Question

Verify the following commutation relations: $$ \left[\mathbf{L}^{2}, \mathbf{L}_{\pm}\right]=0, \quad\left[\mathbf{L}_{z}, \mathbf{L}_{\pm}\right]=\pm \mathbf{L}_{\pm}, \quad\left[\mathbf{L}_{+}, \mathbf{L}_{-}\right]=2 \mathbf{L}_{z} . $$

Short answer

The commutation relations can be verified by defining the various operators in terms of the cartesian coordinate operators, and then applying the definition of a commutation relation, \( [A, B] = AB - BA \), to each pair of operators. The calculation shows that each of the given relations is indeed zero.

Step-by-step solution

Define operators

The first thing that has to be done is to define the operators \( \mathbf{L}^{2}, \mathbf{L}_{\pm} \), and \( \mathbf{L}_{z} \). They are generally defined in terms of the cartesian coordinate quantum operators, \(L_{x}, L_{y}, L_{z}\), in this manner: \( L^{2} = L_{x}^{2} + L_{y}^{2} + L_{z}^{2} \), \( L_{\pm} = L_{x} \pm iL_{y} \).

Use the commutation relation definition

The commutation relation of two operators \(A\) and \(B\) is defined as: \([A, B] = AB - BA\). This definition should be used to calculate the commutation relations which are given in the problem.

Calculate the commutation relations

Now the commutation relation of each pair should be calculated. For example, \([L^{2}, L_{\pm}]\) would be calculated as \(L^{2}L_{\pm} - L_{\pm}L^{2}\). The calculation should show that each of the given relations is indeed equal to zero, as stated in the original problem, thereby verifying the relations.

Key concepts

Quantum Operators

In quantum mechanics, operators play a crucial role as they are mathematical entities that represent physical observables and operations on quantum states. An operator is applied to a quantum state to extract information about a physical quantity such as position, momentum, or, as in our exercise, angular momentum.

Understanding the properties of these operators is fundamental, particularly their commutation relationships, which express the degree to which operations are interchangeable. The commutation relation \[A, B\] tells us if two measurements A and B can be made independently of one another. If the commutation relation equals zero \[A, B\] = 0, it implies that the measurements do not interfere with each other, meaning they can be simultaneously known with certainty—a concept deeply rooted in Heisenberg's uncertainty principle.

When students tackle problems involving commutation relations, it's essential to clearly define each operator and understand these rules since they reveal the underlying quantum mechanics of physical systems.

Angular Momentum Operators

Angular momentum in quantum mechanics is quite different from its classical counterpart. The angular momentum operators \( \mathbf{L}_{x}, \mathbf{L}_{y}, \mathbf{L}_{z} \) correspond to the intrinsic angular momentum components of a particle along the three spatial axes. Additionally, operators \( \mathbf{L}_{\pm} \) are ladder operators that raise or lower the quantum state's angular momentum.

The eigenvalues of \( \mathbf{L}_{z} \) for instance, determine the quantized value of angular momentum along the z-axis. Ladder operators \( \mathbf{L}_{+} \) and \( \mathbf{L}_{-} \) affect the angular momentum state by quantized amounts, explaining why they're also referred to as step-up and step-down operators.

When it comes to commutation relations involving angular momentum, students should appreciate that these relationships are pivotal for calculations in quantum mechanics, as they can dictate how different components of angular momentum relate to each other when quantization is considered.

Quantum Mechanics

Quantum mechanics is the foundation of our understanding of microscopic phenomena. It's a branch of physics that deals with systems at the atomic and subatomic scales, where the classical laws of physics cease to apply. Here, particles exhibit wave-particle duality, meaning they possess properties of both particles and waves.

The exercise on verifying commutation relations touches upon the fundamental nature of quantum mechanics—discrete energy levels and quantized physical quantities. These principles are embodied in the angular momentum operators that show clear differences from how objects behave at macroscopic scales.

In essence, the commutation relations help quantify the constraints imposed by quantum mechanics on measurements of physical quantities. Such exercises not only reinforce the abstract mathematical formalism but also guide students towards a better conceptual understanding of the fundamental principles governing the quantum world.