(a) Write down a simple expression for the internal part of...

(a) Write down a simple expression for the internal part of the partition function for a single isolated hydrogen atom in very weak contact with a reservoir at temperature $T$. Does your expression diverge for $T=0$, for $T \neq 0$ ? (b) Does all or part of this divergence arise from your choice of the zero of energy? (c) Show explicitly any effects of this divergence on calculations of the average thermal energy $U$. (d) Is the divergence affected if the single atom is assumed to be confined to a box of finite volume $L^3$ in order to do a quantum calculation of the full partition function? Explain your answer.

(a) Write down a simple expression for the internal part of the partition function for a single isolated hydrogen atom in very weak contact with a reservoir at temperature $T$. Does your expression diverge for $T=0$, for $T \neq 0$ ?
(b) Does all or part of this divergence arise from your choice of the zero of energy?
(c) Show explicitly any effects of this divergence on calculations of the average thermal energy $U$.
(d) Is the divergence affected if the single atom is assumed to be confined to a box of finite volume $L^3$ in order to do a quantum calculation of the full partition function? Explain your answer.

Key Concepts

Partition Function in Statistical Mechanics

The partition function is a central concept in statistical mechanics that encodes the statistical properties of a system in thermodynamic equilibrium. It is defined as the sum over all states, each weighted by the Boltzmann factor exp(–?E), and it is used to calculate macroscopic quantities such as free energy, entropy, and average energy. It provides a bridge between microscopic states and macroscopic observables.

Internal Partition Function of Bound States

For systems with discrete energy levels, such as a hydrogen atom, the internal partition function is formed by summing the Boltzmann factors over all internal bound states. In the hydrogen atom case, the energy levels approach the continuum as the principal quantum number increases, which leads to an infinite number of states lying arbitrarily close to zero energy, challenging the convergence of the sum.

Divergence in the Partition Function

A divergence in the partition function arises when the sum over states does not converge, often due to an infinite accumulation of states with energies that contribute significantly even at finite temperature. In the hydrogen atom example, because the energy of highly excited states approaches zero and the degeneracy increases, for any nonzero temperature the contributions from these states do not decay sufficiently fast, resulting in a divergent partition function.

Choice of Zero of Energy

The zero of energy in statistical mechanics is arbitrary and, for many calculations, only energy differences are physically meaningful. Although a shift in the energy reference can affect the individual Boltzmann factors, a divergence in the partition function is intrinsic to the density and arrangement of the energy levels rather than a consequence of the chosen energy zero.

Average Thermal Energy and Divergences

The average thermal energy, U, is obtained by differentiating the logarithm of the partition function with respect to temperature. The divergence in the partition function can potentially affect U if the divergence changes with temperature. However, if the divergent part is independent of temperature, it will cancel out upon differentiation, leaving finite, physically meaningful predictions for thermal energy despite an overall divergent partition function.

Finite Volume and Regularization in Quantum Mechanics

Introducing a finite volume, such as confining the atom in a box, modifies the density of states by discretizing the continuum and regularizing the partition function. This constraint limits the number of quantum states available, thereby mitigating the divergence encountered with an infinite system. Finite volume considerations allow for a proper quantum mechanical treatment that avoids the mathematical divergence inherent in the idealized free-space model.

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