Please write this out the traditional paper and pen way, typed solutions really don't help and will result in a thumbs down.
2. Time evolution and operators
Consider a Hilbert space H = C with the standard inner product and the usual basis:
|1⟩
|2⟩
Suppose the Hamiltonian is:
H = hw
where h is the reduced Planck's constant and w is a real constant.
a) Compute the eigenvalues E and the associated normalized eigenstates |E⟩ [6 marks]
b) Suppose the system is prepared in the normalized state |0⟩ = |1⟩ - 3|2⟩. What is the state of the system at time t > 0? Note: You can express your answer on the basis {|E⟩} or on the basis {|1⟩, |2⟩, |3⟩}, whichever way you prefer. [6 marks]
c) Consider an operator A acting on this Hilbert space, such that:
A|1⟩ = 2|1⟩
A|2⟩ = 3|2⟩
Is A Hermitian? Explain carefully. Hint: Compare ⟨1|A|2⟩ with ⟨1|A†|2⟩. You can take duals of the above equations. Also, recall ⟨j|k⟩ = δjk for j, k = 1, 2, 3. [4 marks]
d) Assuming that the operator A from part c satisfies, in addition, A|3⟩ = |1⟩, find A in matrix form and then compute the commutator [AH]. [4 marks]
