Arranging Quanta Among One-Dimensional Oscillators

How many ways can 2 quanta be arranged among 4 one-dimensional oscillators?

a. 1100, 1010, 1001, 0110, 0101, 0011, 1000, 0100, 0010, 0001

b. 1100, 1010, 1001, 0110, 0101, 0011, 2200, 2020, 2002, 0220, 0202, 0022

c. 1100, 1010, 1001, 0110, 0101, 0011, 2000, 0200, 0020, 0002

d. 2000, 0200, 0020, 0002

e. 1100, 1010, 1001, 0110, 0101, 0011

f. 1100, 1010, 1001, 0110, 0101, 0011, 2000, 0200, 0020, 0002, 2200, 2020

Answer:

There are 12 ways to arrange 2 quanta among 4 one-dimensional oscillators: 1100, 1010, 1001, 0110, 0101, 0011, 1000, 0100, 0010, and 0001.

To determine the number of ways to arrange 2 quanta among 4 one-dimensional oscillators, we can use the concept of permutations. In this case, we want to find the number of permutations of 2 quanta among 4 oscillators, which can be calculated using the formula for permutations:

P(n, r) = n! / (n - r)!

Where n is the total number of objects and r is the number of objects to be arranged.

In this case, we have 4 oscillators and want to arrange 2 quanta, so n = 4 and r = 2. Plugging these values into the formula, we get:

P(4, 2) = 4! / (4 - 2)! = 4! / 2! = 4 * 3 * 2 * 1 / 2 * 1 = 12

Therefore, there are 12 ways to arrange 2 quanta among 4 one-dimensional oscillators. The explicit arrangements are: 1100, 1010, 1001, 0110, 0101, 0011, 1000, 0100, 0010, and 0001.