Question

Two identical, non-interacting $^4He_2$ atoms are distributed among 4 different non-degenerate energy levels. The probability that they occupy different energy levels is $p$. Similarly, two $^3He_2$ 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 $\dfrac{p{q}$ (rounded off to one decimal place)?}

• A. 0.6
• B. 0.5
• C. 0.7
• D. 0.8

Hint:

When dealing with indistinguishable particles in quantum mechanics, always take into account the quantum statistics governing the distribution of particles.

Answer 1

The Correct Option is A

For the case of $^4He_2$ atoms, the number of ways the atoms can occupy different energy levels is given by the number of ways to assign two atoms to two distinct levels, which is $\binom{4}{2} = 6$. For $^3He_2$ atoms, the number of ways the atoms can occupy different energy levels is similar, but there is an additional restriction due to the quantum nature of the system. After solving for the probabilities $p$ and $q$, we find that: \[ \frac{p}{q} = 0.6. \] Thus, the correct answer is (A) 0.6.