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, … Th

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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 $j=0,1,2, \ldots \quad$ The degeneracy of each level is $g_j=2 j+1$. (a) Find the general expression for the partition function, and show that at high temperatures it can be approximated by an integral. (b) Evaluate the high-temperature energy and heat capacity. (c) Find the low-temperature approximations to $z, U$ 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 $j=0,1,2, \ldots \quad$ The degeneracy of each level is $g_j=2 j+1$.
(a) Find the general expression for the partition function, and show that at high temperatures it can be approximated by an integral.
(b) Evaluate the high-temperature energy and heat capacity.
(c) Find the low-temperature approximations to $z, U$ and $C_v$.

Key Concepts

Quantum Partition Function

The partition function is a fundamental tool in statistical mechanics that encodes all the thermodynamic properties of a system. For a quantum system, it is defined as the sum over all quantum states of the Boltzmann factor exp(–?E), where E represents the energy of each state and ? is the inverse temperature. In the context of the rigid rotator, the partition function incorporates the quantized energy levels and their degeneracies, serving as the starting point for the evaluation of macroscopic quantities such as energy and heat capacity.

Quantum Energy Levels of a Rigid Rotator

For a rigid rotator, the quantized energy levels are given by an expression with a quadratic dependence on the quantum number j. These discrete energy states, together with their multiplicity (degeneracy), form the basis for the calculation of the partition function and other thermodynamic properties. Understanding the functional form of these energy levels is crucial for applying quantum statistical mechanics to rotational motion.

Degeneracy

Degeneracy refers to the number of distinct quantum states that share the same energy level. In the rigid rotator, the degeneracy is given by a simple function of the quantum number, which must be accounted for when performing the summation in the partition function. This concept is key in determining the weight of each energy level's contribution to thermodynamic quantities.

High-Temperature (Classical) Limit and Integral Approximation

At high temperatures, many rotational states become thermally accessible, and the spacing between the energy levels can be considered small relative to kT. This allows the summation over discrete states in the partition function to be approximated by an integral. This continuum approximation simplifies the evaluation of the partition function, internal energy, and heat capacity, often leading to results compatible with classical statistical mechanics.

Internal Energy and Heat Capacity

Internal energy and heat capacity are macroscopic thermodynamic properties that can be derived from the partition function. The internal energy is obtained by taking the derivative of the logarithm of the partition function with respect to beta (or temperature), while the heat capacity is given by the temperature derivative of the internal energy. These quantities reveal how the system stores and transfers energy, offering insight into the underlying quantum statistics.

Low-Temperature Approximation

At low temperatures, the system is dominated by its lowest energy states because thermal excitation to higher levels is strongly suppressed. This allows for simplifying the partition function to include only the ground state and possibly a few low-lying excited states. The low-temperature analysis yields approximations for the partition function, internal energy, and heat capacity, highlighting the quantum mechanical suppression of thermal fluctuations.

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