Problem 2 - University Physics with Modern 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. (c) 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: 0, \(\sqrt{2}\hbar\), \(\sqrt{6}\hbar\), \(2\sqrt{3}\hbar\), \(2\sqrt{5}\hbar\). Angle range: 0° to 180°. Maximum/minimum: 90° (max), 0°/180° (min).

Step 1: Determine Possible Values of L

For an electron in a hydrogen atom, the quantum number \( n \) determines the principal energy level. Given \( n=5 \), the azimuthal quantum number \( l \) can take values from 0 to \( n-1 \). Thus, the possible values of \( l \) are 0, 1, 2, 3, and 4. The orbital angular momentum \( L \) is given by \( L = \sqrt{l(l+1)} \hbar \), so we calculate \( L \) for each value of \( l \).

Step 2: Calculate L for Each l Value

- For \( l=0 \): \( L = \sqrt{0(0+1)} \hbar = 0 \).- 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 \( l=3 \): \( L = \sqrt{3(3+1)} \hbar = 2\sqrt{3} \hbar \).- For \( l=4 \): \( L = \sqrt{4(4+1)} \hbar = \sqrt{20} \hbar = 2\sqrt{5} \hbar \).

Step 3: Determine Possible Values of Lz

The magnetic quantum number \( m_l \) can have integer values ranging from \( -l \) to \( l \), inclusive. Each value of \( m_l \) gives a possible value of \( L_z = m_l \hbar \). Calculating for each \( l \): - For \( l=0 \): \( m_l=0 \Rightarrow L_z = 0 \)- For \( l=1 \): \( m_l = -1, 0, 1 \Rightarrow L_z = -\hbar, 0, \hbar \)- For \( l=2 \): \( m_l = -2, -1, 0, 1, 2 \Rightarrow L_z = -2\hbar, -\hbar, 0, \hbar, 2\hbar \) - For \( l=3 \): \( m_l = -3, -2, -1, 0, 1, 2, 3 \Rightarrow L_z = -3\hbar, -2\hbar, -\hbar, 0, \hbar, 2\hbar, 3\hbar \)- For \( l=4 \): \( m_l = -4, -3, -2, -1, 0, 1, 2, 3, 4 \Rightarrow L_z = -4\hbar, -3\hbar, -2\hbar, -\hbar, 0, \hbar, 2\hbar, 3\hbar, 4\hbar \)

Step 4: Calculate Possible Angles for each L

The angle \( \theta \) between \( \mathbf{L} \) and the \( z \)-axis is given by \( \cos \theta = \frac{L_z}{L} \). For each value of \( l \), calculate possible \( \theta \) values by using each \( L_z \) calculated. For example, - For \( l = 1 \), \( L = \sqrt{2} \hbar \), possible angles:\( \cos \theta = \frac{-\hbar}{\sqrt{2} \hbar}, \frac{0}{\sqrt{2} \hbar}, \frac{\hbar}{\sqrt{2} \hbar} \Rightarrow \theta = 135^\circ, 90^\circ, 45^\circ \)Repeat similarly for other values of \( l \).

Step 5: Identify Maximum and Minimum Angles

The maximum angle occurs when \( L_z = 0 \), i.e., \( \theta = 90^\circ \) (since \( \cos \theta = 0 \)). The minimum angle occurs when \( L_z = \pm L \), hence \( \theta = 0^\circ \) when aligned with \( L \) along \( z \), and \( \theta = 180^\circ \) when anti-aligned.

Key Concepts

Hydrogen Atom

The hydrogen atom holds a special place in quantum mechanics. It's the simplest atom, consisting of just one proton and one electron, making it an ideal candidate to apply quantum theories. Understanding this atom allows us to explore the broader principles of quantum mechanics, which explain atomic and subatomic processes.
In the quantum model, the hydrogen atom is described by a set of quantum numbers, each describing a specific property of the electron's orbit. The electron moves in distinct energy levels or "shells," defined by the principal quantum number, denoted as \( n \).
The hydrogen atom model is crucial as it introduced the concept of quantization - the idea that energy levels are discrete rather than continuous. It employs the Schrödinger equation to predict the electron's behavior in these energy levels.

Quantum Numbers

Quantum numbers are like an address system for electrons within an atom. They describe the electron's position and movement in a very precise way. There are four quantum numbers, each playing a unique role.

• Principal quantum number \( n \): Determines the energy level and size of the orbital. It is a positive integer.
• Azimuthal quantum number \( l \): Defines the orbital shape and angular momentum. It ranges from 0 to \( n-1 \).
• Magnetic quantum number \( m_l \): Describes the orientation of the orbital. It ranges from \(-l\) to \(+l\).
• Spin quantum number \( m_s \): Represents the electron's spin and can be either \(+\frac{1}{2}\) or \(-\frac{1}{2}\).

When electrons occupy orbitals, these numbers ensure that no two electrons have the same set of all four quantum numbers, adhering to the Pauli exclusion principle.

Angular Momentum

Angular momentum in quantum mechanics refers to the momentum due to the electron's orbit around the nucleus. It's denoted by \( L \) and often linked with the electron's wave-like properties.
In the context of the hydrogen atom, orbital angular momentum is quantized, meaning it can only take specific values determined by the azimuthal quantum number \( l \). The magnitude of the angular momentum is calculated using the formula \( L = \sqrt{l(l+1)} \hbar \), where \( \hbar \) is the reduced Planck's constant.
This quantization shows that classical physics doesn't apply on the atomic scale. Instead, electrons exist in specific states defined by these quantum properties, illustrating that atomic particles behave quite differently from everyday objects.

Azimuthal Quantum Number

The azimuthal quantum number, \( l \), is crucial in defining the shape and type of orbital an electron occupies. It goes hand in hand with the principal quantum number \( n \) and dictates the subshell within a given shell.
For a specific energy level \( n \), \( l \) ranges from 0 to \( n-1 \). Each value of \( l \) corresponds to a different type of orbital:

• \( l = 0 \) is an s-orbital, spherical in shape.
• \( l = 1 \) is a p-orbital, dumbbell-shaped.
• \( l = 2 \) is a d-orbital, exhibiting complex shapes.
• \( l = 3 \) is an f-orbital, even more complex.

These orbitals are an essential part of understanding electron arrangements and chemical behavior, as well as the concept of atomic shape and structure in quantum chemistry.