Are the angular momentum vectors \(\overrightarrow{\mathbf{L}}\) and \(\overrightarrow{\mathbf{S}}\) necessarily aligned?

A body in motion has two types of angular momentum: orbital angular momentum (\(\overrightarrow{\mathbf{L}}\)) and spin angular momentum (\(\overrightarrow{\mathbf{S}}\)). Orbital angular momentum is associated with the movement of a particle around a fixed point, such as an electron orbiting a nucleus, while spin angular momentum is inherent in a particle, much like the rotation of the particle around its own axis.

Two vectors are considered aligned when they are either parallel or antiparallel. In other words, they point in the same or opposite direction. To determine whether \(\overrightarrow{\mathbf{L}}\) and \(\overrightarrow{\mathbf{S}}\) are aligned, we need to analyze the relationship between these two vectors in any given situation.

It is important to note that the alignment of angular momentum vectors depends on the specific system under consideration. In some cases, \(\overrightarrow{\mathbf{L}}\) and \(\overrightarrow{\mathbf{S}}\) may be aligned, while in other cases they may not be aligned. Let's analyze two examples: 1. Planetary motion: In this case, the orbital angular momentum vector \(\overrightarrow{\mathbf{L}}\) points out of the plane of the orbit (in the direction of the angular momentum), while the spin angular momentum vector \(\overrightarrow{\mathbf{S}}\), which is associated with the planet's rotation about its axis, points towards the planet's north or south pole. In general, these two vectors are not aligned unless the planet's rotational axis is perpendicular to the plane of its orbit. 2. Atomic systems: In atomic systems, the orbital angular momentum of an electron (\(\overrightarrow{\mathbf{L}}\)) and its spin angular momentum (\(\overrightarrow{\mathbf{S}}\)) can couple to form a total angular momentum vector (\(\overrightarrow{\mathbf{J}}\)). This total angular momentum vector can have several possible values depending on the quantum numbers associated with the system. In certain cases, the total angular momentum vector is dominated by either \(\overrightarrow{\mathbf{L}}\) or \(\overrightarrow{\mathbf{S}}\), making them effectively aligned. However, in other cases, the total angular momentum vector does not lie along the direction of either \(\overrightarrow{\mathbf{L}}\) or \(\overrightarrow{\mathbf{S}}\), meaning that they are not aligned.

The alignment of the angular momentum vectors \(\overrightarrow{\mathbf{L}}\) and \(\overrightarrow{\mathbf{S}}\) is not guaranteed for all systems. Whether they are aligned or not depends on the specific system and the conditions within that system.