Problem 11 - Fundamentals of Physics

Question

If orbital angular momentum \(\vec{L}\) is measured along, say, \(a z\) axis to obtain a value for \(L_{z}\), show that $$ \left(L_{x}^{2}+L_{y}^{2}\right)^{1 / 2}=\left[\ell(\ell+1)-m_{\ell}^{2}\right]^{1 / 2} \hbar $$ is the most that can be said about the other two components of the orbital angular momentum.

Step-by-Step Solution

Answer

The most we can say is that \(\sqrt{L_x^2 + L_y^2} = \left[\ell(\ell+1) - m_\ell^2\right]^{1/2} \hbar\).

Step 1: Understand the Concept of Angular Momentum Components

Angular momentum \( \vec{L} \) of an object is a vector with components \( L_x \), \( L_y \), and \( L_z \). When \( L_z \) is known to have a value \( m_\ell \hbar \), the other components \( L_x \) and \( L_y \) are indeterminate within the confines set by quantum mechanics.

Step 2: Recall the Total Angular Momentum Formula

The total angular momentum is given by \( L^2 = \ell (\ell + 1) \hbar^2 \), where \( \ell \) is the orbital quantum number and \( \hbar \) is the reduced Planck's constant. This is a key relationship in quantum mechanics for angular momenta.

Step 3: Express the Condition on the Measured Angular Momentum Component

When \( L_z \) is measured, it has a definite value \( m_\ell \hbar \), providing \( L_z^2 = m_\ell^2 \hbar^2 \). This does not change when \( L_x \) and \( L_y \) remain unknown.

Step 4: Use Angular Momentum Magnitude Relation

Since \( L^2 \) is the sum of the squares of its components: \[ L^2 = L_x^2 + L_y^2 + L_z^2 \] and using \( L^2 = \ell(\ell+1)\hbar^2 \), substitute \( L_z^2 \) from above to get: \[ L_x^2 + L_y^2 = L^2 - L_z^2 \] \[ = \ell(\ell+1)\hbar^2 - m_\ell^2\hbar^2 \]

Step 5: Solve for \(\sqrt{L_x^2 + L_y^2}\)

Taking the square root from both sides gives \[ \sqrt{L_x^2 + L_y^2} = \left[ \ell(\ell+1) - m_\ell^2 \right]^{1/2} \hbar \]This expresses the magnitude of the combined indeterminate components of angular momentum \( L_x \) and \( L_y \).

Key Concepts

Quantum Mechanics

Quantum mechanics is a fundamental theory in physics that describes nature at the smallest scales, such as that of subatomic particles. It deviates from classical mechanics by introducing the concept that physical quantities, such as angular momentum, can be quantized. This means these quantities can only take on discrete values.

- **Particles as Waves**: In quantum mechanics, particles can exhibit wave-like behavior. This is exemplified by the wave function, which describes the probability distribution of a particle's position or momentum. - **Uncertainty Principle**: A key principle is the Heisenberg Uncertainty Principle, which states that certain properties, like position and momentum, cannot both be precisely measured at the same time. This principle applies to angular momentum as well.

In the context of angular momentum, quantum mechanics tells us that the angular momentum of particles is not continuous but rather quantized. The quantum mechanical description also restricts the information we can have about certain components of angular momentum, as demonstrated in this exercise.

Angular Momentum Components

Angular momentum in quantum mechanics has three components corresponding to three spatial dimensions: - **\(L_x\)** - **\(L_y\)** - **\(L_z\)**
Together, these components define the overall angular momentum vector \(\vec{L}\). However, due to quantum constraints, we can precisely measure only one of these components, typically \(L_z\), without the other components being affected or remaining unknown.

- **Indeterminacy of \(L_x\) and \(L_y\)**: When we measure \(L_z\), \(L_x\) and \(L_y\) are indeterminate. This means that their exact values are not just unknown; they cannot be precisely known. What quantum mechanics allows us to determine is just their combined magnitudinal effect, not their specific values.

- **Mathematics of Indeterminate Components**: The given formula \(\sqrt{L_x^2 + L_y^2} = \left[\ell(\ell+1) - m_\ell^2 \right]^{1/2} \hbar\) expresses a relationship. It shows that the square root of the sum of their squares gives a permissible solution within the framework defined by the orbital quantum number and the known values of \(L_z\).

Orbital Quantum Number

The orbital quantum number, represented usually as \(\ell\), is an important aspect of the quantum mechanical model of angular momentum. It relates to the shape of the orbital and describes the quantized nature of the angular momentum.

- **Quantized Values**: In quantum mechanics, \(\ell\) can take integer values from 0 up to \(n-1\) for any given principal quantum number \(n\). These integer values dictate the angular part of the wave function, fundamentally affecting the shape and energy of the electron orbitals.- **Role in Angular Momentum**: The value of \(\ell\) is directly related to the magnitude of the total angular momentum \(L\); the magnitude of \(L\) is given by \(\sqrt{\ell(\ell+1)}\hbar\).

The restrictions that \(\ell\) impose, ensure that the angular momentum's magnitude is quantized, consistent with other quantized properties in quantum physics. Thus, the orbital quantum number not only determines the shape of the electron's path around the nucleus but also its angular momentum magnitude, while the component along any chosen axis remains precisely defined only in part, as shown in the specific solution of this exercise.