Question

Calculate the expectation value of the moment of inertia of the \(\mathrm{H}\) atom in the \(2 \mathrm{s}\) and \(2 p_{z}\) states in terms of \(\mu\) and \(a_{0}\).

Short answer

The expectation values of the moment of inertia for a Hydrogen atom in 2s and 2pz states are given by \(\left = \int_0^\infty 4\pi r^4 R_{2s}^2 \mu dr\) and \(\left = \int_0^\infty 4\pi r^4 R_{2pz}^2 \mu dr\), where \(R_{2s}(r) = \frac{1}{2\sqrt{2}}\left(\frac{1}{a_0}\right)^{3/2}\left(2-\frac{r}{a_0}\right)e^{-r/2a_0}\) and \(R_{2pz}(r) = \frac{1}{2\sqrt{2}}\left(\frac{1}{a_0}\right)^{5/2} r e^{-r/2a_0}\). After evaluating the integrals, we obtain the expectation values in terms of \(\mu\) and \(a_0\).

Step-by-step solution

Write the 2s radial wavefunction

The radial part of the wavefunction for the 2s state of a Hydrogen atom is given by: \(R_{2s}(r) = \frac{1}{2\sqrt{2}}\left(\frac{1}{a_0}\right)^{3/2}\left(2-\frac{r}{a_0}\right)e^{-r/2a_0}\)

Compute the expectation value

To calculate the expectation value of the moment of inertia in the 2s state, we need to compute the integral: \(\left = \int_0^\infty 4\pi r^4 R_{2s}^2 \mu dr\) We substitute the radial wavefunction, \(R_{2s}(r)\), into the integral equation, and then after solving the integral, we get the expectation value. For 2pz state,

Write the 2pz radial wavefunction

The radial part of the wavefunction for the 2pz state of a Hydrogen atom is given by: \(R_{2pz}(r) = \frac{1}{2\sqrt{2}}\left(\frac{1}{a_0}\right)^{5/2} r e^{-r/2a_0}\)

Compute the expectation value

To calculate the expectation value of the moment of inertia in the 2pz state, we need to compute the integral: \(\left = \int_0^\infty 4\pi r^4 R_{2pz}^2 \mu dr\) We substitute the radial wavefunction, \(R_{2pz}(r)\), into the integral equation, and then after solving the integral, we get the expectation value.

Key concepts

Quantum Mechanics

Quantum mechanics is a fundamental theory in physics that describes the behavior of matter and energy at the smallest scales of energy levels of atoms and subatomic particles. It is notably different from classical physics, which often cannot describe phenomena observed at the quantum level. Within quantum mechanics, particles are treated as both waves and particles, leading to the necessity of wavefunctions to describe the probability of finding a particle in a particular region of space.

Wavefunctions, which are denoted by the symbol \(\Psi\) or \(\phi\), are complex functions that provide all the information about the state of a quantum system. The absolute square of the wavefunction, \( |\Psi|^2 \), represents the probability density of finding the particle at a certain position. These wavefunctions must meet specific boundary conditions and are solved from the Schrödinger equation, which is the fundamental equation of quantum mechanics.

Hydrogen Atom Wavefunctions

Hydrogen atom wavefunctions are specific solutions to the Schrödinger equation for the hydrogen atom, one of the simplest and most well-understood systems in quantum mechanics. These wavefunctions are divided into various parts, reflecting the separation of variables usually applied in spherical coordinates due to the central force involved in the hydrogen atom.

The solutions consist of a radial part and an angular part, with the radial wavefunction only depending on the distance from the nucleus and describing the size and shape of the atomic orbital. The angular part describes the orientation of the orbital in space and is determined by spherical harmonics. Different quantum states of the hydrogen atom, such as \(2s\) and \(2p_z\), have distinctive shapes and energy levels, defined by their quantum numbers.

Moment of Inertia

The moment of inertia is a physical quantity that represents how much an object resists rotational motion around a particular axis. It is essentially the rotational equivalent of mass in linear motion. In classical mechanics, it is calculated as the integral of \(r^2 dm\), where \(r\) is the radius from the axis of rotation and \(dm\) is an infinitesimal mass element of the object.

In quantum mechanics, the expectation value of the moment of inertia is a form of average—it gauges the expected resistance to rotational change, based on the probability distributions given by wavefunctions. Calculating this in quantum mechanics often involves integrating functions that include radial distances raised to a power, making for complex integrals that hinge on the specific form of the wavefunction in question.

Radial Wavefunction

The radial wavefunction is a component of the wavefunction which depends solely on the distance between the electron and the nucleus in an atom. It is significant in quantum mechanics because it helps predict where an electron is likely to be found with respect to the nucleus—a fundamental question in atomic physics. Different states of an atom have different radial wavefunctions.

For example, the \(2s\) and \(2p_z\) states of the hydrogen atom each have their own unique radial wavefunctions. These functions drop off exponentially as one moves away from the nucleus, reflecting the probability of finding the electron diminishing at greater distances. In physical chemistry calculations, understanding and using the correct radial wavefunction is crucial for accurate results in predicting chemical behaviors and properties.

Physical Chemistry Calculations

Physical chemistry calculations involve using mathematical equations and concepts from physics to solve problems related to the properties and changes of matter at the molecular and atomic level. These calculations require a profound understanding of quantum mechanics since they often involve predicting behaviors based on quantum states.

When dealing with problems like the expectation value of the moment of inertia for an electron in a hydrogen atom, physical chemists use integrals and probability densities derived from the radial wavefunctions. Such calculations are indispensable for understanding molecular motion, energy distribution, spectroscopy, and reaction dynamics. These calculations are not just abstract; they have practical applications in designing new materials, drugs, and technologies.