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Statistical Mechanics

by: Quinn Larkin

Statistical Mechanics PHY 831

Marketplace > Michigan State University > Physics 2 > PHY 831 > Statistical Mechanics
Quinn Larkin
GPA 3.67

Scott Pratt

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Scott Pratt
Class Notes
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This 3 page Class Notes was uploaded by Quinn Larkin on Saturday September 19, 2015. The Class Notes belongs to PHY 831 at Michigan State University taught by Scott Pratt in Fall. Since its upload, it has received 63 views. For similar materials see /class/207609/phy-831-michigan-state-university in Physics 2 at Michigan State University.

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Date Created: 09/19/15
PRACTICE When you are not practicing remember someone somewhere is practicing and when you meet him he will win 7 E Macauley THINGS TO MEMORIZE S Zpi 1HPi i TdSdEPdVEodN TSEPVEMN ddpddr ddhddr dNee Eo sass 1 1 fr 1 bdy tt lt2Whgtd f QWW new fpbosonsfermions W d5 d5 0 E T7 C E T7 V dT MM P dT PN 3H 3H 7 T 7 T i E i ltq aqgt 7 P 6pgt 7 ltqpgt ltpqgt Don7t memorize factors but know dependencies of more complicated expressions For instance you wouldnt be expected to memorize all the 212 factors in PpT co n71 r 32 p 27 1 3 7 T WT 1 An 7 E B d 6 i 39 i l W 1 was 6 p mag2 dN 7 1 d6 7 32 2i1d6 6 E 261E A2 2 gdeTEe 6 i Eaph a scattering length but you would be expected to know what happens to the second virial coef cient if a repulsive or attractive interaction is added or in what way the pressure would change if a resonance was included 1 Consider two neutrons spin 12 particles They occupy a two level system where the single particle energies are 6 and 6 which is thermalized at a temperature T a What is the average energy of the system as a function of T b What is the chance that the ground state is occupied as a function of T c What is the entropy S of the system as a function of T D 00 4 CT CT Using the last Maxwell relation from Beginning with the expression TdS dE PdV 7 udN derive the Maxwell relations i dT Q T2 d u E dE MT39 and 7 2 dogN g p dp T dT p the previous problem show that if P and SN are 2 LP T T p pZOV39 functions of T and p that E dp LP 5 SN T Consider a system with an order parameter denoted by z along with a particle number N energy E and volume V The system will maximize entropy if E dz EVN Show that if the system has a xed volume and is connected to a bath at temperature TB and chemical potential MB that can can exchange both particles and energy that the total entropy of both the system and the bath will be maximized if the pressure is maximized ie dP 7 0 dz TTB gtMMB V You may use the identity PV TS 7 E MN Consider a particle moving in one dimension according to the Hamiltionian H xp2m2 Bz4 Using the equipartion generalized equi partition or virial theorems nd lt964gt Consider a thermalized two dimensional gas of charged non interacting massless spin zero bosons whose energies are given by e p0 Find the density number per area required for Bose condensation Give answer in terms of c T and h H 00 H O H H Consider the one dimensional case for the massless particles from the previous problem Can you solve for the critical density number per length in this case For massless particles what is the minimum number of dimensions for Bose condensation and how does this compare with the massive case Consider a two dimensional gas of non interacting non relativistic electrons of mass m a What is the density of single particle states DE at the Fermi surface in terms of 6f and m b What is the energy per area for T 0 in terms of m and 6f c In terms of Def and derivatives of Def nd the lowest order expanding in T non zero correction to the energy density at low temperature NT N 2 17qu0 a V 39 Solve for the critical point To and pc in terms of 110 and a Consider the equation of state PN7T7V Consider a two dimensional lattice of three dimensional oscillators where the speed of sound for both longitudinal and transverse modes is 05 a What is the Debye frequency Give wD in terms of cs and NA the number of lattice sites per area b What is the energy of a system of N sites as a function of T if T gtgt hwp Consider the mean eld approximation to the lsing model where the Hamiltonian is H Z ipBai 7 qJltUgtUi where 0 takes on values of 1 or 1 a Beginning with the de nition of ltUgt Ear Uiei HW ltUgt derive a transcendental expression for ltUgt as a function of MB and 1 b What is the critical temperature To c Find the susceptibility X dltUgtdB for T gtgt To Give answer to lowest non zero order in 1T


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