x + ω 2 y = const. 2. ENERGY OPERATOR FOR A 2D OSCILLATOR IN A MAGNETIC FIELD
This “elliptic” vector potential is related to the stan dard vector potential with circular vector lines A0(x, y) by the gradient transformation
Let us consider the motion of a spinless particle of mass m and charge q in the field of a 2D anisotropic 2
A ( x, y ) = A 0 ( x, y ) + ∇F ( x, y ),
(4)
2
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