Problem 4 - College Physics

Question

An electron is in the hydrogen atom with \(n = 5 .\) (a) Find the possible values of \(L\) and \(L _ { z }\) for this electron, in units of \hbar. (b) For each value of \(L ,\) find all the possible angles between \(L\) and the \(z\) axis. (d) What are the maximum and minimum values of the magnitude of the angle between \(L\) and the \(z\) axis?

Step-by-Step Solution

Answer

Possible \( L \) values are 0, \( \sqrt{2} \hbar \), \( \sqrt{6} \hbar \), \( \sqrt{12} \hbar \), \( \sqrt{20} \hbar \). Max angle is 90°, min angle varies by \( l \).

Step 1: Determine Values of l for n = 5

For an electron in a hydrogen atom, the principal quantum number is denoted as \( n \). The azimuthal or angular momentum quantum number \( l \) can take integer values from 0 to \( n-1 \). Thus, for \( n = 5 \), the possible values for \( l \) are \( 0, 1, 2, 3, \) and \( 4 \).

Step 2: Calculate L for Each l

The magnitude of the angular momentum \( L \) is given by the formula \( L = \sqrt{l(l+1)}\hbar \). Placing each value of \( l \) from step 1 into this formula, we calculate:- For \( l = 0 \), \( L = \sqrt{0(0+1)}\hbar = 0 \hbar \).- For \( l = 1 \), \( L = \sqrt{1(1+1)}\hbar = \sqrt{2}\hbar \).- For \( l = 2 \), \( L = \sqrt{2(2+1)}\hbar = \sqrt{6}\hbar \).- For \( l = 3 \), \( L = \sqrt{3(3+1)}\hbar = \sqrt{12}\hbar \).- For \( l = 4 \), \( L = \sqrt{4(4+1)}\hbar = \sqrt{20}\hbar \).

Step 3: Determine Possible Values of Lz for Each l

The magnetic quantum number \( m_l \), which gives \( L_z \), can take integer values from \( -l \) to \( +l \). Thus, \( L_z = m_l\hbar \). For each \( l \), we have:- For \( l = 0 \), \( L_z = 0 \hbar \) (since \( m_l = 0 \)).- For \( l = 1 \), \( L_z = -1\hbar, 0\hbar, +1\hbar \).- For \( l = 2 \), \( L_z = -2\hbar, -1\hbar, 0, +1\hbar, +2\hbar \).- For \( l = 3 \), \( L_z = -3\hbar, -2\hbar, -1\hbar, 0, +1\hbar, +2\hbar, +3\hbar \).- For \( l = 4 \), \( L_z = -4\hbar, -3\hbar, -2\hbar, -1\hbar, 0, +1\hbar, +2\hbar, +3\hbar, +4\hbar \).

Step 4: Calculate Cosine of Angle Between L and z-axis

The angle \( \theta \) between \( \mathbf{L} \) and the \( z \)-axis is given by \( \cos \theta = \frac{L_z}{L} \). Using the \( L \) and \( L_z \) values from the previous steps, you calculate \( \theta \) for each \( l \) value.- For each \( L \) calculated in Step 2 and corresponding \( L_z \) from Step 3, compute \( \theta \) using \( \cos \theta = \frac{L_z}{L} \).

Step 5: Identify Maximum and Minimum Values for Angle

The cosine values range from \( -1 \) to \( +1 \). As \( L_z \) varies, the maximum angle \( \theta \) corresponds to the smallest \( |\cos \theta| \) and the minimum angle \( \theta \) corresponds to largest \( |\cos \theta| \). For given \( l \), the maximum and minimum \( \theta \) correspond to the extremes of \( m_l \): - Maximum angle for each \( l \) corresponds to positive or negative \( m_l = \pm l \) (\( \cos \theta = \pm 1 \)). - Minimum angle corresponds to \( m_l = 0 \) (\( \cos \theta = 0 \)).

Key Concepts

Angular Momentum

Angular momentum is a fundamental concept in quantum mechanics, especially when analyzing the behavior of electrons in atoms. It refers to the rotational momentum of a particle, which in the case of an electron, is influenced by both its orbital motion and intrinsic spin.
In quantum mechanics, the angular momentum of an electron is quantized, which means it can only take on certain distinct values. The total angular momentum is described by the quantum number \( l \), known as the azimuthal or angular momentum quantum number.
For an electron in an atom, the magnitude of angular momentum \( L \) is calculated using the formula:\[L = \sqrt{l(l+1)}\hbar\]where \( \hbar \) is the reduced Planck's constant. This quantization introduces a rich structure and set of rules that electrons must follow, influencing their distribution around a nucleus.

Quantum Numbers

Quantum numbers are essential in quantum mechanics, as they provide a descriptive framework for electrons in atoms. There are four main quantum numbers that describe an electron's energy level, position, and spin:

• The principal quantum number \( n \) indicates the energy level and relative size of the electron's orbit.
• The azimuthal (or angular momentum) quantum number \( l \) defines the shape of the electron's orbital and is dependent on \( n \).
• The magnetic quantum number \( m_l \) describes the orientation of the orbital in space, with values ranging from \(-l\) to \(+l\).
• The spin quantum number \( m_s \) specifies the electron's intrinsic spin direction, which can be either \(+\frac{1}{2}\) or \(-\frac{1}{2}\).

These quantum numbers determine the allowable electron configurations, ensuring they adhere to the Pauli exclusion principle, where no two electrons in an atom can possess identical quantum numbers simultaneously.

Hydrogen Atom

The hydrogen atom is the simplest atom and a fundamental topic in quantum mechanics. It consists of just one proton as its nucleus, with a single electron orbiting around it.
The behavior and energy states of the electron in a hydrogen atom are well understood through the solutions to Schrödinger's equation, which predicts quantized energy levels. These discrete energy states are described by the principal quantum number \( n \), with higher \( n \) values corresponding to increasingly higher energy levels.
Hydrogen's electron configuration is directly influenced by its quantum numbers, providing insights into more complex atomic structures and molecular behavior. This makes hydrogen an import model for understanding more complicated multi-electron atoms.

Electron Configuration

Electron configuration is a representation of the distribution of an atom's electrons in its atomic orbitals. It follows a set of rules derived from the quantum numbers, which ensure that electrons fill the lowest available energy levels first (known as the Aufbau principle).
For the hydrogen atom, with only one electron, the electron configuration is simple: 1s¹. However, as atoms become more complex, a systematic approach using quantum numbers ensures correct electron placement, avoiding rule violations like the Pauli exclusion principle and Hund's rule.
This configuration plays a critical role in determining an atom's chemical properties and reactions, influencing how atoms bond and form compounds. Mastery of electron configurations is crucial for predicting and understanding the behavior of elements in the periodic table.