By Klaus Ziock
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Extra resources for Basic Quantum Mechanics
1 2(kj2 + 1) 1 k ~- Since the same ratio for Eq. 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.
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. For more than two particles, the mathematical difficulties are often considerable, just as in the case of the classical many-body problem. There exist, however, very powerful approximative methods to deal with more complicated problems.
A square wave is not a solution of the Schrodinger equation. 3 one can show, however, that the following theorem holds true. THEOREM The smaller the interval x ± ~x is over wnfch'aju~ction 1p(x) differs substantially from zero, the larger is the interval k over which its Fourier amplitude cp(k) differs substantially from zero. This theorem whose proof can be found in the literaturel l applies to any kind of function and thereby also to solutions of the Schrodinger equation. Relying on this theorem, we can now state with confidence that the momentum spectrum of a well localized particle extends to very high momenta.
Basic Quantum Mechanics by Klaus Ziock