Question

The statistical mechanics of a basketball. Consider a basketball of mass \(m=1 \mathrm{~kg}\) in a basketball hoop. To simplify, suppose the hoop is a cubic box of volume \(V=1 \mathrm{~m}^{3}\). (a) Calculate the lowest two energy states using the particle-in-a-box approach. (b) Calculate the partition function at \(T=300 \mathrm{~K}\). Show whether quantum effects are important or not. (Assume that they are important only if \(q\) is smaller than about \(10 .\) )

Short answer

The lowest two energy states are \(3.30 \times 10^{-68} \mathrm{~J}\) and \(6.60 \times 10^{-68} \mathrm{~J}\). The partition function is \(q = 2\). Quantum effects are important because \(q < 10\).

Step-by-step solution

Understand the Particle-in-a-Box Model

In the particle-in-a-box model, a particle is confined in a box with infinitely high walls. The energy levels of a particle in a three-dimensional box are given by the equation \[ E_{n_x, n_y, n_z} = \frac{h^2}{8mL^2} (n_x^2 + n_y^2 + n_z^2) \] where \(h\) is Planck's constant, \(m\) is the mass of the particle, \(L\) is the length of the side of the box, and \(n_x, n_y, n_z\) are quantum numbers (1, 2, 3, ...).

Calculate the First Energy Level

For the lowest energy state, \(n_x = n_y = n_z = 1\):\[ E_{111} = \frac{h^2}{8mL^2} (1^2 + 1^2 + 1^2) = \frac{3h^2}{8mL^2} \]Plugging in the values, \(h = 6.626 \times 10^{-34} \mathrm{~J} \cdot \mathrm{s}\), \(m = 1 \mathrm{~kg}\), and \(L = 1 \mathrm{~m}\) gives:\[ E_{111} = \frac{3 \times (6.626 \times 10^{-34})^2}{8 \times 1 \times (1)^2} = 3.30 \times 10^{-68} \mathrm{~J} \]

Calculate the Second Energy Level

For the next lowest energy state, higher quantum numbers such as \(n_x = 1\), \(n_y = 1\), and \(n_z = 2\) can be used:\[ E_{112} = \frac{h^2}{8mL^2} (1^2 + 1^2 + 2^2) = \frac{6h^2}{8mL^2} \]Using the same values for \(h\), \(m\), and \(L\):\[ E_{112} = \frac{6 \times (6.626 \times 10^{-34})^2}{8 \times 1 \times (1)^2} = 6.60 \times 10^{-68} \mathrm{~J} \]

Calculate the Partition Function

The partition function \(q\) at temperature \(T\) is given by:\[ q = \sum_{i} e^{-E_i / k_B T} \]Where \(k_B\) is the Boltzmann constant \(1.38 \times 10^{-23} \mathrm{~J/K}\) and \(T\) is the temperature. Using the first two energy levels calculated,\[ q \approx e^{-E_{111} / k_B T} + e^{-E_{112} / k_B T} \]For \(T = 300 \mathrm{~K}\):\[ q \approx e^{-3.30 \times 10^{-68} / (1.38 \times 10^{-23} \times 300)} + e^{-6.60 \times 10^{-68} / (1.38 \times 10^{-23} \times 300)} \]Given \(E_i \ll k_B T\), the exponential terms are approximately 1, so:\[ q \approx 1 + 1 = 2 \]

Determine Relevance of Quantum Effects

Quantum effects are considered important if \(q\) is smaller than 10. Calculated \(q = 2\), which is smaller than 10, therefore quantum effects are important in this scenario.

Key concepts

particle-in-a-box model

The particle-in-a-box model is a classic concept in quantum mechanics. It describes a particle confined within an idealized box with infinitely high walls. In such a situation, the particle can only occupy specific energy levels. The energy levels for a particle in a three-dimensional box are given by the formula: \( E_{n_x, n_y, n_z} = \frac{h^2}{8mL^2} (n_x^2 + n_y^2 + n_z^2)\) \( h\) is Planck's constant, \( m\) is the mass of the particle, and \( L\) is the length of the side of the box. \( n_x, n_y, n_z\) are the quantum numbers corresponding to each dimension. These numbers can only be positive integers (1, 2, 3, ...). By understanding this model, we can start to calculate the energy levels of the particle.

energy levels

Energy levels in the context of the particle-in-a-box model are discrete and depend on the quantum numbers \( n_x, n_y, \) and \( n_z \). For the basketball scenario:

• The lowest energy state is when all quantum numbers are 1 (\( n_x = n_y = n_z = 1\)). The corresponding energy is: \ ( E_{111} = \frac{3h^2}{8mL^2} = 3.30 \times 10^{-68} \text{J} ) \
• The next energy level can be found by slightly increasing one of the quantum numbers, like \( n_x = n_y = 1\) and \( n_z = 2 \), yielding: \( E_{112} = \frac{6h^2}{8mL^2} = 6.60 \times 10^{-68} \text{J}\)

These energy levels are essential for calculating the partition function and analyzing the system's quantum nature.

partition function

The partition function \( q\) is a key concept in statistical mechanics. It helps us connect microscopic properties (like energy levels) to macroscopic observables (like temperature). In our scenario, the partition function is given by:\( q = \sum_i{e^{-E_i / k_B T}} \) where \( k_B\) is the Boltzmann constant, and \( E_i\) is the energy level. For a temperature of 300 K and using the energy levels we calculated, we find:
\( q \approx e^{-3.30 \times 10^{-68} / (1.38 \times 10^{-23} \times 300)} + e^{-6.60 \times 10^{-68} / (1.38 \times 10^{-23} \times 300)} \). Given that \( E_i \ll k_B T\), the exponentials are approximately 1, leading to \( q\approx 2\). Quantum effects are important since \( q < 10\). This result shows the importance of discrete energy levels and quantum mechanics in the system's behavior.