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 M a) M) M) M: :1). () Mrs) Recall that W is the size operator. with eigenvalues, 2 and Ill, 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 . 4 u H=E\"(0 1)' where E\" is a constant with the units of energy. [a] Does H commute with either W or A? How do you know? (b) Find the energy eigenvalues and eigenstates in the size basis. How do these relate to the eigenstates of A and ii"? (c) At time I = (i. you have a small mouse. 0 Find the state of the mouse at a later time f. o What is the probability that you will have a small mouse at a later time t? 0 Can you say with certainty what the energy of a small mouse is? (d) At time t = (L you have a large mouse. 0 Find the state of the mouse at a later time t. o What is the probability that you will have a large mouse at a later time t? 0 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 i : (i you have a happy mouse. c Find the state of the mouse at a later time i. o What is the probability that you will have a happy mouse at a later time i? o Is there any later time t at which the mouse will be guaranteed to be unhappy? {f} At time l'. : (i you have an unhappy mouse. 0 Find the state of the mouse at a later time t. o What is the probability that you will have an unhappy mouse at a later time t? o 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