2. (5 points) Suppose we have a bid deck of cards consisting only of: 1 2 3 1 2 3 3 1200 500 200 400 3 The 1200, 500, 200, and 400 refer to how many of each card are in the deck. (a) We start to take one card out at a time and we read its numerical value. What are the eigenvalues of this system? (b) What is the average value of the numbers for the cards in the full deck? (c) Will the "operator" that reads each number ever measure the average value? What values will the card-reading operator always obtain? Explain. (d) Suppose there is another operator that can read whether the card is a heart or a club. Does the heart/club operator commute with the number-reading operator? Why or why not? (e) We want to use symbols to write what we found and express it quantum mechanically. Write symbols for the wave functions of each card. Also make up a symbol for each operator. Show how the mathematics of these operators would work. (f) Can we determine definitively whether the number-reading operator is Hermitian? Why or why not? (g) Let's assume that the wave functions of the number operator are orthogonal and the number operator is Hermitian, so that the wave functions and the number operator do correspond to a proper quantum mechanical variable. Take a superposition of the 4 cards and show what the coefficient values cn will be in the superposition to give the correct number of cards in the deck when the superposition wave function is squared.