By Ahn D., Park S.-H.

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2) n This fundamental law is the essence of statistical mechanics, with the concept of thermal equilibrium and temperature. If a system is very weakly coupled to a heat bath at a given temperature, if the coupling is indefinite or not known precisely, if the coupling has been on for a long time, and if all the “fast” processes have happened and all the “slow” processes have not, the system is said to be in thermal equilibrium. Consider two different states of a system that has the same energy, E j = Ek .

157) For this reason, we call L± the ladder operators, in particular, the angular momentum-raising and -lowering operators. 151): c± = lm L∓ L± lm 2 = 2 (l(l + 1) − m(m ± 1)) . 158) If we take c± as positive, then we have the following expressions: c± = l(l + 1) − m(m ± 1). 160) which gives θ, φ lm = e imφ θ, 0 lm . 162) ⎛ ∂ − l cot θ⎞ θ, φ lm = 0. 163), the orbital angular momentum eigenfunction has the following functional form: l Yll (θ, φ) = θ, φ ll = αe ilφ (sin θ ) . (2l + 1) which gives Yll (θ, φ) = (−1)l 2l l !

N1n2 = ⎛⎜ 1 2 ⎝ N! 24) P 4/20/2011 10:35:27 AM 52 BASIC QUANTUM STATISTICAL MECHANICS where the pi denotes the different states, and the sum is taken over all possible permutations of different numbers pi. 25) n where fn(1) is the operator acting on functions of xn. When such an operator acts on the function Φ N1N2 it carries it either into the same function or into another function in which the state of one particle has been changed. 26) =f ( 1) ik ni nk for nondiagonal elements, where ∫ fik(1) = ϕi *( x) f (1)ϕk ( x)dx.