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final

# final - N noninteracting spin-1/2 fermions in an area A...

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Statistical Mechanics — PHY 5524 Final Exam April 27, 2009 1. (50 pts) Consider a three-dimensional gas of N noninteracting bosons in a volume V with energy dispersion E ( k ) = ¯ hv | k | where v has units of velocity. (a) Determine the one-particle density of states for this gas. (b) Show that this gas exhibits Bose condensation at a finite temperature T c ( N/V ) γ and determine the exponent γ . (c) For T < T c obtain an expression for the pressure, P , of this gas. Use the fact that Σ = - PV where Σ is the grand potential, Σ = - k B T ln Z , and Z is the grand partition function. Express your answer in terms of an integral over energy E . (Do not try to perform this integral). (d) For T < T c find an expression for the total energy, E , of this gas, expressed as an integral over energy E (Again, do not try to perform this integral). (e) For T < T c , using the results of Parts (c) & (d), show that PV = αE and determine the value of α . ( Hint: Perform an integration by parts.)

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2. (50 pts) Consider a two-dimensional gas of
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Unformatted text preview: N noninteracting spin-1/2 fermions in an area A with non-relativistic energy dispersion E ( k ) = ¯ h 2 | k | 2 / (2 m ). (a) Find an expression for the Fermi energy, E F , of this gas expressed as a function of n = N/A , m , and fundamental constants. (b) Obtain an exact expression for the chemical potential, μ , of this gas valid at arbitrary temperature. (c) Show that in the limit T → 0 the expression for μ obtained in Part (b) becomes equal to the Fermi energy obtained in Part (a). (d) For k B T ¿ E F the speciﬁc heat of this gas is C V = π 2 3 Nk B ˆ k B T E F ! , (you do not need to show this). In one or two sentences explain qualitatively why this speciﬁc heat is so much lower than the speciﬁc heat of a classical 2D monatomic gas ( C cl . V = Nk B ). Hint : For Part (b) you may ﬁnd the following integral useful: Z ∞ 1 z-1 e y + 1 dy = ln(1 + z ) . 2...
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