Question

8.4 The Two-Dimensional Central-Force Problem The 2D harmonic oscillator is a 2D central force problem (as discussed in TZD Many physical systems involve a particle that moves under the influence of a central force; that is, a force that always points exactly toward, or away from, a force center O. In classical mechanics a famous example of a central force is the force of the sun on a planet. In atomic physics the most obvious example is the hydrogen atom, in which the electron is held to the proton by the central Coulomb force. Other examples where the force is at least approximately cen- tral include the motion of any one electron in a multielectron atom, and the mo- tion of either atom as it orbits around the other atom in a diatomic molecule. section 8.4) with the potential where M is the particle mass and wc is the classical resonant frequency, as for the 1D harmonic oscillator. An unusual feature of this problem is that it can be separated in both Cartesian and polar coordinates. If you separate it in Cartesian coordinates Central force If the force on a particle is central, t does no work when the particle moves in any direction perpendicular to the radius vector, as shown in Fig. 8.6. FIGURE 8.6 This means that the particle's potential energy U is constant in any such dis A central force points exactly placement. Thus U may depend on the particle's distance. r. from the force toward or away from O. If the center O, but not on its direction, Therefore, instead of writing the potentialt undergoes a displacement energy as U(x, y,z), we can write simply U(r) when the force is central. This perpendicular to the radius vector property of central forces will allow us to solve the Schrödinger equation using separation of variables y(x,y)-x(x)Y(y) you will find that the solutions for X and γ are just the wavefunctions of the 1D harmonic oscillator. This implies that the ground state is OP, the force does no work and the potential energy U is therefore As an introduction to the three-dimensional central-force problem, we consider first a two-dimensional particle moving in a central-force field. We will not present a complete solution of the Schrödinger equation for this system since is fairly complicated and we are not really interested in two- dimensional systems here. Nevertheless, we will carry it far enough to see two important facts: First, like the energies for a square box, the allowed energies of a two-dimensional particle in a central-force field are given by two quantumy numbers Second, we will find that one of the two quantum numbers is closely connected with the angular momentum of the particle. with energy Eoh The ground state energy is the sum of the energies of the X and Y oscillators, so it is twice the ground state energy of a 1D harmonic oscillator. The length b is the same as for the 1D oscillator Since the potential energy Udepends only on r, the distance of the parti- Wc cle from the force center O. it is natural to adopt r as one of our coordinates In two dimensions the simplest way to do this is to use polar coordinates (r. φ) as defined in Fig. 8.7. It is a simple trigonometric exercise to express x and y in FIGURE 8.7 terms of r and ф as in Fig. 8.7. or vice versa Problem 8.16). Note that r is Definition of the polar coordinates defined as the distance from O to the point of interest and is therefore always and d in two dimensions. positive:0 r < 00, If we increase the angle ф by 2" (one complete revolu- tion). we come back to our starting direction. Therefore, you must remember that φ = φ) and φ = φ@ + 2π represent exactly the same direction. Use these results to write the ground state wavefunction in the form and show that this wavefunction is a solution to (8.38) and (8.39), the equations separated in polar coordinates. (Hint:0(p) is a very simple function.) Be sure to use the correct values of m and E in (8.39). The wave function which depends on x and y, can just as well be expressed as a function of r and d: and the Schrödinger equation can similarly be rewritten in terms of r and ф. When one rewrites the partial derivatives of the Schröding terms of r and &. one finds that equation in If you have had some experience with handling partial derivatives, you should be able to verify this rather messy identity (Problem 8.19). Otherwise, it is probably simplest for now to accept it without proof. In any case is helpful to note that all three terms are dimensonally consistent. Given the identity (8.33), we can rewrite the Schrödinger equation (8.6) in terms of r and ф as (8.34) (8.38) (8.39)

Answer 1

wihn it in te polar co-ordinte AgScho linger esuatiox im pl gal n lar la-ordi ate o) Fom Rt) eqhation 2m (u- dd r d P Al- fr γ (2b 11-en@-(. 2rn 下(2. b 4 2mue Hevce our wave funchon o a solution to (i) and Hence our wave funchon (2) en mco and