Problem 36 - University Physics with Modern Physics

Question

(a) Show all the distinct states for an electron in the \(N\) shell of hydrogen. Include all four quantum numbers. (b) For an f-electron in the \(N\) shell, what is the largest possible orbital angular momentum and the greatest positive value for the component of this angular momentum along any chosen direction (the \(z\) -axis)? What is the magnitude of its spin angular momentum? Express these quantities in units of \(\hbar\) . (c) For an electron in the \(d\) state of the \(N\) shell, what are the maximum and minimum angles between its angular momentum vector and any chosen direction (the \(z\) -axis)? (d) What is the largest value of the orbital angular momentum for an \(f\) -electron in the \(M\) shell?

Step-by-Step Solution

Answer

(a) 32 states. (b) Largest orbital \(2\sqrt{3}\hbar\), component \(3\hbar\), spin \(\frac{\sqrt{3}}{2}\hbar\). (c) Angles are \(\cos^{-1}(1/\sqrt{6})\) to \(90^\circ\). (d) Orbital angular momentum is \(2\sqrt{3}\hbar\).

Step 1: Understanding Quantum Numbers

Electrons in an atom are described by four quantum numbers: the principal quantum number\(n\), the azimuthal quantum number \(l\), the magnetic quantum number \(m_l\), and the spin quantum number \(m_s\). For the \(N\) shell, \(n=4\). The possible values of \(l\) are \(0, 1, 2,\) and \(3\). Each \(l\) value corresponds to terms \(s, p, d,\) and \(f\) respectively for hydrogen-like atoms.

Step 2: Listing Distinct States of an Electron in N Shell

For each \(l\) value, the possible \(m_l\) values are \(-l, -(l-1), \ldots, l\). The spin quantum number \(m_s\) can be \(+\frac{1}{2}\) or \(-\frac{1}{2}\). Calculate the distinct states for each \(l\): \(s\) has 2 states, \(p\) has 6 states, \(d\) has 10 states, and \(f\) has 14 states, totaling 32 distinct states.

Step 3: Finding Orbital Angular Momentum for f-electron in N Shell

For \(l=3\) (f-electron), the orbital angular momentum \(|L|\) is given by \(|L| = \sqrt{l(l+1)}\hbar\). Substituting \(l\), we get \(|L| = \sqrt{3(4)}\hbar = 2\sqrt{3}\hbar \).

Step 4: Calculating Maximum Orbital Angular Momentum Component

The maximum \(z\)-component of orbital angular momentum is \(m_l\hbar\). For \(l=3\), the largest \(m_l\) is 3, so the maximum component is \(3\hbar\).

Step 5: Determining Spin Angular Momentum

Spin angular momentum \(|S|\) is defined as \(|S| = \sqrt{s(s+1)}\hbar\) where \(s=\frac{1}{2}\) for electrons. Therefore, \(|S| = \sqrt{\frac{1}{2}(\frac{3}{2})} \hbar = \frac{\sqrt{3}}{2}\hbar\).

Step 6: Finding Angular Momentum Angles for d-state

For \(l=2\), the minimum angle between the angular momentum vector and the \(z\)-axis is \(\cos^{-1}(\frac{|m_l|}{|L|})\). For maximum \(|m_l=2|\), this gives minimum angle as \(\cos^{-1}(1/\sqrt{6})\) and for \(m_l=0\), maximum angle is \(\cos^{-1}(0) = 90^\circ\).

Step 7: Calculating Largest Orbital Angular Momentum for f-electron in M Shell

For \(n=3\) in the \(M\) shell and \(l=3\), the largest orbital angular momentum \(|L|\) is \(|L| = \sqrt{l(l+1)}\hbar = 2\sqrt{3}\hbar\), same as in the \(N\) shell, as it depends only on \(l\).

Key Concepts

Principal Quantum Number

The principal quantum number, denoted by the symbol \(n\), is a fundamental concept in quantum mechanics. It signifies the main energy level or shell of an electron within an atom. In simpler terms, it's like the floor number where an electron "lives" within an atom. The principal quantum number dictates the overall size and energy of the electron's orbit:

• A larger \(n\) value implies a larger orbit and higher energy level.
• For hydrogen, the shells are often labeled with letters: K (\(n=1\)), L (\(n=2\)), M (\(n=3\)), and N (\(n=4\)).

In the provided exercise, the electron is situated in the \(N\) shell, meaning \(n = 4\). This indicates that the electron is on one of the higher energy levels, with three possible lower levels beneath it. The \(n\) value sets a foundation for determining the remaining quantum numbers that further define the electron's position and behavior within the atom.

Angular Momentum

Angular momentum in quantum mechanics is an essential property that describes the rotation of an electron around the nucleus of an atom. This property is quantified by the azimuthal quantum number, \(l\), which is also commonly referred to as the angular momentum quantum number. Here's what you should know:

• The value of \(l\) ranges from 0 to \(n-1\), where \(n\) is the principal quantum number.
• Each \(l\) value corresponds to specific electron subshells and shapes: \(s\) (\(l=0\)), \(p\) (\(l=1\)), \(d\) (\(l=2\)), \(f\) (\(l=3\)), and so on.
• The angular momentum of an electron provides insights into its orbital shape and stability within the atom.

For example, an f-electron in the \(N\) shell has \(l = 3\), which provides it with a substantial amount of orbital angular momentum. This is calculated using the formula \( \sqrt{l(l+1)}\hbar \), leading to significant effects on the electron's behavior.
This mathematical expression of angular momentum illustrates the intrinsic "spin" that electrons have due to their wave-like nature. Ultimately, analyzing angular momentum helps us understand the impact of electron motion on atomic properties.

Spin Quantum Number

The spin quantum number \(m_s\) is an intrinsic property of electrons that gives a measure of their angular momentum in a given direction, much like a spinning top. Unlike other quantum numbers, \(m_s\) is not derived from the electron's position but from its intrinsic nature:

• It has only two possible values: \(+\frac{1}{2}\) or \(-\frac{1}{2}\).
• These values reflect whether the electron spin is "up" or "down," affecting magnetic properties.

Although a small value, the spin quantum number significantly impacts the behavior of electrons due to quantum mechanical properties. In the exercise, the given spin angular momentum for electrons is calculated as \( \sqrt{\frac{1}{2}(\frac{3}{2})}\hbar = \frac{\sqrt{3}}{2}\hbar \).
This showcases the unique nature of quantum spin—it has no classical analog and remains one of the intriguing aspects of particle physics.
Understanding spin is crucial for fields such as quantum computing and magnetism, where electron alignment and behavior form the basis for advanced technological applications.

Magnetic Quantum Number

The magnetic quantum number, symbolized by \(m_l\), dictates the orientation of an electron's orbital in three-dimensional space. This number is crucial for understanding how electrons behave in the presence of a magnetic field:

• It can take values ranging from \(-l\) to \(+l\), including zero. For each value of \(l\), there are \(2l+1\) possible values of \(m_l\).
• Each value of \(m_l\) aligns with a specific orbital orientation.

For instance, in an \(f\) subshell, where \(l=3\), \(m_l\) can range from \(-3\) to \(+3\), allowing for seven different orientations. The exercise describes the maximum component of orbital angular momentum, which is calculated as \(3\hbar\) for an \(f\)-electron when \(m_l=3\). This number indicates the levels of degeneracy and affects the spectral lines in presence of external magnetic fields.
The magnetic quantum number helps provide understanding beyond just where an electron is, describing more explicitly its three-dimensional interaction and response to external forces. In essence, this enriches the portrayal of electron dynamics in an atom's complex environments.