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==Problem== |
==Problem== |
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| − | (a)Find the Bose-Einstein condensation temperature <math>T_{BEC}</math> for a large number, <math>N</math>, of non-interacting atoms of mass <math>M</math> confined to a volume <math>V</math>. Assume the atoms have spin 0, or have integer spin but the spin degeneracy is lifted by an applied magnetic field. A derivation of the full result is required (8 points), but you can get partial credit by just answering such questions as: What is the momentum of an atom in the condensate? What is the value of the chemical potential for <math>T \le T_{BEC}</math> <math>T_{BEC}</math>? What can be said about the value of <math> |
+ | (a)Find the Bose-Einstein condensation temperature <math>T_{BEC}</math> for a large number, <math>N</math>, of non-interacting atoms of mass <math>M</math> confined to a volume <math>V</math>. Assume the atoms have spin 0, or have integer spin but the spin degeneracy is lifted by an applied magnetic field. A derivation of the full result is required (8 points), but you can get partial credit by just answering such questions as: What is the momentum of an atom in the condensate? What is the value of the chemical potential for <math>T \le T_{BEC}</math> <math>T_{BEC}</math>? What can be said about the value of <math>T_{BEC}</math> by dimensional analysis alone? |
(b)How can one create, in practice, such a BE condensate? (1 point) |
(b)How can one create, in practice, such a BE condensate? (1 point) |
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Useful integral: |
Useful integral: |
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<math>\int\limits_0^\infty {dx{{\sqrt x } \over {e^x - 1}}} = {{\sqrt \pi } \over 2}\zeta \left( {3/2} \right) \cong 2.3152</math> |
<math>\int\limits_0^\infty {dx{{\sqrt x } \over {e^x - 1}}} = {{\sqrt \pi } \over 2}\zeta \left( {3/2} \right) \cong 2.3152</math> |
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==Solution== |
==Solution== |
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Revision as of 02:19, 26 February 2007
Problem
(a)Find the Bose-Einstein condensation temperature for a large number, , of non-interacting atoms of mass confined to a volume . Assume the atoms have spin 0, or have integer spin but the spin degeneracy is lifted by an applied magnetic field. A derivation of the full result is required (8 points), but you can get partial credit by just answering such questions as: What is the momentum of an atom in the condensate? What is the value of the chemical potential for ? What can be said about the value of by dimensional analysis alone?
(b)How can one create, in practice, such a BE condensate? (1 point)
(c)Estimate the number density necessary to obtain microKelvin if the atoms are (1 point).
Useful integral:
