Question

In its ground state, nitrogen's \(2 p\) electrons interact to pro duce \(j_{T}=\frac{3}{2}\). Given Hund's rule, how might the orbital angular momenta of these three electrons combine?

Short answer

Hund's rules dictate that each electron in a sub-shell occupy a unique orbital before any is doubly occupied and that they align their spins. Thus, in this case, the three \(2 p\) electrons' spins are aligned which gives rise to the angular momentum of \( j_{T} = \frac{3}{2} \).

Step-by-step solution

Understanding Hund's rule

Hund's rule: Every orbital in a sub-shell is singly occupied before any orbital is doubly occupied, and all electrons in singly occupied orbitals have their spins aligned. For the \(2p\) shell, this means there are three orbitals, which can be singly occupied before any is doubly occupied.

Determine the orbital angular momentum of each electron

The orbital angular momentum (l) of each electron in a P orbital is 1. Every electron has spin magnetic quantum number (ms = +/-1/2). With all three electrons having parallel spins, each contributes 1/2 units of angular momentum. We regard the three electrons as three vectors each of length 1/2 and try to combine them to get a resultant vector of length 3/2.

Combining the individual angular momenta

Considering the one electron in the z direction, it has an angular momentum of 1 along the z-direction. Adding the next electron's angular momentum, still 1 unit but in a direction making an angle with the z-direction, gives rise to a total angular momentum of magnitude \( \sqrt{1+1+2cos(\Theta)} = \sqrt{2(1+cos(\Theta))} \). This situation is most likely to occur when the angle \( \Theta \) between them is smallest i.e., they are parallel because the electrons would prefer to be as far apart as possible due to electron-electron repulsion. Hence, \( cos(\Theta) = 1 \), which gives us \( \sqrt{2(1+1)} = \sqrt{4} = 2 \). Therefore, as per Hund’s rules, the angular momentum (j) when two p-electrons combine is 2.

Combine the Total Angular Momentum

The third electron's spin adds to this to give a resultant \( j_{T} \) of 3/2. Hence, the three \(2 p\) electrons spin parallel to each other, giving rise to the Total Angular Momentum (\( j_{T} \)) = 3/2. This represents the coupling of the three electrons' orbital angular momenta, conforming to Hund's rule.

Key concepts

Understanding Quantum Mechanics

Quantum mechanics is a fundamental theory in physics that describes the behavior of energy and matter on the atomic and subatomic scale. In this theory, particles such as electrons are not depicted as precise points, but rather as a cloud of possibilities, each with its own probability. It introduces the wave-particle duality, where objects can exhibit both wave-like and particle-like characteristics.

To grasp quantum mechanics, one must be comfortable with its probabilistic nature and the idea that exact predictions of certain measurements are not always possible. Instead, we can determine the probability of finding a particle in a particular state or position. This departure from classical physics requires an understanding of concepts like uncertainty principles and quantization, where certain properties, such as energy and angular momentum, can only take on discrete values known as quanta.

Additionally, quantum mechanics incorporates complex mathematical functions and operators to appropriately describe the quantum states of particles. These are vital in predicting the outcomes of experiments and comprehending the behavior of particles at the quantum level, an area where our everyday intuition does not always serve us well.

Exploring Orbital Angular Momentum

Orbital angular momentum is one of the quintessential elements in quantum mechanics, referring to the angular momentum of an electron as it 'orbits' around the nucleus of an atom. It comes from the classical concept of angular momentum which is seen in objects rotating around a fixed point, but in the quantum realm, it takes on a more abstract meaning.

Unlike in the macroscopic world where any rotation can occur, in quantum mechanics, the orbital angular momentum for electrons in an atom is quantized. This means that electrons can only possess certain 'allowed' values of angular momentum, determined by quantum numbers. For the electron in a P orbital, as referred to in the exercise, this quantum number (l) has a value of 1, which signifies the angular momentum level.

The quantization also means that the angular momentum vector can only have certain orientations in space. This is particularly important when considering how multiple electrons within an atom interact with each other, which brings us to Hund's rule and the arrangement of electrons in orbitals to minimize repulsion.

Unveiling Electron Spin

The concept of electron spin is another vital principle in quantum mechanics and represents a form of intrinsic angular momentum carried by electrons, independent of any actual 'spinning' motion, since quantum particles do not rotate in the conventional sense.

Electron spin is quantized, with electrons having a spin quantum number (\( s \) of 1/2. This results in two possible orientations for the spin: 'up' (\( m_{s} = +1/2 \) and 'down' (\( m_{s} = -1/2 \) which coincide with the possible spin angular momentum values that an electron can exhibit. These two spin states are fundamentally important for understanding the magnetic properties of atoms and the way electrons fill up the available electron shells and subshells within an atom.

When applying Hund's rule, as presented in the exercise, we see how spin plays a significant role in electron configuration. Electrons will fill orbitals so that their spins are aligned before pairing up, which maximizes the total spin and minimizes repulsion between the electrons. This principle helps explain the electronic structure of atoms, the nature of chemical bonds, and the behaviour of materials in a magnetic field.