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Bose Einstein condensation

  1. Apr 17, 2015 #1

    CAF123

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    1. The problem statement, all variables and given/known data
    Explain the conditions under which Bose-Einstein condensation occurs and show this happens for density ##\rho \equiv N/V > \rho_C(T)##, where $$\rho_C(T) = A(kT)^{3/2} \int_0^{\infty} \frac{x^{1/2}}{ e^x − 1} dx .$$

    Suppose the energy of the particles on the lattice is now ## \epsilon_j → \epsilon_{j\chi} = E\chi+ \frac{\hbar^2 k^2_j}{(2m)}##, where ##\chi = 0, 1## and ##E > 0##. Obtain an expression for ##\rho \equiv N/V##.

    2. Relevant equations
    ##\rho = \rho_o + \rho_+## where ##\rho_o## is density of ground state and ##\rho_+## density of all others.

    3. The attempt at a solution

    The first part is ok, in the integral, ##x = \beta \epsilon##. In the second part, are we just shifting the ground state energy to ##E\chi##? and then evalaute the integral ##\rho_C(t)## using that expression for ##\epsilon_{j \chi}##? I'm not sure what the conditions ##\chi = 0,1## and ##E>0## imply yet either. Thanks!
     
  2. jcsd
  3. Apr 20, 2015 #2

    CAF123

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    Bump!
     
  4. Apr 23, 2015 #3

    CAF123

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    Bump
     
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