- #1
physicus
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Homework Statement
Suppose an ideal bose gas sees a periodic potential with a period a in both x and y directions. Its eigenstates are altered from the free-particle form. The lowest band has energies
[itex]\epsilon_\vec{k}=2t(2-cos(k_xa)-cos(k_ya))[/itex]
where t is an energy scale that depends on the amplitude of the periodic potential, and the wavevectors [itex]\vec{k}[/itex] are restricted to the first Brillouin zone: [itex]\pi/a <k_{x,y}<\pi/a[/itex].
(i) Show that the density of states [itex]g(\epsilon)[/itex] is zero for [itex]\epsilon>8t[/itex] and [itex]\epsilon<0[/itex]
(ii) Show that [itex]g(\epsilon) \simeq L^2/4\pi ta^2[/itex] at low energies [itex]0\leq\epsilon\ll 8t[/itex]
Hint: Sketch the dispersion relation above consider the approximate form of [itex]\epsilon_{\vec{k}}[/itex] at lom wavelengths ([itex]ka \ll 1[/itex]).
Homework Equations
The Attempt at a Solution
Number (i) is quite obvious since the cosine can only take values between -1 and 1. Therefore the expression in brackets can only take value between 0 and 4 which shows that there are no states with energies < 0 or energies > 8t.
(ii) For long wavelengths we can approximate
[itex] \epsilon_\vec{k}=2t(2-cos(k_xa)-cos(k_ya)) = -2t(k_x^2 a^2+k_y^2a^2) = -2ta^2\vec{k}^2[/itex]
But how does that help me to get the density of states? I don't know how to begin. Can someone give me an ansatz?
Cheers, physicus