GATE 2023 PH – Question 64
Two identical, non-interacting 4He2 atoms are distributed among 4 different non- degenerate energy levels. The probability that they occupy different energy levels is p. Similarly, two 3He2 atoms are distributed among 4 different non-degenerate energy levels, and the probability that they occupy different levels is q. What is the value of p q (rounded off to one decimal place)?
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Correct answer: 0.59 to 0.61
Explanation
Two identical particles distributed over 4 non-degenerate energy levels.
**Bosons ($^4$He atoms).** Two bosons can share a level. The number of distinct states is the number of ways to choose 2 levels with repetition allowed:
$$\binom{4+2-1}{2}=\binom52=10 .$$
Of these, the states with both bosons in the **same** level number 4. So there are $10-4=6$ states with the bosons in **different** levels. Each state is equally likely:
$$p=\frac6{10}.$$
**Fermions ($^3$He atoms).** Two fermions cannot occupy the same level. The allowed states are the $\binom42=6$ choices of two different levels, and in every one of them the particles are in different levels:
$$q=\frac66=1 .$$
$$\frac pq=\frac{6/10}{1}=\mathbf{0.6}.$$