3.14.1 Suppose we have a particle in a wavepacket, where the spatial wavefunction at some time t is (r) = A(r) exp (ik ·r). Here, A(r) is a function that varies very slowly in space compared to the function exp (ik ·r), describing the envelope of the wavepacket.

(i)   Given that the particle current density is given by  show that  where p is the (vector) expectation value of the momentum.

(ii)  With similar approximations, evaluate the expectation value of the energy on the assumption that the potential energy is constant in space.

(iii) Hence, show that the velocity of the probability density corresponds to the velocity we would expect classically.

For each of the following English statements, (i) translate it into predicate logic, (ii) write the negation of that statement in predicate logic with the negation symbols pushed as far in as possible so that any negation symbols are directly in front of a predicate, and then (iii) translate the result of (ii) back to (natural) English.

For the logic, let the domain of discourse be cats and food. Use only the predicates Loves(x, y) and Likes(x, y) which say that cat x loves or likes (respectively) food y; the predicates Cat(x) and Food(x), which say whether x is a cat or food (respectively); and the predicates x = y and x 6= y, which say whether x and y are the same object. (Use the constant “Garfield” to refer to Garfield in your logical statements.)

(a) If a cat loves fish and lasagna, then it’s Garfield.

(b) There is some food that Garfield likes but no other cat likes.

(c) Every cat likes some food that Garfield does not like.