Reading Quiz

Question 1:

Suppose you flip two coins. What is the multiplicity of the macrostate defined by getting one head and one tail? What is the probability of arriving at this macrostate?

Answer:

The multiplicity = 2; (Coin 1 = H; Coin 2 = T) or (Coin 1 = T; Coin 2 = H).
Since there are four distinct macrostates to describe the toss of two coins,
the probability of the (HT) macrostate = multiplicity of HT/total multiplicity = 2/4 = 1/2.
  1. Macrostate(1H 1T)=2 50%
  2. the multiplicity is 2, so the probability is 50%
  3. Multiplicity is 2. Probability is 1/2.
  5. The multiplicity is 2. The probability of arriving at this macrostate is 50%.
  6. 2 - 50%
  7. Two. One-half.
  8. The multiplicity is 2 and the probability is 1/2.
  9. The multiplicity is two. The probability is two over how many total macrostates are possible, which is three.

Question 2:

Consider flipping 10 coins. What is the multiplicity of the macrostate with 9 heads?

Answer:

The multiplicity = 10. There are 10 different ways (coins) that could turn up tails while the remaining 9 are heads.
  1. 10
  2. 10
  3. Multiplicity is 10
  5. The multiplicity is 10.
  6. 10 choose 9 or 10
  7. 10
  8. The multiplicity is 10.
  9. 10 factorial/((9 factorial)*(10-9)factorial)=10

Question 3:

Consider flipping 10 coins. What is the multiplicity of the macrostate with 8 heads?

Answer:

The multiplicity = (10 x 9)/2 = 45. Note that 8 heads is the same as getting 2 tails. There are 10 ways to get the first tail, and 9 possible coins that can yield the second tail, but we do not want to double count so we divide by 2.
  1. 45
  2. 45
  3. Multiplicity is 45
  5. The multiplicity is 45.
  6. 10 choose 8 or 45
  7. 45
  8. The multiplicity is 45.
  9. 45

Question 4:

What is the numerical value of ``5 choose 3''?

Answer:

5!/[3! x (5-3)!] = 5!/(3! x 2!) = 120/(6 x 2) = 10.
  1. 10
  2. 10
  3. 10
  5. 10.
  6. 10
  7. 10
  8. 10.
  9. 10

Question 5:

Suppose you flip a few hundred coins, and want to use the result as an analogy for a two-state paramagnet. How would you determine my system's energy by looking at the coins?

Answer:

The heads would be analogous to a spin aligned with the B field and tails a spin anti-aligned (or vice versa). So the ``energy'' would be determined by the difference between the number of heads and the number of tails.
  1. The face of the coin is either Heads or Tails representing either Up or Down magnetic moment. The energy of the system is then determined by the total number of up and down dipoles.
  2. in the two state paramagnet, it is assumed that the magnets are not affected by their neighbors, only the applied magnetic field, so a random sample would represent the group as a whole.
  3. you would determine which macrostate the system is in- up or down
  5. The total energy of the system is determined by the total number of up and down dipoles for a two state paramagnet. So by looking at the total number of heads and tails for all the coins flipped will tell us the system's energy. All we have to do is say what heads and tails stand for in terms of an up or down dipole and then knowing the energy for each dipole we can get the total energy.
  6. number up minus number down.
  7. Since the energy is dependent upon the direction of the field, you would want to determine the net field...thus take the multiplicity of up states minus the multiplicity of down states...the absolute value of this should yield an overall value proportional to the total energy.
  8. I would associate heads with an up-state and tails with a down-state, and then determine the multiplicity of the system as the multiplicity tells the energy of the system.
  9. The total energy of the system is the total number of heads up or down combinations, which then specifies which macrostate the system is in.

Question 6:

Was there anything you had particular difficulty with in this reading? Is so, describe briefly.
  • not really
  • No, probability is not too hard and so for now the concepts are not too difficult.
  • no i don't think so.
  • No, I find it to be very straightforward.