Homework due February 11.

A PDF version of these exercises is available here.

Exercise 1Consider 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 satisfieswhere .

Suppose with . Find the trajectory provided and .

Exercise 2Let be a subspace.

Given prove that there is a unique element such thatProve that is a linear map and thatfor all .

Prove thatis a subspace and that

where is the identity map.

Prove that .

Exercise 3ThePauli spin matricesare

Prove the commutation relationsLet be a unit vector, . Let denote the observableShow that the spectrum of is and find its eigenvectors.

Compute the commutatorfor distinct unit vectors .

Consider the state with density matrix . Compute the expectation and variance of in this state for any .

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