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Unformatted text preview: ive modes is given by
Ep = p( p + 2n0 U0 ) (3.20) We can deﬁne the healing length from
2 = n0 U0
mξh (3.21) In the long wavelength limit, qξh << 1, we ﬁnd sound-like dispersion Eq =
vs |q |. Sound velocity
1/2 n0 U0
m vs = (3.22) We can interpret the appearance of the gapless mode as manifestation of spontaneously broken symmetry: this mode arises because the superﬂuid state spontaneously breaks the U (1) symmetry corresponding to the conservation in the
number of of particles. However sound mode by itself does not imply supeﬂuidity. As we know, sound modes exist in room temperature gases.
In the short wavelength limit, qξh >> 1, we ﬁnd free particle dispersion
Eq = q 2 /2m.
It is natural to ask about the change in the wavefunction (3.8) implied by
the Bogoliubov analysis. From the form of the mean-ﬁeld Hamiltonian (3.12)
we expect that it should have coherent superpositions of p, −p pairs. So we
expect the wavefunction to be of the form
† P |ΨBog = C eαbp=0 + p fp b† b† p
p− |vac (3.2...
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- Fall '10