Three-Dimensional Potential Well

Lyudvig Dmitrievich Faddeev, O. Yakubovskiĭ

Zhao Su-qin

The Approximate Solution and the Exact Solution of three dimensional harmonic oscillator in Potential V=λxy is derived.The energy level and degeneracy of three dimensional harmonic oscillatorin in Potential V=λxy is discussed.

Carlo Jacoboni

Tapan Kumar Das

Vladimir Mitin, Dmitry I. Sementsov, Nizami Z. Vagidov

In the previous chapter we have analyzed the peculiarities of quantized electron motion in layered structures with one-dimensional potential wells. From the mathematical point of view, the solution of the Schrödinger equation for one-dimensional potential profiles is much simpler. However, many quantum objects, such as atoms, molecules, and quantum dots, are three-dimensional objects. Thus, in order to analyze electron motion in such objects we need to find solutions of the Schrödinger equation for three-dimensional potential profiles. The electron motion in spaces with dimensionality higher than one, especially for rectangular potential profiles with infinite potential barriers, is not so difficult to analyze. At the same time we have to keep in mind that such potential profiles frequently represent some approximation of the more complex, real potential profiles. Depending on the type of structure and on the form of the potential profile, the electron motion may be limited in two directions (two-dimensional quantization) or in three directions (three-dimensional quantization). In this chapter we will show that the existence of discrete energy levels in the electron spectrum is an intrinsic feature of electron motion in potential wells of any form and dimensionality. An electron in a rectangular potential well (quantum box) In the previous chapter we studied the electron motion in one-dimensional potential wells. An electron's motion along one direction was confined by the potential profile and the momentum in this direction was quantized.

Zhao Su-qin

The approximate solution and the exact solution of three dimensional harmonic oscillator in potential H′=(λμω_0~2/2)(x~2+y~2+z~2) is derived. The energy level and degeneracy of three dimensional harmonic oscillatorin in potential H′=(λμω_0~2/2)(x~2+y~2+z~2) is discussed.