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homework9

homework9 - d Calculate the low-temperature speciﬁc heat...

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Physics 481: Condensed Matter Physics - Homework 9 due date: Friday, March 25, 2011 Problem 1: Copper nanowire (20 points) Consider a long thin wire of copper along the z -direction with length L = 1 cm and a square cross section (width a is a few ˚ A). Treat the wire as a free electron gas, demanding that the wave function vanishes outside of the wire. Use periodic boundary conditions along the length of the wire. The electron density for copper is 8.49 × 10 22 electrons/cm 3 . a) Solve the Schr¨ odinger equation for this geometry. b) Which states are occupied at zero temperature? Qualitatively describe the dependence on the width a . c) Calculate the maximum possible width a of the wire such that only the ground state in x and y directions is occupied.
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Unformatted text preview: d) Calculate the low-temperature speciﬁc heat for the case when only the ground state in x and y directions is occupied (algebraic answer in terms of a and the electron density OK). Problem 2: Electronic density of states (10 points) Calculate the electronic density of states of the free electron gas in one and two spatial dimensions. Discuss the character of the van-Hove singularity at ± = 0 in these cases. Problem 3: Pressure of ideal Fermi gas (Marder, problem 6.3, 10 points) Find the pressure of the free electron gas in three dimensions at zero temperature. Hint: Start from a thermodynamic relation, e.g., p =-( ∂E/∂V ) N (at zero temperature)....
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