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A system of N one-dimensional localized oscillators, at a given temperature T , is associated with the Hamiltonian
where
with x > 0.
(a)Obtain the canonical partition function of this classical system. Calculate the internal energy per oscillator, u = u (T ). What is the form of u (T ) in the limits ε→0 and ε→ ∞?
(b)Consider now the quantum analog of this model in the limit ε→ ∞ . Obtain an expression for the canonical partition function. What is the internal energy per oscillator of this quan- tum analog?
Physics
The classical partition function is given by Z = ZN , where
Note that u→kBT at both limits, ε→0 and ε→ ∞
The quantum results are simple in the limit of an infinite potential barrier (ε→ ∞). in this limit, we have to discard all of the harmonic oscillator states with even values of n (which are associated with even wave functions; wave functions that do not vanish at the origin, q = 0).
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