Problem 40 - College Physics

Question

An electron in hydrogen is in the 5\(f\) state. (a) Find the largest possible value of the \(z\) component of its angular momentum. (b) Show that for the electron in part (a), the corresponding \(x\) and \(y\) components of its angular momentum satisfy the equation \(\sqrt { L _ { x } ^ { 2 } + L _ { y } ^ { 2 } } = \hbar \sqrt { 3 }\)

Step-by-Step Solution

Answer

(a) Largest \(L_z\) is \(3\hbar\). (b) \(\sqrt{L_x^2 + L_y^2} = \hbar\sqrt{3}\).

Step 1: Understand the Quantum Numbers

The electron is in the 5\(f\) state. For an electron in a hydrogen atom, the principal quantum number \(n=5\) and the azimuthal (orbital) quantum number \(l\) for an \(f\) state is 3 (since \(s\), \(p\), \(d\), and \(f\) correspond to \(l = 0, 1, 2, 3\) respectively). The magnetic quantum number \(m_l\) can take values from \(-l\) to \(+l\).

Step 2: Determine the Largest \(z\) Component of Angular Momentum

The largest possible value of \(m_l\) is \(l = 3\). The \(z\) component of angular momentum, \(L_z\), is given by \(m_l \hbar\). Thus, the largest \(L_z\) is \(L_z = 3\hbar\).

Step 3: Calculate Total Angular Momentum

The magnitude of the total angular momentum \(L\) is given by the formula \(L = \sqrt{l(l+1)}\hbar\). For \(l = 3\), \(L = \sqrt{3(3+1)}\hbar = 2\sqrt{3}\hbar\).

Step 4: Calculate \(L_x\) and \(L_y\) Components

To find \(L_x\) and \(L_y\), use the relation \(L^2 = L_x^2 + L_y^2 + L_z^2\). With \(L_z = 3\hbar\), then \((L_x^2 + L_y^2) = L^2 - L_z^2 = (2\sqrt{3}\hbar)^2 - (3\hbar)^2\).

Step 5: Simplify and Verify the Given Equation

Calculate \((2\sqrt{3}\hbar)^2 = 12\hbar^2\) and \((3\hbar)^2 = 9\hbar^2\). Thus, \(L_x^2 + L_y^2 = 12\hbar^2 - 9\hbar^2 = 3\hbar^2\). Therefore, \(\sqrt{L_x^2 + L_y^2} = \hbar\sqrt{3}\), verifying that \(\sqrt{L_x^2 + L_y^2} = \hbar\sqrt{3}\).

Key Concepts

Angular Momentum

Angular momentum in quantum mechanics is a fundamental concept that relates to the rotational symmetry of a system. In the context of an electron in an atom, it deals with the motion of the electron around the nucleus. Angular momentum is quantized, meaning it can only take specific values. This is described using quantum numbers, which help define the state of an electron in an atom.

The three components of angular momentum are typically denoted as \(L_x\), \(L_y\), and \(L_z\). In the quantum mechanical view, the exact position and velocity of an electron can't be known simultaneously, leading to an inherent uncertainty in the values of these components. However, the magnitude of angular momentum \(L\) can be determined using the formula \(L = \sqrt{l(l+1)}\hbar\), where \(l\) is the azimuthal quantum number and \(\hbar\) is the reduced Planck constant.

For an electron in the hydrogen atom, this angular momentum arises due to its orbital motion around the nucleus. Calculating the \(z\) component's maximum value reveals how angular momentum affects the possible orientations of these orbitals.

Quantum Numbers

Quantum numbers are integral to understanding the state of electrons in atoms. These numbers provide a set of parameters that systematically describe the unique quantum state of an electron.

• Principal Quantum Number (\(n\)): Dictates the energy levels and size of the electron orbitals, with higher numbers representing orbitals that are farther from the nucleus.
• Azimuthal Quantum Number (\(l\)): Determines the shape of the orbital. Different values indicate different subshells (\(s, p, d, f\)), corresponding to \(l = 0, 1, 2, 3\), respectively.
• Magnetic Quantum Number (\(m_l\)): Specifies the orientation of the orbital in space, with values ranging between \(-l\) and \(+l\).

In our exercise, an electron in a 5\(f\) state has a principal quantum number \(n = 5\) and an azimuthal quantum number \(l = 3\). From this, one can determine that the possible \(m_l\) values are \(-3, -2, -1, 0, 1, 2, 3\). The largest \(z\) component of angular momentum corresponds to the largest \(m_l\), which is \(3\).

The quantum numbers, therefore, play a critical role in determining the quantum states and transition possibilities of electrons.

Hydrogen Atom

The hydrogen atom serves as a classic example in quantum mechanics due to its simplicity of having only one electron. It provides significant insights into atomic structures and electron behavior due to the relatively simple interactions between its electron and proton.

In the hydrogen atom, quantum numbers play a crucial role in defining the states of the electron. The energy levels of the electron are primarily determined by the principal quantum number \(n\). Each energy level corresponding to \(n\) consists of subshells unique to each azimuthal quantum number \(l\).

The hydrogen atom model helps us understand the distribution of electron probability densities in various orbitals. For a given principal quantum number, different values of \(l\) and \(m_l\) create distinct shapes and orientations of orbitals. These shapes govern the likelihood of finding an electron in a particular volume of space around the nucleus.

Exploring an electron in different states, like the 5\(f\) state, helps demonstrate how these quantum probabilities manifest in real-world electron configurations.

Electron States

Electron states in quantum mechanics describe the potential energy levels and spatial distribution of electrons within an atom. Each state is a unique combination of quantum numbers, which specify the electron's characteristics and behavior in its atom.

For instance, in the hydrogen atom, the 5\(f\) electron state includes a principal quantum number of 5 and an azimuthal quantum number of 3. This configuration marks the electron's presence within a high-energy and complex shaped orbital. The orbitals characterize where electrons are most likely to be found, and they are not fixed paths but rather probability zones.

The quantization of electron states imposes limits on the possible positions and energies that an electron can have. It is beyond classical physics, where any energy value would be theoretically possible. Quantum mechanics allows the precise calculation of the probabilities of electrons occupying certain states, which is foundational for modern chemistry and physics.

Understanding these electron states is critical for interpreting atomic transitions that occur during phenomena like light emission and absorption, and it reveals the intricate nature of atomic and subatomic processes.