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Gravitational Force and the Planck Scale

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Introduction to the Standard Model

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Quantum Flavordynamics and Quantum Chromodynamics

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Special Unitary Groups and the Standard Model - Part 1 .

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Special Unitary Groups and the Standard Model - Part 2

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Special Unitary Groups and the Standard Model - Part 3 .

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Standard Model Lagrangian

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Units, Constants and Useful Formulas

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Last modified: January 26, 2018


The basic equations:



P_i = (1/Z)exp(-βE_i) 

Z = Σ_ie^-βE_i 

E  = -∂lnZ/∂β 

K_BT = 1/β

S = βE + lnZ

A = -K_BTlnZ



The Harmonic Oscillator
-----------------------

Consider the classical harmonic oscillator.

E = p²/2m + kx²/2  (KE + PE)

Z = ∫dxdpexp(-β[p²/2m +  kx²/2]

  = ∫dpexp(-βp²/2m) ∫exp(-βkx²/2)dx

  = √(2πm/β)√(2π/βk)

  = 2πm/βω where ω = √(k/m)

Now,

-lnZ = -ln(2π/ω) + lnβ

and.

E = -∂lnZ/∂β 

  = 1/β

  = K_BT

If we only consider the kinetic energy term then,

-lnZ = -(1/2)ln(2πm) + (1/2)lnβ

and.

E = -∂lnZ/∂β 

  = (1/2)β

  = K_BT/2

Likewise, if we only consider the potential energy term 
then,

-lnZ = -(1/2)ln(2π/k) + (1/2)lnβ

and.

E = -∂lnZ/∂β 

  = (1/2)β

  = K_BT/2

This is in agreement with the EQUIPARTION OF ENERGY theorem
that states in a system in thermal equilibrium there will be an 
average energy of K_BT/2 associated with each degree of freedom.  
Thus, for a system of n kinetic degrees of freedom and m 
associated potential energy degrees of freedom, the total 
average energy of the system will be (1/2)(n + m)K_BT.

Now consider the quantum harmonic oscillator.

E = nhω

Z = Σ_nexp(-βnhω) 

  = 1 + exp(-βhω) +(exp(-βhω))² + ...

This is the geometric series:

  = 1/(1 - exp(-βhω))

Now,

-lnZ = ln(1 - exp(-βhω))

and,

E = -∂lnZ/∂β 

  = hω/(exp(βhω) - 1)