Problem 22 - University Physics with Modern Physics

Question

(a) What is the angular momentum \(L\) of the electron in a hydrogen atom, with respect to an origin at the nucleus, when the atom is in its lowest energy level? (b) Repeat part (a) for the ground level of He'. Compare to the answer in part (a).

Step-by-Step Solution

Answer

Both hydrogen and He⁺ have zero angular momentum in their ground state.

Step 1: Understanding Angular Momentum in Quantum Mechanics

The angular momentum \( L \) of an electron in an atom is quantized and given by the formula \( L = \sqrt{l(l + 1)} \hbar \), where \( \hbar \) is the reduced Planck's constant and \( l \) is the azimuthal quantum number. In the lowest energy level (ground state), also known as the \( n=1 \) state, \( l = 0 \).

Step 2: Calculate Angular Momentum for Hydrogen Atom

For a hydrogen atom in its ground state, \( l = 0 \). The angular momentum \( L \) is calculated as \( L = \sqrt{0(0 + 1)} \hbar = \sqrt{0} \hbar = 0 \hbar \). Thus, \( L = 0 \).

Step 3: Calculate Angular Momentum for Hein Ground State

For a He⁺ ion (singly ionized helium) in its ground state, the principal quantum number \( n \) is still 1, as it only contains one electron similar to hydrogen. Therefore, \( l = 0 \) for the lowest energy level. The angular momentum \( L \) is \( L = \sqrt{0(0 + 1)} \hbar = 0 \hbar = 0 \). Hence, \( L = 0 \).

Step 4: Compare Angular Momentum Between Hydrogen and He⁺

Both the hydrogen atom and the He⁺ ion, when in their ground states, have an azimuthal quantum number \( l = 0 \). This results in zero angular momentum \( L = 0 \) for both systems.

Key Concepts

Quantized Angular Momentum

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.

Azimuthal Quantum Number

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’s 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⁺ 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.

Ground State Hydrogen Atom

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⁺ 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.