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A Fermi gas of only two particles, assume that each particle can be in one of three possible quanta states, s = 1, 2, 3. The possible number of states of the whole gas is?
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Possible Number of States in a Fermi Gas of Two Particles


Introduction

In a Fermi gas, each particle obeys the Pauli exclusion principle, which states that no two identical fermions can occupy the same quantum state simultaneously. The total number of possible states for a system of two particles can be determined by considering the number of possible combinations for each particle.

Number of States for Each Particle

Since each particle can be in one of three possible quanta states (s = 1, 2, 3), the number of states for each particle is 3.

Combinations of States for Two Particles

To determine the number of possible combinations of states for the whole gas, we need to consider the possible combinations of states for the two particles.

Since the particles are indistinguishable, we cannot simply multiply the number of states for each particle. Instead, we need to consider all possible combinations while avoiding double counting.

Combination 1: Particle 1 in State s1 and Particle 2 in State s2

In this combination, there are 3 possibilities for Particle 1 (s1 = 1, 2, 3) and 2 possibilities for Particle 2 (s2 = 1, 2, 3 excluding the state occupied by Particle 1). Therefore, there are 3 x 2 = 6 possible combinations for this case.

Combination 2: Particle 1 in State s2 and Particle 2 in State s1

Since the particles are indistinguishable, this combination is equivalent to Combination 1. Therefore, there are also 6 possible combinations for this case.

Combination 3: Particle 1 in State s1 and Particle 2 in State s3

In this combination, there are 3 possibilities for Particle 1 (s1 = 1, 2, 3) and 2 possibilities for Particle 2 (s3 = 1, 2, 3 excluding the state occupied by Particle 1). Therefore, there are 3 x 2 = 6 possible combinations for this case.

Combination 4: Particle 1 in State s3 and Particle 2 in State s1

Since the particles are indistinguishable, this combination is equivalent to Combination 3. Therefore, there are also 6 possible combinations for this case.

Combination 5: Particle 1 in State s2 and Particle 2 in State s3

In this combination, there are 3 possibilities for Particle 1 (s2 = 1, 2, 3) and 2 possibilities for Particle 2 (s3 = 1, 2, 3 excluding the state occupied by Particle 1). Therefore, there are 3 x 2 = 6 possible combinations for this case.

Combination 6: Particle 1 in State s3 and Particle 2 in State s2

Since the particles are indistinguishable, this combination is equivalent to Combination 5. Therefore, there are also 6 possible combinations for this case.

Total Number of Possible States

To calculate the total
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A Fermi gas of only two particles, assume that each particle can be in one of three possible quanta states, s = 1, 2, 3. The possible number of states of the whole gas is?
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