Spring 2023 Homework Assignments
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In class, we showed that an engine cannot have an efficiency higher than that of a Carnot engine, or the Kelvin or the Clausius statements of the second law would be violated. (a) Now, consider a Carnot engine working in reverse, acting as a refrigerator. Show that a general refrigerator (working between the same temperatures) needs a larger amount of work than the Carnot refrigerator to put an equivalent amount of heat QH into the high temperature heat bath. (i.e., show that otherwise, the Kelvin or the Clausius statements of the second law would be violated.) (b) Taking the heat going into the refrigerator as positive, and that leaving it as negative, what is the sign of Σi Qi/Ti ?
Find the number of microstates Ω(E) ΔE of a particle confined to a square box with side L and energy between E and E+ΔE (a) for a classical particle, (b) for a quantum particle.
1. Each electron in an ensemble is known to have its spin oriented in the +x direction with probability 1/2 or the +y direction with probability 1/2. (a) Find the density matrix for this ensemble and check that its trace is 1. (b) Find the ensemble averages for the expected value of the spin component in the x, y, and z directions. Could you have guessed these results? 2. A collection of N harmonic oscillators have the Hamiltonian H = p12/2m + p22/2m + ... + pN2/2m + k x12/2 + k x22/2 + ... + k xN2/2 Find the ensemble average of the quantity x14 + x24 + ... + xN4 (a) if the oscillators are classical oscillators, (b) if the oscillators are quantum oscillators.
A student has three states in the classroom: (1) attentively listening to the lecture, (2) awake but daydreaming, (3) asleep. The transition rates between these states are given as (1) -> (2) 2.0/hour (2) -> (1) 1.0/hour (2) -> (3) 1.0/hour (3) -> (2) 2.0/hour (a) Write down the transition matrix and the corresponding master equation. (b) Find the eigenvalues and the left and right eigenvectors of the transition matrix. (c) If the state of the student is known to be (1) at the start of the lecture, what are the probabilities at the end of one hour? (d) What are the probabilities if the lecture extends "forever"? This is what I got:
Four electrons are placed in a one dimensional harmonic oscillator potential with single-particle enegy levels given by εn = ħω (n + 1/2). (a) List the three lowest enegy levels for the four electron system, along with their degeneracies. (b) If the system is kept at sufficiently low temperatures so that it is sufficient to consider only these three lowest energy states in the ``sum over states'', write down the canonical partition function.
The scaled Hamiltonian of the strip-Ising model in the figure has the form -βH = K Σi ( SiSi+1 + σiσi+1 + Siσi ) with periodic boundary conditions so that SN+1 = S1 and σN+1 = σ1. (a) Write down the transfer matrix for the system. (b) Calculate the largest eigenvalue of this matrix to obtain the partition function. (You may use computer code if necessary.) (c) Plot (1/2N) K2 ∂2/∂K2 ln Z = cv/kB as a function of K. From the maximum of this curve, identify a "critical coupling" Kc. Note: ∂2/∂K2 f(K) ≈ [ f(K+ΔK)+f(K-ΔK)-2f(K) ]/(ΔK)2
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