p847ps2 - ) = B (independent oF ǫ ) and fnd B . (c). In...

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Physics 847: Problem Set 2 Due Thursday, April 15 at 11:59 P. M. Each problem is worth 10 points unless otherwise specifed. 1. (20pts.) Density of states in k space and in energy. Consider a Free particle in a d-dimensional box oF edge L. The normalized eigen- states oF the Free-particle Schr¨odinger equation are V - 1 / 2 exp( i k · r ), where V is the volume, the allowed values oF k are k = (2 π/L )( n 1 , n 2 , .... n d ) (a d-dimensional vector), and n 1 , n 2 ... are positive or negative integers (this assumes periodic boundary conditions in all d directions). Thus, the density oF allowed points in k-space is 1 / [(2 π/L ) d ] = L d / (2 π ) d . The corresponding energies are ǫ ( k = ¯ h 2 k 2 / (2 m ). Assume that the particles are spinless. Now defne the density of states ρ ( ǫ ) so that ρ ( ǫ ) represents the num- ber oF states between energies ǫ and ǫ + . (a). In three dimensions, show that ρ ( ǫ ) = 1 / 2 , and fnd A in terms oF the volume V and various Fundamental constants. (b). In two dimensions, show that ρ ( ǫ
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Unformatted text preview: ) = B (independent oF ǫ ) and fnd B . (c). In one dimension, show that ρ ( ǫ ) = Cǫ-1 / 2 and fnd C . 2. Pathria, Prob. (7.2). However, just veriFy that the virial expansion is valid through ℓ = 2, and derive the frst two virial coe±cients, a 1 and a 2 . 3. Pathria,Prob. (7.13). Omit the part oF the problem beginning “Refne your argument. ..” In proving the absence oF Bose-Einstein condensa-tion in two dimensions, consider the limit A → ∞ , N → ∞ , but N/A a fnite constant. 4. (20 pts.) Pathria, Prob. (7.14). Interpret the sentence beginning “Dis-cuss. ..” to mean “²or which values oF n and s does Bose-Einstein condensation occur at a non-zero temperature?” Replace the phrase “Study the thermodynamic behavior oF the system and show that, quite generally,. ..” by “Show that, quite generally,. ..” 1...
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This note was uploaded on 07/15/2011 for the course PHYSICS 847 taught by Professor Stroud during the Spring '10 term at Ohio State.

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