Problem 48 - University Physics with Modern Physics

Question

(a) What is the lowest possible energy ( in electron volts) of an electron in hydrogen if its orbital angular momentum is \(\sqrt{12 \hbar ?}\) (b) What are the largest and smallest values of the \(z\) -component of the orbital angular momentum (in terms of \(\hbar )\) for the electron in part (a)? (c) What are the largest and smallest values of the spin angular momentum (in terms of \(\hbar\) ) for the electron in part (a)? (d) What are the largest and smallest values of the orbital angular momentum (in terms of \(\hbar\) ) for an electron in the \(M\) shell of hydrogen?

Step-by-Step Solution

Answer

(a) -0.85 eV; (b) 3ℏ, -3ℏ; (c) ±0.5ℏ (spin); (d) √6ℏ, 0 ℏ (M-shell).

Step 1: Determine the Orbital Angular Momentum Quantum Number

The orbital angular momentum is given by \( L = \sqrt{l(l+1)} \hbar \). We know \( L = \sqrt{12} \hbar \), so we equate: \( \sqrt{l(l+1)} = \sqrt{12} \). Solving for \( l(l+1) \), we get \( l(l+1) = 12 \). The integers \( l \) and \( l+1 \) that satisfy this equation are \( l=3 \) and \( l+1=4 \).

Step 2: Find the Lowest Possible Energy Level (Principal Quantum Number)

The smallest value of the principal quantum number \( n \) such that \( l < n \) is \( n = 4 \). The energy of an electron in a hydrogen atom is given by \( E_n = -\frac{13.6}{n^2} \text{ eV} \). Substituting \( n = 4 \), we find \( E_4 = -\frac{13.6}{4^2} = -\frac{13.6}{16} = -0.85 \text{ eV} \).

Step 3: Largest and Smallest Z-component of Orbital Angular Momentum

The z-component of the orbital angular momentum is given by \( L_z = m_l \hbar \), where \( m_l \) ranges from \( -l \) to \( l \). Therefore, the maximum value is \( L_{z,max} = 3\hbar \) and the minimum value is \( L_{z,min} = -3\hbar \).

Step 4: Largest and Smallest Value of Spin Angular Momentum

Electrons have a fixed spin quantum number of \( s = \frac{1}{2} \). The spin angular momentum is given by \( S = \sqrt{s(s+1)} \hbar = \sqrt{\frac{1}{2}(\frac{1}{2}+1)} \hbar = \sqrt{\frac{3}{4}} \hbar = \frac{\sqrt{3}}{2} \hbar \). The z-component is \( S_z = m_s \hbar = \pm \frac{1}{2} \hbar \), giving maximum and minimum values of \( \frac{1}{2}\hbar \) and \( -\frac{1}{2}\hbar \).

Step 5: Orbital Angular Momentum in the M-shell

In the M-shell (\( n=3 \)), the possible values of \( l \) are \( 0, 1, 2 \). The largest orbital angular momentum is \( L = \sqrt{2(2+1)} \hbar = \sqrt{6} \hbar \) for \( l=2 \), and the smallest is \( L = 0 \) for \( l=0 \).

Key Concepts

Orbital Angular Momentum

In quantum mechanics, orbital angular momentum represents the motion of electrons around the nucleus of an atom, akin to the classical idea of an orbit but in a quantum context. It's quantified by the quantum number \( l \), where \( l \) can be any integer from 0 to \( n-1 \), with \( n \) being the principal quantum number. Angular momentum \( L \) is calculated with the formula \( L = \sqrt{l(l+1)} \hbar \), where \( \hbar \) is a reduced Planck constant used in quantum calculations.
For example, in the exercise, an electron in a hydrogen atom is given an orbital angular momentum of \( \sqrt{12} \hbar \), which suggests \( l(l+1) = 12 \). The solution for \( l \) calculates to 3, indicating a somewhat complex orbital path involving higher energy levels, typical in outer electron shells.

Spin Angular Momentum

Electrons possess an intrinsic form of angular momentum called spin, which is distinct from the angular momentum due to their motion around the nucleus. This spin is quantified by the spin quantum number, \( s \), which is always \( \frac{1}{2} \) for electrons.
The spin angular momentum \( S \) is computed using \( S = \sqrt{s(s+1)} \hbar \). This calculation results in \( S = \frac{\sqrt{3}}{2} \hbar \). The z-component of spin angular momentum can take only two possible values, \( S_z = \pm \frac{1}{2} \hbar \), reflecting the binary nature of electron spin, often referred to as "spin-up" and "spin-down."
This intrinsic property is crucial as it significantly influences magnetic properties and interactions between particles.

Quantum Numbers

Quantum numbers are fundamental to describing the state of an electron in an atom. They provide a unique identifier for electrons, defining their energy level, shape, orientation, and spin.
There are four types of quantum numbers:

• **Principal quantum number (\( n \))**: Determines the energy level or shell and can take any positive integer value.
• **Azimuthal quantum number (\( l \))**: Describes the shape of the orbital and ranges from 0 to \( n-1 \).
• **Magnetic quantum number (\( m_l \))**: Defines the orientation of the orbital in space, with possible values from \( -l \) to \( l \).
• **Spin quantum number (\( s \))**: Represents the intrinsic spin of the electron, which is \( \pm \frac{1}{2} \).

These numbers collectively determine the electron's location and energy within an atom, with the configuration influencing chemical behavior and bonding.

Principal Quantum Number

The principal quantum number, represented by \( n \), is the first and most critical quantum number in determining an electron's energy in an atom. It essentially labels the energy level or shell that an electron occupies.
In the exercise, regarding the calculations around \( l \) and energy, the smallest possible \( n \) where \( l < n \) was found to be 4. This means the electron is in the fourth energy level, which is further from the nucleus and higher in energy.
The principal quantum number directly affects the electron's energy, following the formula for hydrogen: \( E_n = -\frac{13.6}{n^2} \text{ eV} \). In this equation, the energy becomes more negative (and thus lower in absolute value) as \( n \) increases, indicative of less tightly bound electrons located further from the nucleus. For \( n=4 \), the energy calculates to \( -0.85 \text{ eV} \).
This concept is pivotal in understanding atomic structure and why electrons reside in varied shells, affecting atoms' chemical properties and interactions.