Problem 8 - 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 \(\vec{L}\) and the \(z\) -axis. (c) What are the maximum and minimum values of the magnitude of the angle between \(\vec{L}\) and the \(z\) -axis?

Step-by-Step Solution

Answer

Possible values of \(L\) are for \(l=0\) to 4; compute \(\theta\) using \(\cos \theta = \frac{L_z}{L}\). Maximum angle at \(L_z = 0\), minimum at \(L_z = \pm L\).

Step 1: Determine possible values of L

The quantum number for angular momentum, \(L\), is determined by the formula \(L = \sqrt{l(l+1)}\, \hbar\), where \(l\) is the azimuthal quantum number and \(l\) can take integer values from \(0\) to \(n-1\). Since \(n = 5\), \(l\) can be \(0, 1, 2, 3,\) or \(4\). Consequently, \(L\) can take the values \(\sqrt{0(1)}\, \hbar, \sqrt{1(2)}\, \hbar, \sqrt{2(3)}\, \hbar, \sqrt{3(4)}\, \hbar,\) and \(\sqrt{4(5)}\, \hbar\).

Step 2: Find possible values for Lz

The quantum number for the z-component of angular momentum, \(L_z\), is \(m \hbar\), where \(m\) ranges from \(-l\) to \(l\), including 0. For each \(l\), \(L_z\) can take values \(-l \hbar, -(l-1) \hbar, ..., 0, ..., (l-1) \hbar, l \hbar\).

Step 3: Calculate angles between L and the z-axis

The angle \(\theta\) between \(\vec{L}\) and the \(z\)-axis is found using the relation \(\cos \theta = \frac{L_z}{L}\). For each \(l\), use the possible \(L_z\) values to calculate \(\cos \theta\) and then solve for \(\theta\).

Step 4: Determine maximum and minimum angles

The maximum angle occurs when \(|L_z|\) is minimized (\(L_z = 0\)), giving the largest \(\theta\), and the minimum angle occurs when \(|L_z|\) is maximized (\(|L_z| = L\)), giving \(\theta = 0\). Calculate these angles for the different values of \(l\).

Key Concepts

Angular Momentum

In quantum mechanics, angular momentum is a crucial property of particles that tells us about their rotational motion. Unlike classical physics, where angular momentum can take any real value, quantum mechanics dictates that angular momentum is quantized. This means that it comes in discrete values. The total angular momentum, represented by \(L\), is calculated using the formula \(L = \sqrt{l(l+1)}\, \hbar\), where \(l\) is the azimuthal quantum number.

• \(\hbar\) is a fundamentally important constant in quantum mechanics, known as the reduced Planck's constant.
• The quantum number \(l\) can range from 0 up to \(n-1\), where \(n\) is the principal quantum number.

Each value of \(l\) determines a different possible value for \(L\). This quantization leads to interesting and specific behaviors of particles, especially electrons within atoms.

Quantum Numbers

Quantum numbers are like an address system for electrons within an atom, specifying a particular electron's position and properties. They are essential to understanding the electron configurations and describing the unique quantum state of an electron.

• The principal quantum number \(n\) specifies the energy level or shell of an electron, ranging from 1 to infinity.
• The azimuthal quantum number \(l\) relates to the angular momentum of the electron and affects its orbital shape.
• The magnetic quantum number \(m\) defines the orientation of the orbital in space and can range between \(-l\) and \(l\), including 0.
• Additionally, there is the spin quantum number, which can either be \(+1/2\) or \(-1/2\), describing the intrinsic spin of the electron.

Each set of these numbers gives us a comprehensive way to describe and predict the behavior of electrons in an atom.

Hydrogen Atom

The hydrogen atom is often used as a simple model in quantum mechanics because it consists of only one electron orbiting a single proton. This simplicity allows for a clear exploration of fundamental quantum concepts without the added complications present in larger atoms.

• In hydrogen, the principal quantum number \(n\) indicates the electron's energy level, which increases as the electron moves farther from the nucleus.
• The quantum numbers related to the electron's angular momentum and magnetic properties also offer predictions about its position and behavior.

This model demonstrates how quantum numbers and quantization lead to defined, discrete energy levels, often visualized as concentric circles around the nucleus representing energy shells.

Azimuthal Quantum Number

The azimuthal quantum number, represented by \(l\), plays a vital role in defining the shape of an electron's orbital. It is also directly tied to the angular momentum of the electron. For any given principal quantum number \(n\), \(l\) can take an integer value from 0 up to \(n-1\).

• When \(l = 0\), the orbital is spherical in shape, often referred to as an \(s\)-orbital.
• When \(l = 1\), the orbital shapes into paraboloids, commonly referred to as \(p\)-orbitals, indicating a greater angular variation than \(s\)-orbitals.
• Higher values of \(l\) correspond to \(d\)- and \(f\)-orbitals, each with more complex shapes.

This number also influences the energy levels within the same principal quantum level. Different shapes dictated by \(l\) provide distinct probable locations for finding an electron, contributing to the rich structure of atoms.