Question

*Problem 6.73. Bose-Einstein condensation in low-dimensional traps As we found in Problem 6.70, Bose-Einstein condensation does not occur in ideal one and two- dimensional systems. However, this result holds only if the system is confined by rigid walls. In the following, we will show that Bose-Einstein condensation can occur if a system is confined by a spatially varying potential. For simplicity, we will treat the system semiclassically Let us assume that the confining potential has the form $V(r) \sim r^n$. (6.256) Then the region accessible to a particle with energy $\epsilon$ has a radius $L \sim \epsilon^{1/n}$. Show that the corresponding density of states behaves as $g(\epsilon) \sim L^d \epsilon^{\frac{d}{2}-1} \sim \epsilon^{d/n} \epsilon^{\frac{d}{2}-1} \sim \epsilon^\alpha$, (6.257) where $\alpha = \frac{d}{n} + \frac{d}{2} - 1$ (6.258) What is the range of values of $n$ for which $T_c > 0$ for $d = 1$ and 2? More information about experiments on Bose-Einstein condensation can be found in the references.

          *Problem 6.73. Bose-Einstein condensation in low-dimensional traps
As we found in Problem 6.70, Bose-Einstein condensation does not occur in ideal one and two-
dimensional systems. However, this result holds only if the system is confined by rigid walls. In
the following, we will show that Bose-Einstein condensation can occur if a system is confined by a
spatially varying potential. For simplicity, we will treat the system semiclassically
Let us assume that the confining potential has the form
$V(r) \sim r^n$.
(6.256)
Then the region accessible to a particle with energy $\epsilon$ has a radius $L \sim \epsilon^{1/n}$. Show that the
corresponding density of states behaves as
$g(\epsilon) \sim L^d \epsilon^{\frac{d}{2}-1} \sim \epsilon^{d/n} \epsilon^{\frac{d}{2}-1} \sim \epsilon^\alpha$,
(6.257)
where
$\alpha = \frac{d}{n} + \frac{d}{2} - 1$
(6.258)
What is the range of values of $n$ for which $T_c > 0$ for $d = 1$ and 2? More information about
experiments on Bose-Einstein condensation can be found in the references.

*Problem 6.73. Bose-Einstein condensation in low-dimensional traps
As we found in Problem 6.70, Bose-Einstein condensation does not occur in ideal one and two-
dimensional systems. However, this result holds only if the system is confined by rigid walls. In
the following, we will show that Bose-Einstein condensation can occur if a system is confined by a
spatially varying potential. For simplicity, we will treat the system semiclassically
Let us assume that the confining potential has the form
V(r) ∼ r^n.
(6.256)
Then the region accessible to a particle with energy ϵ has a radius L ∼ϵ^1/n. Show that the
corresponding density of states behaves as
g(ϵ) ∼ L^d ϵ^(d)/(2)-1∼ϵ^d/nϵ^(d)/(2)-1∼ϵ^α,
(6.257)
where
α = (d)/(n) + (d)/(2) - 1
(6.258)
What is the range of values of n for which Tc > 0 for d = 1 and 2? More information about
experiments on Bose-Einstein condensation can be found in the references.

Added by Jesse P.

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