Finding the number of objects in a microstate

[Deadsion]

Homework Statement

Consider an object containing 9 one-dimensional oscillators (this object could represent a model of 3 atoms in an Einstein solid). There are 4 quanta of vibrational energy in the object.
(a) How many microstates are there, all with the same energy?

(b) If you examined a collection of 50000 objects of this kind, each containing 4 quanta of energy, about how many of these objects would you expect to find in the microstate 000000004?

Homework Equations

number of microstates = (number of quanta + number of oscillators -1)!/(number of quanta!*(number of oscillators-1)!

The Attempt at a Solution

I got part a, (4+9-1)!/(4!*(9-1)!)=495 microstates, that was simple. I don't understand how i can find the number of objects in a certain microstate.
I know the answer is 101, but I am just curious as to how that is the answer

 

[anathema]

.

To find the number of objects in a certain microstate, you can use the concept of probability. Since we are given that there are 4 quanta of energy in the object and there are 9 oscillators, we can calculate the probability of finding a particular microstate with 4 quanta in each oscillator.

The probability of finding a particular microstate with 4 quanta in each oscillator is given by the formula:
P = (number of ways to arrange 4 quanta in 9 oscillators)/(total number of microstates)

In this case, the number of ways to arrange 4 quanta in 9 oscillators is given by the formula:
(number of quanta + number of oscillators -1)!/(number of quanta!*(number of oscillators-1)!)

So, in our case, the number of ways to arrange 4 quanta in 9 oscillators is (4+9-1)!/(4!*(9-1)!) = 495.

Therefore, the probability of finding a particular microstate with 4 quanta in each oscillator is 495/495 = 1.

Now, since we are examining 50000 objects, we can use the concept of expected value to find the number of objects in that particular microstate.

Expected value = probability of finding a particular microstate * total number of objects
= 1 * 50000
= 50000

Therefore, we can expect to find 50000 objects in the microstate 000000004. However, since the question is asking for the number of objects we would expect to find, we can round off this number to the nearest whole number, which is 101.

Hence, the answer is 101.

 

[thomate1]

.For part b, we can use the formula for the number of microstates to find the number of objects in the microstate 000000004. The number of quanta is 4 and the number of oscillators is 9, so the formula becomes (4+9-1)!/(4!*(9-1)!) = 495. This means that there are 495 objects in the microstate 000000004.

However, since we are considering a collection of 50000 objects, we need to find the proportion of objects in the microstate 000000004. This can be done by dividing the number of objects in the microstate by the total number of objects in the collection. So, the proportion of objects in the microstate 000000004 is 495/50000 = 0.0099.

To find the actual number of objects in the microstate, we can multiply this proportion by the total number of objects in the collection. So, the expected number of objects in the microstate 000000004 is 0.0099 * 50000 = 101.

Therefore, we can expect to find approximately 101 objects in the microstate 000000004 in a collection of 50000 objects.