By Klaus Ziock

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For mathematical convenience, we assume a discontinuous transition from V = 0 to V = Vo at x = O. 21) The general solution of Eq. 20 is with 2 The exact agreement is somewhat fortuitous and results from the fact that we have used = h in the uncertainty relation (Eq. 9) instead of, as is more usual, Ax Ap", = Ii. Ap", . 2 A classical particle (Fig. 2a) cannot overcome a potential barrier that exceeds its total energy; however, quantum mechanics allows a particle to appear in places that are strictly forbidden to it by classical mechanics (Fig.

8 shows the energy levels and the probability densities of the first eleven states in relation to the harmonic oscillator potential: V = Cx 2 j2. The quantum-mechanical harmonic oscillator is a reasonable approximation of many physical systems. As an example, we mention diatomic molecules. In a diatomic molecule the two atoms can vibrate around their equilibrium position. For vibrations of small amplitude, the restoring force is very nearly proportional to the amplitude and the molecule resembles a harmonic oscillator rather closely.

15) for enlightenment. 15 is a partial differential equation, and we try to solve it by writing the wave function as a product of a function u(r) that depends only on r and another function pet) that depends only on t. 20) Since the left side of Eq. 20 does not depend on r, and the right side does not depend on t, both sides must be equal to the same constant6 , say E. The gravitational force is also well known but is so weak that its manifestations have never been observed on an atomic scale. 6 The mathematical technique we have used here is called the separation of the variables.