Problem 5 - Physics for Scientists and Engineers with Modern Physics

Question

(I) A hydrogen atom has \(\ell=5 .\) What are the possible values for \(n, m_{\ell},\) and \(m_{s} ?\)

Step-by-Step Solution

Answer

\( n \geq 6 \), \( m_{\ell} = -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5 \), \( m_s = \pm \frac{1}{2} \).

Step 1: Identify quantum numbers' relationships

In quantum mechanics, the values of the quantum numbers are interrelated. The azimuthal quantum number \( \ell \) determines the shape of the electron's orbital within an atom.

Step 2: Determine possible values for principal quantum number \( n \)

The principal quantum number \(n\) must be greater than \( \ell \). Since \( \ell = 5 \), the minimum value for \( n \) is 6. Hence, \( n \) can take on any integer value from 6 upwards, but commonly the smallest value is used in many contexts. So, \( n = 6 \) is the most likely choice.

Step 3: Identify possible values for magnetic quantum number \( m_{\ell} \)

The magnetic quantum number \( m_{\ell} \) determines the orientation of the orbital. Its possible values range from \(-\ell\) to \(+\ell \). Therefore, for \( \ell = 5 \), \( m_{\ell} \) can take any integer value from \(-5\) to \(+5\), inclusive.

Step 4: Analyze possible values for spin magnetic quantum number \( m_{s} \)

The spin magnetic quantum number \( m_s \) determines the electron spin direction. It can have a value of either \(-\frac{1}{2}\) or \(+\frac{1}{2}\), regardless of other quantum numbers.

Key Concepts

Principal Quantum Number

The principal quantum number, denoted as \( n \), is fundamental in quantum mechanics as it describes the energy level of an electron in an atom. It is an integer that determines the size and energy of an electron's orbit. The higher the value of \( n \), the larger and more energetic the electron's orbit. This number must always satisfy the condition \( n > \ell \), where \( \ell \) is the azimuthal quantum number. In the case of \( \ell = 5 \), the smallest possible value for \( n \) is 6. Consequently, \( n \) can be any integer starting from 6, but in practical terms, we often consider \( n = 6 \) as the first energy level for this case.

The principal quantum number plays a crucial role in:

• Determining the total number of orbitals within an energy level, which is \( n^2 \).
• Influencing the atom's electron configuration and corresponding chemical properties.

Understanding \( n \) is essential for predicting how electrons behave and interact in different energy states.

Magnetic Quantum Number

The magnetic quantum number, represented as \( m_{\ell} \), relates to the orientation of an electron's orbital in three-dimensional space. Each value of \( m_{\ell} \) corresponds to a different orientation of the orbital for a given azimuthal quantum number \( \ell \).

For a given \( \ell \), \( m_{\ell} \) can take on any integer value from \(-\ell\) to \(+\ell\). In the scenario where \( \ell = 5 \), \( m_{\ell} \) has a range of possible values from \(-5\) to \(+5\). Hence, there are 11 possible values for \( m_{\ell} \), reflecting the different orientations that the electron orbitals can assume.

The magnetic quantum number is significant because it:

• Diversifies the spatial distribution of electrons in orbitals.
• Influences how atoms interact with magnetic fields, contributing to their magnetic properties.

Understanding \( m_{\ell} \) is vital for visualizing the actual spatial orientation of atomic orbitals.

Spin Magnetic Quantum Number

The spin magnetic quantum number, denoted as \( m_{s} \), describes the intrinsic spin of an electron within an atom. Unlike other quantum numbers, \( m_{s} \) is independent of \( n \), \( \ell \), or \( m_{\ell} \). Instead, it specifically concerns the electron's spin, which can be visualized as the electron's rotation around its own axis.

The possible values for \( m_{s} \) are \(+\frac{1}{2}\) and \(-\frac{1}{2}\). These values represent the two fundamental spin states that an electron can have, often described as "spin-up" and "spin-down."

The spin magnetic quantum number is particularly important because:

• It influences the magnetic properties of electrons and atoms.
• Determines the electron's behavior according to the Pauli Exclusion Principle, which states that no two electrons in the same atom can have identical sets of quantum numbers.

Comprehending \( m_{s} \) helps explain the unique behavior of electrons in magnetic fields and their interactions in multi-electron systems.