By J. M. Cassels FRS (auth.)
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Extra info for Basic quantum mechanics
In either situation the unwanted terms disappear. Sometimes it is not convenient to consider a region of space sufficiently large to embrace the entire wave function. For example, t Sometimes, though not in this book, a complex and non-Hermitian Vis used as a calculational device, to describe the disappearance of particles into states that the physicist does not want to discuss. Only a partial picture of the situation can be gained from such calculations. 20 BASIC QUANTUM MECHANICS a beam of particles may be accelerated (x very negative), scattered by a field of force (x = 0), and finally detected (x very positive).
The product (~x )t ~p, which at t = 0 had the minimum value allowed by the uncertainty principle, increases without limit as time goes on. 5. 2)) plays a central role when fi does not mention t. A qualitative discussion of its solutions is therefore appropriate now. 1) where This linear and homogeneous equation is of the second order, which means that it involves the second but no higher derivative of u with respect to x. Consequently it must have two linearly independent solutions before boundary conditions are taken into account.
P, and yields unity when integrated over all p'. Time dependence of wave packets Some important facts about the development with time of the wave packet can be seen without examining the wave function in detail. The momentum is conserved, since p is an operator which commutes with H and does not depend tIn shorthand, _[ eilk'-klx dx = 21T8(k- k') = 21T8(k'- k). ONE-DIMENSIONAL APPLICATIONS 45 on t. Therefore, d - dt d (p)=- (p2)=0 dt and (p ), (p 2), and hence ~p preserve at all times the values already computed for t = 0.