Lecture Notes2

# Lecture Notes2 - 1 Formula Sheet of Statistical Mechanics...

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1 Formula Sheet of Statistical Mechanics Ch. 1 Review 1. Lagrangian equation and Hamilton’s equation 2. Energy Levels of some quantum systems a) 1-D infinite well: , 2 2 2 2 x m H - = η 2 2 2 8 ma n h n = e . b) 1-D oscillator: , 2 1 2 2 2 2 2 kx x m H + - = η w n n η + = 2 1 e . c) Rigid rotor: , sin 1 sin sin 1 2 2 2 2 2 + - = q q q I H η ( 29 I J J n 2 1 2 η + = e . 3. Calculation of Degeneracy e ¶e e e e w d d ) ( ) ( Φ = 3-D infinite well: ) ( 8 2 2 2 2 2 z y x n n n ma h + + = e 2 2 2 2 2 2 8 R h ma n n n z y x = = + + e 2 / 3 2 2 2 8 6 ) 3 4 ( 8 1 ) ( = = Φ h ma R e p p e . 4. Thermodynamics equations + - = j j j dN pdV TdS dE m , j j j dN Vdp TdS dH + + = m + - - = j j j N d pdV SdT dA m , j j j dN Vdp SdT dG + + - = m 5. Useful mathematical formula N nN N nN - = λ ! , ( 29 = = = + + 0 1 0 2 1 ! ! 1 1 r r N r i i N r N N N r N x x N x x x + - = + Γ = - = = 0 2 / ) 1 ( 2 / 1 2 / , 2 ) 2 1 ( , ) 2 ( 2 ) 1 ( 5 3 1 2 odd n a n even n a a n dx e x I n n ax n n p 6. Constants: Stefan-Boltzman constant σ =5.67 × 10 -8 W/m 2 K 4 , Bohr Magneton: 9.273 × 10 -21 erg gauss -1 , j q L q L dt dL j = j q L p j = j j j p q H q p H j - = =

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2 Ch. 2 The Canonical Ensemble = k k a A a W ! ! ) ( 0 ) ( = - - E a a a nW a k k k k k j b a λ N j V j E p - = - - - = = j j j j j j e e V E P p p j j e e b b p E Ep p V E N - - + - = b b b , Ep p E p V N - - - - = , ¶b p p V E V N N - = + , , ¶b b b p T p T V E V N N T - = - , , = k k j j b B a A b a W ! ! ! ! ) , ( const nQ k T E S + + = λ rev rev j j j j j j w q dE P P d E E d d d - = + = V N T nQ kT E , 2 = λ T N V nQ kT p , = λ nQ k T nQ kT S V N λ λ + = , ) , , ( ) , , ( T V N nQ kT T V N A λ - = A a a W a a a W A A a P j a a j j j * ) ( ) ( ) ( 1 = = =
3 Ch. 3 Formula Table Microcanonical ensemble, ) , , ( E V N = degeneracy = ln k S dN T dV T p dE T dS m - + = 1 V N E kT , ln 1 = E N V kT p , ln = E V N kT , ln - = m Canonical ensemble, - = j V N E j e T V N Q ) , ( ) , , ( b Q kT A ln - = dN pdV SdT dA m + - - = V N T Q kT Q k S , ln ln + = T N V Q kT p , ln = T V N Q kT , ln - = m V N T Q kT E , 2 ln = Grand canonical ensemble, kT N j E N e e T V j m b m - = Ξ ) , , ( Ξ = ln kT pV pdV Nd SdT pV d + + = m ) ( m , ln ln V T kT k S

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