Basic Quantum Mechanics by Klaus Ziock

By Klaus Ziock

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17) This is the famous SchrOdinger equation of the hydrogen atom. We shall solve it later (Chapter 5) and find complete agreement between the experimental values for the energy levels of the hydrogen atom and the values calculated, using Eq. 17. 1. Assuming the proton to be stationary. 20 ~ATTER VVAVES Schrodinger's postulates are the key to a complete description of quantum phenomena: (a) If we can overcome the mathematical difficulties involved in solving the Schrodinger equation. (b) If we know the force law applicable to the situation.

58 approaches 2jk, it follows thatf(x) in Eq. 59 does not stay smaller than e x2 / 2 for large x. Fortunately, there is a way out of this dilemma. If in Eq. 62) and hence all following coefficients vanish. 63) This is a polynomial and therefore certainly smaller than ex2 / 2 for large values of x and finite n. 64) A = 2n + 1 and if, at the same time (depending on whether n is even or odd), either all the odd-numbered or all the even-numbered Ak vanish. We thus have two series of eigenfunctions with ascending values of the quantum number n: n = 0,2,4, ..

39 THE LINEAR HARMONIC OSCILLATOR In other words, a classical harmonic oscillator in one dimension can have any amplitude and, hence, any total energy but only one frequency. 41) Now we write the Schrodinger equation for the one-dimensional quantum mechanical harmonic ascillator. , we have expressed the potential energy in terms of the freq uency w that a classical harmonic oscillator with the same restoring force C would have. To simplify Eq. 44) For convenience, we shall drop the primes henceforth and write x instead, keeping in mind that our unit of length is now: ~ li/mOJ.

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