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Problem 1: This problem concerns a collection of N identical harmonic oscillators (perhaps an Einstein solid) at temperature T. The allowed energies of each oscillator are 0, hf, 2hf, and so on. a) Prove =1+x + x² + x³ + .... Ignore Schroeder's comment about proving 1-x the formula by long division. Prove it by first multiplying both sides of the equation by (1 – x), and then thinking about the right-hand side of the resulting expression. b) Evaluate the partition function for a single harmonic oscillator. Use the result of (a) to simplify your answer as much as possible. c) Use E = - дz to find an expression for the average energy of a single oscillator. z aB Simplify as much as possible. d) What is the total energy of the system of N oscillators at temperature T?

Problem 1: This problem concerns a collection of N identical harmonic oscillators (perhaps an Einstein solid) at temperature T. The allowed energies of each oscillator are 0, hf, 2hf, and so on. a) Prove =1+x + x² + x³ + .... Ignore Schroeder's comment about proving 1-x the formula by long division. Prove it by first multiplying both sides of the equation by (1 – x), and then thinking about the right-hand side of the resulting expression. b) Evaluate the partition function for a single harmonic oscillator. Use the result of (a) to simplify your answer as much as possible. c) Use E = - дz to find an expression for the average energy of a single oscillator. z aB Simplify as much as possible. d) What is the total energy of the system of N oscillators at temperature T?

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Problem 1: This problem concerns a collection of N identical harmonic oscillators (perhaps an Einstein solid) at temperature T. The allowed energies of each oscillator are 0, hf, 2hf, and so on. a) Prove =1+x + x² + x³ + .... Ignore Schroeder's comment about proving 1-x the formula by long division. Prove it by first multiplying both sides of the equation by (1 – x), and then thinking about the right-hand side of the resulting expression. b) Evaluate the partition function for a single harmonic oscillator. Use the result of (a) to simplify your answer as much as possible. c) Use E = - дz to find an expression for the average energy of a single oscillator. z aB Simplify as much as possible. d) What is the total energy of the system of N oscillators at temperature T?
Transcribed Image Text:Problem 1: This problem concerns a collection of N identical harmonic oscillators (perhaps an Einstein solid) at temperature T. The allowed energies of each oscillator are 0, hf, 2hf, and so on. a) Prove =1+x + x² + x³ + .... Ignore Schroeder's comment about proving 1-x the formula by long division. Prove it by first multiplying both sides of the equation by (1 – x), and then thinking about the right-hand side of the resulting expression. b) Evaluate the partition function for a single harmonic oscillator. Use the result of (a) to simplify your answer as much as possible. c) Use E = - дz to find an expression for the average energy of a single oscillator. z aB Simplify as much as possible. d) What is the total energy of the system of N oscillators at temperature T?
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