Advanced Quantum Mechanics Schwabl · Problem 3.6

Correlation Functions, Scattering, and Response

Physics — Quantum Mechanics · bose-gas-excitation-spectrum-spherical-well

Problem

a) Determine the excitation spectrum

\omega_\mathbf{k}

of the weakly interacting Bose gas for the spherical well potential

V(\mathbf{x}) = \lambda'\Theta(R-|\mathbf{x}|)

. Analyze the limiting case

R \to 0

and compare the result with the excitation spectrum for the contact potential

V_\mathbf{k} = \lambda

. The comparison yields

\lambda = \frac{4\pi}{3}\lambda' R^3

. b) Determine the range of the parameter

k_0 R

, where

k_0^2 = 4mn\lambda

, in which the excitation spectrum displays a "roton minimum". Discuss the extent to which this parameter range lies within or outside the range of validity of the Bogoliubov theory of weakly interacting bosons. *Hints:* Rewrite the spectrum in terms of the dimensionless quantities

x = k/k_0

and

y = k_0 R

and consider the derivative of the spectrum with respect to

x

. The condition for the derivative to vanish should be investigated graphically.

Browse the problem library