1Understanding questions C Give an example of what is possib

1.)Understanding questions

C) Give an example of what is possible with Hei senberg ’ s uncertainty principle that would not be classically allowed.

2.)A property of the boundary conditions for wave functions specifies that must be coninuous in order to avoid discontinuous probability values. Why can’t we have such probabilities?

Solution

a) Zero point energy

The wave function must be continuous everywhere. so that  there are no sudden jumps in the probability density.If a function has a discontinuity such as a sharp step, this can be seen as a limiting case of a very rapid change in the function. Such a rapid change would mean that the derivative of the function was very large (either a very large positive or negative number). In the limit of a step function, this would imply an infinite derivative. Since the momentum operator is a derivator, this would lead to infinite momentum which would lead to a physically unrealistic system. Also, a discontinous first derivation would lead to an infinite second derivation leading to infinite energy which also not realistic.