Practice Problem Set 2 Solutions

1. To answer this question, work out the commutator.

   

   Therefore, and K are compatible observables.

2. For simplicity, write the exponential in the form

   

   In this notation, the kinetic energy operator is

   

   Now note that

   

   Therefore

   

   is therefore a kinetic energy eigenfunction with eigenvalue .

3. Using the given data, we calculate

   

   The second part of the problem asks us to solve an inverse problem: Given a target probability, how close can the ``wires'' be?

   

   It should shortly be possible to manufacture circuits whose traces are this far apart. Such devices will not be classical digital computers but will have to account for quantum effects.