By Professor Dr. Walter Greiner (auth.)
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Additional resources for Quantum Mechanics: An Introduction
Heisenberg's uncertainty principle is a consequence of the wave character of the particles (more exactly: of the guiding field of the particles). Using the superposition principle, the probability field is a wave packet superposed of waves with a definite momentum (plane waves). The particle guided by this wave packet can be found with a high probability within L\x. It is said to be localized in L\x. e. a momentum packet of width nL\k, is required. In classical physics, uncertainty relations of a similar form appear in processes involving waves.
6). 57) will also be a solution because of the linearity of the equation. EXAMPLE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Super position of Plane Waves, Momentum Probability with The representation of a wave field tP(r, t) by superposition of de Broglie waves, (1) is an example of such a superposition. The normalization factor in (1) results from 7 tP;tPp,d3 r = g~oo N 2 j exp [-*(p - p'). -~":--(py - py)/n (pz - pz)f1i = N 2 (271i 8 ( p ;t') 1 h(x) = - 7r sin(gx) 1 .
Hm - - and h(ax) = a- h(x) g-+oo X The 8 functions play an important role in various mathematical treatments encountered in quantum mechanics; they will be discussed in greater detail in Chap. 4). e. JtPp,tP;d r = 8(p - p') 00 3 . -00 With this normalization we get 39 (2) The wave function for ail arbitrary state 'l/J(r, t) can be expanded into de Broglie waves (1) according to 1 = = V(27rn)3 c(p, t) exp ( -i E;t) Now using (2), it can easily be proved that 00 'l/J(r, t)
Quantum Mechanics: An Introduction by Professor Dr. Walter Greiner (auth.)