By Carlo F. Barenghi
The goal of this primer is to hide the fundamental theoretical details, quick and concisely, for you to permit senior undergraduate and starting graduate scholars to take on tasks in topical study components of quantum fluids, for instance, solitons, vortices and collective modes.
The number of the cloth, either concerning the content material and point of presentation, attracts at the authors research of the luck of suitable examine initiatives with newbies to the sector, in addition to of the scholars suggestions from many taught and self-study classes at the topic matter.
Starting with a short historic review, this article covers particle information, weakly interacting condensates and their dynamics and eventually superfluid helium and quantum turbulence.
At the tip of every bankruptcy (apart from the 1st) there'll be a few workouts. targeted ideas may be made to be had to teachers upon request to the authors.
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Extra resources for A Primer on Quantum Fluids
G. 1007/978-3-319-42476-7_4 53 54 4 Waves and Solitons Fig. 1 a One dimensional sound waves, that is, sinusoidal perturbations of the background density n 0 , of wavelength λ and amplitude δn 0 (x, t) n 0 . b The dispersion relation ω(k) of the homogeneous (weakly-interacting) condensate, according to Eq. 5), for g > 0 (black line), g = 0 (blue line) and g < 0 (red line). Solid lines plot the real part of ω and dashed lines plot the imaginary part. For g < 0, ω becomes imaginary for small k; everywhere else ω is real.
A) Normalize the wavefunction, and hence determine an expression for the ThomasFermi radius Rr in terms of N , as and r . (b) Determine an expression for the peak density in terms of N and Rr . (c) Find an expression for the ratio Rr / r , and comment on its behaviour for large N . (d) What is the energy of the condensate? 3 Derive the expression for the variational energy of a three-dimensional trapped condensate, Eq. 35). Repeat in two dimensions (for a potential V (x, y) = mωr2 (x 2 + y 2 )/2) and in one dimension (for a potential V (x) = mωr2 x 2 /2).
Similarly, the total energy of the gas at T = 0 is, U= N (E)E dE = 4πV 5 2m h2 3/2 5/2 EF = 3 N EF . 37) P = n EF . 5 This pressure is finite even at T = 0, unlike the Bose and classical gases, and does not arise from thermal agitation. Instead it is due to the stacking up of particles in energy levels, as constrained by the quantum rules for fermions. This degeneracy pressure prevents very dense stars, such as neutron stars, from collapsing under their own gravitational fields. 28 2 Classical and Quantum Ideal Gases Fig.
A Primer on Quantum Fluids by Carlo F. Barenghi