GATE 2026 PH – Question 20
A gas of non-interacting $^4$He atoms (of mass $m$) is in a three-dimensional trap whose energy levels can be approximated by those of a harmonic oscillator potential $V(x,y,z)=\frac12m\omega^2(x^2+y^2+z^2)$. The chemical potential of the gas at $T=0$ K is
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Correct answer: (C) $3\hbar\omega/2$
Explanation
The trap potential is a three-dimensional isotropic harmonic oscillator, with single-particle energies
$$\varepsilon_{n_x,n_y,n_z}=\hbar\omega\left(n_x+n_y+n_z+\tfrac32\right).$$
For non-interacting bosons ($^4$He atoms) at $T=0$ K, **all particles occupy the single-particle ground state**, $n_x=n_y=n_z=0$, with energy
$$\varepsilon_0=\frac32\hbar\omega .$$
The chemical potential is the energy needed to add one particle at $T=0$. The added boson also goes into the ground state, so
$$\mu=\varepsilon_0=\frac32\hbar\omega\quad(\text{option C}).$$