The quantum energy levels of a rigid rotator are =j(j+1) h^2 / 8 π^2 m a^2, where j=0,1,2, …, m

step 1 Question number 39 and we need to evaluate the integral and in this question we are going to let

The quantum energy levels of a rigid rotator are =j(j+1) h^2...

The quantum energy levels of a rigid rotator are $$ \varepsilon_j=j(j+1) h^2 / 8 \pi^2 m a^2, $$ where $j=0,1,2, \ldots, m$ and $a$ are positive constants. The degeneracy of each level is $g_j=2 j+1$. (a) Find the general expression for the partition function $z_0$. (b) Show that at high temperatures it can be approximated by an integral. (c) Evaluate the high-temperature energy $U$ and heat capacity $C_v$. (d) Also, find the low-temperature approximations to $z_0, U$ and $C_v$.

The quantum energy levels of a rigid rotator are
$$
\varepsilon_j=j(j+1) h^2 / 8 \pi^2 m a^2,
$$
where $j=0,1,2, \ldots, m$ and $a$ are positive constants. The degeneracy of each level is $g_j=2 j+1$.
(a) Find the general expression for the partition function $z_0$.
(b) Show that at high temperatures it can be approximated by an integral.
(c) Evaluate the high-temperature energy $U$ and heat capacity $C_v$.
(d) Also, find the low-temperature approximations to $z_0, U$ and $C_v$.

Key Concepts

Thermodynamic Quantities from the Partition Function

The internal energy and heat capacity of a system can be derived from the partition function by taking appropriate derivatives with respect to temperature. This process links microscopic energy levels, through their statistical distribution, to macroscopic observables, providing a bridge between quantum mechanics and thermodynamics.

Rigid Rotator Model in Quantum Mechanics

The rigid rotator model describes a system with fixed internuclear distance that rotates freely in space, leading to quantized rotational energy levels. It serves as a fundamental model in quantum mechanics for understanding rotational spectra and molecular rotation, illustrating how quantum energy quantization and degeneracy directly impact thermodynamic behavior.

Low-Temperature Approximations

In the low-temperature regime, only the lowest energy quantum states are significantly populated, as thermal energy is too low to excite higher energy levels. This allows for approximations that focus on the contribution of the ground state and a few low-lying excited states, which is critical in evaluating partition functions and deriving low-temperature behavior of quantities like internal energy and heat capacity.

Partition Function

The partition function is a central quantity in statistical mechanics that encodes the statistical properties of a system at thermal equilibrium. It serves as a generating function for deriving thermodynamic observables by summing over all possible states weighted by their Boltzmann factors, and allows for the calculation of quantities such as internal energy, free energy, entropy, and heat capacity.

Quantum Energy Levels and Degeneracy

Quantum energy levels arise from the quantization of observable parameters in systems that obey quantum mechanics. In many cases, these energy levels have degeneracies, meaning multiple distinct quantum states share the same energy. Understanding degeneracy is essential since it contributes to the statistical weight of each energy level within the partition function and influences thermodynamic properties.

High-Temperature (Classical) Limit

At high temperatures, many quantum systems can be approximated by classical physics, as the thermal energy exceeds the spacing between quantized energy levels. This leads to the transition from a sum over discrete states to an integral over a continuous variable, simplifying calculations of the partition function and associated thermodynamic properties.

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