# lec23 - MIT OpenCourseWare http/ocw.mit.edu 8.044...

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Unformatted text preview: MIT OpenCourseWare http://ocw.mit.edu 8.044 Statistical Physics I Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms . Canonical Ensemble CLASSICAL QUANTUM p ( { p, q } ) = e −H ( { p,q } ) /kT /Z p ( state ) = e − E state /kT /Z Z = e −H /kT { dp, dq } Z = e − E state /kT states 8.044 L23B1 EXAMPLE 2 LEVEL SYSTEM: STATES OF AN IMPURITY IN A SOLID E = ε 1 g g-FOLD DEGENERACY * EXCITED E = 0 GROUND STATE LOCATION INTERNAL ENERGY LEVELS PHYSICAL DIFFERENCE 8.044 L23B2 STATES: | 0 > , | 1 >, g > · · · | E =0 E = e − E state /kT = 1 × e + g × e − /kT = 1 + ge − /kT Z 1 = states p ( state ) = e − E state /kT /Z 1 1 = 1+ ge − /kT for | 0 > = e − /kT 1+ ge − /kT for | i > i = 1 , · · · g 8.044 L23B3 g 1+ g 1 p(E= ε ) 1+ g p(state = i = 0) ε / k T 8.044 L23B4 ε Assume • N impurities ( N 1) ε 0 = ( V /V ) − γ • V V Z = Z N F ( T, V, N ) = − kT ln Z = − NkT ln Z 1 1 g ( kT 2 ) e − /kT S = − ∂F = Nk ln Z 1 + NkT 1+ ge − /kT ∂T V 8.044 L23B5 e...
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lec23 - MIT OpenCourseWare http/ocw.mit.edu 8.044...

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