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In quantum mechanics, angular momentum is not continuous but quantized. This means it can only take on specific discrete values. For electrons in atoms, the quantized nature of angular momentum ensures that certain rotational states are allowed. The formula for calculating the angular momentum is given by \[ L = \sqrt{l(l + 1)} \hbar \]where:

• \( L \) is the angular momentum.
• \( l \) is the azimuthal quantum number.
• \( \hbar \) is the reduced Planck's constant.

The quantization signifies that for each electron within an atom, the angular momentum is restricted to these set values depending on the quantum number \( l \). As a consequence, the electron's behavior exhibits discrete rather than continuous changes relative to angular momentum levels.

The azimuthal quantum number, represented by \( l \), is crucial to understanding the electron's angular momentum. It determines the shape of the orbital the electron occupies and is directly related to its angular momentum. This quantum number depends on the principal quantum number \( n \), with values ranging from 0 to \( n-1 \). Here鈥檚 how it works:

• \( l = 0 \) corresponds to an s orbital (spherical shape).
• \( l = 1 \) corresponds to a p orbital (dumbbell shape).
• \( l = 2 \) corresponds to a d orbital, and so forth.

In the context of the hydrogen atom or singly ionized helium, the ground state or lowest energy level occurs at \( n = 1 \). Here, \( l \) can only be zero, which means the electron does not possess angular momentum in this state. Thus, both hydrogen and He鈦� ions, in their ground states, have no angular momentum due to the quantum number \( l = 0 \). This absence of orbital angular momentum is a direct consequence of the properties defined by the azimuthal quantum number.

The ground state of an atom is its lowest energy level. For a hydrogen atom, this is known as the \( n=1 \) state. In this state, the quantum numbers defining an electron's position and behavior are very specific. The principal quantum number \( n = 1 \) entails that the azimuthal quantum number \( l \), which determines angular momentum, must be 0. This implies:

• The electron is in the s orbital, a spherical region around the nucleus.
• The angular momentum \( L \) is zero since \( l = 0 \).

The implications of this are profound as it means the electron's behavior is bound by quantization in such a way that its angular motion is non-existent in the ground state. This concept is directly translated to the He鈦� ion as well, where despite being a different element, it shares this ground state angular momentum property due to having similar energy level configurations (singly ionized shares \( n=1 \)). Therefore, understanding the ground state of hydrogen offers insights into foundational quantum mechanics principles.