Questions: The Ideal Bose Gas and Critical Temperature

Question 1

Below the critical temperature T_c in a Bose-Einstein condensate, where are the condensed particles located?

• A. They cluster together in the center of the container, forming a spatially localized dense droplet
• B. They settle to the lowest spatial points in the container under gravity, like a classical liquid
• C. They occupy the zero-momentum quantum ground state but are delocalized throughout the entire volume of the gas
• D. They become distinguishable classical particles and arrange themselves in a crystal lattice

Answer: (C)

Bose-Einstein condensation is not spatial localization—it is occupation of a single quantum state, specifically the zero-momentum ground state. A particle in a zero-momentum state has a de Broglie wavelength equal to the system size, meaning it is spread throughout the entire volume. The condensate is a macroscopic quantum state that fills the container. This is the central misconception the topic warns against: confusing BEC with ordinary condensation (vapor to liquid), where molecules do cluster spatially. In BEC, 'condensation' refers to condensation in momentum space, not real space.

Question 2

In an ideal Bose gas below T_c, what happens to the chemical potential μ as temperature is lowered further below T_c?

• A. μ decreases (becomes more negative) as fewer excited states are thermally accessible at lower temperatures
• B. μ remains pinned at zero for all T < T_c, because the macroscopically occupied ground state acts as an infinite particle reservoir
• C. μ increases above zero once condensation begins, reflecting the increased binding energy of the condensate
• D. μ oscillates around zero as particles redistribute between the condensate and excited states

Answer: (B)

Below T_c, μ is pinned exactly at zero (the ground state energy). Here is why: the ground-state occupation is ⟨n₀⟩ = 1/[exp(−μ/kT) − 1]. For this to accommodate a macroscopic number of particles without diverging to infinity, μ must equal exactly zero. If T decreases further below T_c, more particles spill into the ground state, but μ stays at zero—the ground state absorbs the excess. This pinning is the thermodynamic signature of condensation, and it causes the distinct kink in heat capacity at T_c and explains why C_V has a different functional form above and below the transition.

Question 3

Bose-Einstein condensation in an ideal gas is a purely quantum statistical phase transition driven by quantum indistinguishability, occurring without any interparticle interactions.

• T. True
• F. False

Answer: True

Correct. The ideal Bose gas model has no interparticle interactions—particles are non-interacting point bosons. BEC emerges purely from quantum statistics: bosons can occupy the same state without restriction, so when the thermal de Broglie wavelength becomes comparable to the interparticle spacing (quantum wave packets overlap), the ground state becomes macroscopically occupied. No attractive force or intermolecular potential is required. This is what makes BEC fundamentally different from ordinary phase transitions like condensation of water vapor, which depends on intermolecular attractions. The effect is purely statistical.

Question 4

Bose-Einstein condensation is essentially the same physical process as the condensation of water vapor into liquid water, just occurring at very low temperatures.

• T. True
• F. False

Answer: False

False. Ordinary condensation (vapor → liquid) is a first-order phase transition driven by intermolecular attractions: molecules with sufficient kinetic energy escape the liquid surface, while sufficiently attractive interactions cause vapor molecules to cluster and form a denser liquid phase. BEC is a second-order phase transition (continuous, with no latent heat) that occurs in an ideal gas with no interactions, driven purely by quantum statistics—specifically, the bosonic property that particles can accumulate without limit in a single quantum state when thermal wavelengths become comparable to interparticle spacing. The only common feature is the word 'condensation'; the underlying physics is entirely different.

Question 5

What physical condition determines the critical temperature T_c for Bose-Einstein condensation, and why does condensation occur when this condition is met?

Model answer: T_c is reached when the thermal de Broglie wavelength λ = h/√(2πmkT) becomes comparable to the mean interparticle spacing (n^{−1/3}), i.e., when nλ³ ≈ ζ(3/2) ≈ 2.612. At this point, quantum wave packets of neighboring particles begin to overlap significantly. Below T_c, the excited states can no longer accommodate all N particles—the maximum number of particles in excited states is N(T/T_c)^{3/2}, which is less than N—so the excess spills into the single zero-momentum ground state, which then becomes macroscopically occupied. Condensation occurs because quantum indistinguishability means bosons 'prefer' already-occupied states, and the ground state, once slightly favored, attracts an unlimited number.

The condition nλ³ ~ 1 has an intuitive interpretation: it is when quantum mechanics becomes unavoidable. At high temperature, λ is tiny compared to interparticle spacing, so particles behave classically (distinguishable, independently moving). As T drops, λ grows, and when wave packets overlap, the quantum identity of particles—their indistinguishability and bosonic statistics—starts to dominate thermodynamics. BEC is the extreme limit of this quantum regime.