Question

Consider a particle free to translate in one dimension. The classical Hamiltonian is \(H=p^{2} / 2 m\). a. Determine \(q_{\text {classical}}\) for this system. To what quantum system should you compare it in order to determine the equivalence of the classical and quantum statistical mechanical treatments? b. Derive \(q_{\text {classical}}\) for a system with translational motion in three dimensions for which \(H=\left(p_{x}^{2}+p_{y}^{2}+p_{z}^{2}\right) / 2 m\).

Short answer

The classical partition function \(q_{\text{classical}}\) for a one-dimensional system with Hamiltonian \(H = \frac{p^2}{2m}\) is \(\infty\). For a three-dimensional system with Hamiltonian \(H = \frac{p_x^2 + p_y^2 + p_z^2}{2m}\), the classical partition function is given by \(\infty^3\times\left(\frac{2m\pi}{\beta}\right)^{\frac{3}{2}}\), which is not finite. The classical and quantum mechanical treatments are not equivalent in these cases.

Step-by-step solution

Recall the classical partition function formula

The classical partition function, q_classical, is given by the following integral: \(q_{classical} = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} e^{-\beta H(q, p)} dq dp\) where \(\beta = 1/k_B T\). Here, the Hamiltonian H is a function of position 'q' and momentum 'p'; 'k_B' is the Boltzmann constant, and 'T' is the temperature.

Substitute the Hamiltonian for a 1D system

The one-dimensional Hamiltonian is given as: \(H = \frac{p^2}{2m}\) Substituting this into the partition function formula, we get: \(q_{classical} = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} e^{-\beta \frac{p^2}{2m}} dq dp\)

Perform integration and determine q_classical

The above integral can be split into two separate integrals as follows: \(q_{classical} = \int_{-\infty}^{\infty} e^{-\beta \frac{p^2}{2m}}dp \int_{-\infty}^{\infty} dq\) The integral over 'q' is a trivial one, and it should evaluate to infinity because there are no limits on the particle's position in one dimension: \(\int_{-\infty}^{\infty} dq = \infty\) The integral over 'p' is a Gaussian integral, which evaluates to: \(\int_{-\infty}^{\infty} e^{-\beta \frac{p^2}{2m}}dp = \sqrt{\frac{2m\pi}{\beta}}\) Multiplying both integrals, we get: \(q_{classical} = \sqrt{\frac{2m\pi}{\beta}} \times \infty = \infty\) Comparing this with quantum systems, such as the infinite square well, the classical partition function is not finite; thus, classical and quantum mechanical treatments will not be equivalent in this case. #Part_b#

Substitute the Hamiltonian for a 3D system

The three-dimensional Hamiltonian is given as: \(H = \frac{p_x^2 + p_y^2 + p_z^2}{2m}\) Substituting this into the partition function formula, we get: \(q_{classical} = \int e^{-\beta \frac{(p_x^2 + p_y^2 + p_z^2)}{2m}} dq_x dq_y dq_z dp_x dp_y dp_z\)

Perform integrations

Like before, we can separate the integrations. This time there will be three position integrals and three momentum integrals. Each position integral is unbounded and will evaluate to infinity. Each momentum integral is again a Gaussian integral: \(q_{classical} = \left(\int_{-\infty}^{\infty} dq_x \right) \left( \int_{-\infty}^{\infty} dq_y \right) \left( \int_{-\infty}^{\infty} dq_z \right) \left( \int_{-\infty}^{\infty} e^{-\beta \frac{p_x^2}{2m}}dp_x \right) \left( \int_{-\infty}^{\infty} e^{-\beta \frac{p_y^2}{2m}}dp_y \right) \left( \int_{-\infty}^{\infty} e^{-\beta \frac{p_z^2}{2m}}dp_z \right)\) Thus, we get: \(q_{classical} = (\infty) (\infty) (\infty) \left(\sqrt{\frac{2m\pi}{\beta}}\right) \left(\sqrt{\frac{2m\pi}{\beta}}\right) \left(\sqrt{\frac{2m\pi}{\beta}}\right)\) Simplifying we now receive: \(q_{classical} = \infty^3 \times \left(\frac{2m\pi}{\beta}\right)^{\frac{3}{2}}\) The classical partition function for the given three-dimensional system is thus proportional with cubic infinity times the factor \(\left(\frac{2m\pi}{\beta}\right)^{\frac{3}{2}}\).

Key concepts

Statistical Mechanics

Statistical mechanics is a branch of theoretical physics that uses probability theory to study the average behavior of a mechanical system where the state of the system is uncertain. This discipline is particularly important for explaining the thermodynamic behavior of large systems. In the context of our exercise, statistical mechanics helps us to compute the partition function, which plays a crucial role in determining the macroscopic properties of the system from its microscopic constituents.

The partition function, denoted as \( q_{classical} \), is a sum over all possible states of the system, each weighted by the Boltzmann factor \( e^{-\beta H(q, p)} \), where \(\beta = 1/k_B T\) is the inverse temperature multiplied by the Boltzmann constant, \(H(q, p)\) is the Hamiltonian representing the total energy of a state, 'q' represents coordinates, and 'p' represents momenta. For a classical system where positions and momenta can take any continuous values, this sum turns into an integral over all possible values of \((q, p)\).

Hamiltonian Mechanics

Hamiltonian mechanics is a reformulation of classical mechanics that predicts the same outcomes as Newtonian mechanics but works with different mathematical tools. It is especially useful for problems in statistical mechanics.

In Hamiltonian mechanics, the Hamiltonian function \(H\) represents the total energy of the system, which is the sum of kinetic and potential energies. For a particle in free space, like in our exercise, the Hamiltonian depends only on the kinetic energy since there is no potential. The kinetic energy for one-dimensional motion is given by \( \frac{p^2}{2m} \) for momentum \(p\) and mass \(m\), and it's straightforwardly extended for three-dimensional motion to \( \frac{p_x^2 + p_y^2 + p_z^2}{2m} \).

The classical partition function involves integrating over this Hamiltonian function multiplied by the Boltzmann factor, and in the absence of potential energy, it simplifies the problem significantly. However, we see from the exercise that this simplification results in an infinite value when the system's degrees of freedom are not confined.

1D and 3D Translational Motion

Translational motion refers to the movement of a particle or system of particles in space. In our exercise, we are considering a particle free to move in either one dimension (1D) or three dimensions (3D).

In the case of 1D motion, the particle is free to move along a single axis, and the partition function reflects this unconstrained motion as it integrates over all possible positions and momenta along that axis. The result, as indicated by the exercise, is an infinite partition function due to the integration over all possible positions.

Similarly, for 3D motion, the particle can move in three mutually perpendicular directions, and the partition function calculation requires integrating over all possible positions and momenta in three-dimensional space. Again, the contributions from the spatial integrals are infinite for free particles. Only the momentum integrals give finite Gaussian results, but when multiplied by the infinite spatial integrals, the overall partition function becomes infinite.

The infinite values in the classical partition function for free translational motion in both 1D and 3D hint at the limitations of classical physics in describing quantum systems. Quantum mechanics introduces a discrete nature to energy states, which often results in a finite partition function, whereas the classical approach leads to divergences when systems are not confined.