4. We've talked a good bit about quantum mice, but we're not done yet. Let's add time dependence to the discussion. We'll work in the size basis, where we have the operators and eigenstates W = (10), = 1 ( 33), 14) = - A (s) = (66). 1/35 (-2), Recall that W is the size operator, with eigenvalues, 2 and 10, corresponding to small and large mice, and A is the attitude operator, with eigenvalues, +1 and -1, corresponding to happy and unhappy mice. To have time dependence, we need a Hamiltonian, which in the size basis we will take to be (69). H = Eo where Eo is a constant with the units of energy. (a) Does H commute with either W or A? How do you know? (c) At time t = 0, you have a small mouse. (b) Find the energy eigenvalues and eigenstates in the size basis. How do these relate to the eigenstates of A and W? (2)= = (1), (u) = (). Find the state of the mouse at a later time t. What is the probability that you will have a small mouse at a later time t? Can you say with certainty what the energy of a small mouse is? (d) At time t = 0, you have a large mouse. Find the state of the mouse at a later time t. What is the probability that you will have a large mouse at a later time t? Can you say with certainty what the energy of a large mouse is? Can we say with certainty that small mice have more or less energy than large mice? Why or why not? (e) At time t= 0 you have a happy mouse. . Find the state of the mouse at a later time t. What is the probability that you will have a happy mouse at a later time t? Is there any later time t at which the mouse will be guaranteed to be unhappy? (f) At time t = 0) you have an unhappy mouse. Find the state of the mouse at a later time t. What is the probability that you will have an unhappy mouse at a later time t? Is there any later time t at which the mouse will be guaranteed to be happy? Can we say with certainty that happy mice have more or less energy than unhappy mice? Why or why not?