By Klaus Ziock
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Extra info for Basic Quantum Mechanics
Example text
Before we set out to solve the Schrodinger equation for this problem, we try to find out as much as we can about the general character of its solutions. 84) The fact that the shape of the potential is the same in all cells of our onedimensional crystal lattice does not mean that the eigenfunctions have to be Fig. 9 52 SoME SIMPLE PROBLEMS periodic with the lattice period. True, the Schrodinger equation remains unchanged if we replace V(x) with V(x + d), but it leaves a constant factor in its eigenfunctions undetermined.
Oo? ,2 dx Since the integrand changes sign as we go from x to -x the integral must vanish between symmetrical limits. , the two functions are not orthogonal over the interval meaningless to say that two functions are orthogonal unless we specify the interval, area, or volume in which they are orthogonal. o < x < 00. It is thus 3 SOME SIMPLE PROBLEMS At this point only some mathematics stands between us and our intermediate goal: the calculation of the parameters of the hydrogen atom. A look at the Schrodinger equation in the form in which we shall later solve it (Eq.
2 0.. 0 Displacement from equilibrium position (h) Fig. 6 (concluded) Fig. 1. 48 SOME SIMPLE PROBLEMS To sum it up, the wave function of the three-dimensional harmonic oscillator is the product of the wave functions of three one-dimensional harmonic oscillators in the x, y, and z direction. The total energy is the sum of the energies of the three one-dimensional oscillators. This results in a very complicated behavior since a solution for the x-coordinate with, for example, n = 2 can combine with any solution for the y and z coordinate.