Homework due February 11.
Exercise 1 Consider the Hamiltonian on the phase space , where is bounded from above: for all for some . Suppose the initial momentum and that the initial position is such that .
- Show that the position satisfies
- Suppose with . Find the trajectory provided and .
Exercise 2 Let be a subspace.
- Given prove that there is a unique element such that
- Prove that is a linear map and that
for all .
- Prove that
is a subspace and that
where is the identity map.
- Prove that .
Exercise 3 The Pauli spin matrices are
- Prove the commutation relations
- Let be a unit vector, . Let denote the observable
Show that the spectrum of is and find its eigenvectors.
- Compute the commutator
for distinct unit vectors .
- Consider the state with density matrix . Compute the expectation and variance of in this state for any .