Question

Equation \((9-27)\) gives the density of states for a system of oscillators but ignores spin. The result, simply one state per energy change of \(\hbar \omega_{0}\) between levels. is incorrect if particles are allowed diff erent spin states at each level. but modification to include spin is easy. From Chapter 8 , we know that a particle of spin s is allowed \(2 s+1\) spin orientations, so the number of states at each level is simply multiplied by this factor. Thus, $$ D(E)=(2 s+1) / \hbar \omega_{0} $$ (a) Using this density of states, the definition $$ \begin{array}{l} \text { Nheud }(2 s+1)=\delta, \text { and } \\ \qquad N=\int_{0}^{\infty} \mathcal{N}(E) D(E) d E \end{array} $$ calculate the parameter \(B\) in the Boltzmann distribution \((9-31)\) and show that the distribution can thus be tewritten as $$ \mathcal{N}(E)_{\text {Bolu }}=\frac{\varepsilon}{k_{\mathrm{B}} T} \frac{1}{e^{E / \mathrm{L}_{\mathrm{B}} T}} $$ (b) Algue that if \(k_{\mathrm{B}} T \gg \delta\), the occupation number is much less than I for all \(E\).

Short answer

The Boltzmann distribution can be rewritten as \[\mathcal{N}(E)_{\text {Bolu }}=\frac{\varepsilon}{k_{\mathrm{B}} T} \frac{1}{e^{E / \mathrm{L}_{\mathrm{B}} T}}\] under the consideration of spin of the particles. In the limit of high temperatures where \(k_B T >> \delta\), the equation implies the occupation number is much less than 1 for all \(E\).

Step-by-step solution

Understanding the density of states \[ D(E) = \frac{(2s+1)}{\hbar \omega_0} \]

In a system of oscillators where the particles can have different spin states at each energy level, each particle with spin \(s\) is allowed to have \(2s +1\) spin orientations. The equation represents the density of states which is the number of quantum states available to be occupied.

Applying the definition of total number of oscillators

The total number of oscillators \(N\) is given by \[N=\int_{0}^{\infty} \mathcal{N}(E) D(E) d E\] where \(\mathcal{N}(E)\) is the quantity characterizing the number of particles at each energy level, and \(D(E)\) is the density of states. Plug the definition of \(D(E)\) into the equation and integrate.

Calculating the parameter \(B\)

Even though not given directly in the exercise, we get the form of distribution function from the equation \(\mathcal{N}(E) = B e^{-E / k_{B}T}\). Here \(B\) can be calculated by setting the condition that the total population is conserved. Replace \(\mathcal{N}(E)\) with \(B e^{-E / k_BT}\) in the integral for \(N\) and solve it for \(B\). Substituting \(B\) in the equation for \(\mathcal{N}(E)\), we get the final formula.

Interpreting the distribution at high temperatures

It is argued that if \(k_B T >> \delta\), the occupation number is much less than 1 for all \(E\). This can be proved by expanding the exponential in the derived Boltzmann distribution and ignoring high order terms for small \(E / k_BT\).

Key concepts

Quantum Oscillators

In the realm of quantum mechanics, oscillators are a foundational concept used to describe and understand various physical systems. A quantum oscillator typically refers to a quantum system that exhibits wave-like behavior, and it includes the quantization of energy levels. The simplest example of a quantum oscillator is the quantum harmonic oscillator, which models the energy of a particle trapped in a potential well with forces that restore it to an equilibrium position.

A crucial aspect of quantum oscillators is that the energy levels are discrete, meaning particles can only inhabit specific energy states, with the difference between these states being proportional to the fundamental frequency of the oscillator, represented by \( \hbar \omega_0 \). This discrete nature of quantum systems stands in contrast to classical systems where the oscillators' energy levels can vary continuously.

Understanding quantum oscillators is essential as they serve as prototypes for more complex quantum systems, and their energy levels can be affected by factors like spin orientations. This influence on energy levels leads to an important concept called the density of states, which quantifies the number of quantum states available at each energy level, factoring in particle spin.

Spin Orientations

Spin, a fundamental property of quantum particles, describes the intrinsic angular momentum that particles possess. For a particle with spin \( s \), which can be a fraction or whole number, there exist a finite number, precisely \( 2s+1 \) possible spin orientations that the particle can exhibit. From an atomic perspective, these orientations can be perceived as the 'directions' in which a particle's spin can be aligned.

When considering systems such as quantum oscillators, recognizing the impact of spin is crucial because it multiplies the number of available states for a particle to occupy. Therefore, when calculating the density of states, \( D(E) \) for a system, each energy level must be modified by this factor to reflect the correct number of quantum states, leading to the modified density of states expression \( D(E) = \frac{(2s+1)}{\hbar \omega_0} \). This accounts for all possible orientations the spin can take, thereby impacting statistical properties like distribution functions in quantum statistics.

As such, spin plays a non-trivial role in many quantum phenomena and is vital to consider when predicting and explaining the behavior of systems at the quantum level.

Boltzmann Distribution

The Boltzmann distribution is a cornerstone of statistical mechanics which describes the distribution of particles over different energy states in thermal equilibrium. It is essential for understanding how energy is distributed among particles in a system at a given temperature. The basic assumption of the Boltzmann distribution is that the probability of finding a particle in a particular state is proportional to \( e^{-E / k_{B}T} \), where \( E \) is the energy of the state, \( k_B \) is the Boltzmann constant, and \( T \) is the temperature.

To compute any statistical mechanical property of a system, you need to determine the parameter \( B \) in the Boltzmann distribution, which ensures that the total number of particles \( N \) is preserved across all energy states, as per the principle of conservation of particles. The exercise involves integrating the distribution over all energy states, weighted by the density of states, to find \( B \), thereby obtaining the explicit Boltzmann distribution function for the system in consideration.

The relationship between temperature and energy levels, as described by the Boltzmann distribution, is particularly insightful. At high temperatures, where \( k_B T \) is much greater than the energy difference between states (\( \delta \)), the occupation numbers are significantly less than one. This implies that each state is sparsely occupied, which aligns with the classical prediction that energy is more evenly spread across states when the system has more thermal energy available.