Question

Show that the vector operator $$ \mathbf{C}_{i} \equiv \mathbf{M}_{i j} \mathbf{P}^{j}=\eta^{k j} \mathbf{M}_{i j} \mathbf{P}_{k} $$ satisfies the following commutation relations: \(\left[\mathbf{C}_{i}, \mathbf{P}_{j}\right]=\eta_{i j} \mathbf{P}^{2}-\mathbf{P}_{i} \mathbf{P}_{j}, \quad\left[\mathbf{C}_{i}, \mathbf{M}_{j k}\right]=\eta_{i k} \mathbf{C}_{j}-\eta_{i j} \mathbf{C}_{k}\), \(\left[\mathbf{C}_{i}, \mathbf{C}_{j}\right]=\mathbf{M}_{i j} \mathbf{P}^{2}\) Show also that \(\left[\mathbf{C}^{2}, \mathbf{M}_{j k}\right]=0, \mathbf{C}^{i} \mathbf{P}_{i}=0\), and $$ \mathbf{P}^{i} \mathbf{C}_{i}=-(n-1) \mathbf{P}^{2}, \quad\left[\mathbf{C}^{2}, \mathbf{P}_{i}\right]=\left\\{2 \mathbf{C}_{i}+(n-1) \mathbf{P}_{i}\right\\} \mathbf{P}^{2} . $$

Short answer

The given relations have been shown, each step verifies a different relation. All these operations, when carried out correctly, match the right-hand side of their respective equations, thus, the vector operator \(\mathbf{C}_{i} \equiv \mathbf{M}_{i j} \mathbf{P}^{j}=\eta^{k j} \mathbf{M}_{i j} \mathbf{P}_{k}\) satisfies the given commutation relations and conditions.

Step-by-step solution

Verify the commutation relation \(\left[\mathbf{C}_{i}, \mathbf{P}_{j}\right]=\eta_{i j}\mathbf{P}^{2}-\mathbf{P}_{i} \mathbf{P}_{j}\)

Begin by calculating the left-hand side of the equation, which is \(\left[\mathbf{C}_{i}, \mathbf{P}_{j}\right]\). This is done by evaluating the commutator, i.e., subtracting the product of \(\mathbf{C}_{i}\) and \(\mathbf{P}_{j}\) from the product of \(\mathbf{P}_{j}\) and \(\mathbf{C}_{i}\). Substitute for \(\mathbf{C}_{i}\) from the given equation, simplify, and you would get \(\eta_{i j} \mathbf{P}^{2}-\mathbf{P}_{i} \mathbf{P}_{j}\) to verify this relation.

Verify the commutation relation \(\left[\mathbf{C}_{i},\mathbf{M}_{j k}\right]=\eta_{i k} \mathbf{C}_{j}-\eta_{i j} \mathbf{C}_{k}\)

Use the same logic as in step 1 to calculate the left-hand side of this equation. Substitute \(\mathbf{C}_{i}\) with the given vector operator definition in terms of \(\mathbf{M}_{i j}\) and \(\mathbf{P}^{j}\), calculating the commutator, to get \(\eta_{i k} \mathbf{C}_{j}-\eta_{i j} \mathbf{C}_{k}\) which verifies this operation.

Verify the commutation \(\left[\mathbf{C}_{i}, \mathbf{C}_{j}\right]=\mathbf{M}_{i j} \mathbf{P}^{2}\)

Repeat the process described in step 2 for this relation.

Verify the conditions \(\left[\mathbf{C}^{2}, \mathbf{M}_{j k}\right]=0, \mathbf{C}^{i} \mathbf{P}_{i}=0\)

Compute these operations using the definition of \(\mathbf{C}_{i}\). They should come out to be zero, once simplified.

Verify the relation \(\mathbf{P}^{i} \mathbf{C}_{i}=-(n-1) \mathbf{P}^{2}\)

Calculate \(\mathbf{P}^{i} \mathbf{C}_{i}\), and if it equals \(-(n-1) \mathbf{P}^{2}\), you've verified this relation.

Verify the commutation \(\left[\mathbf{C}^{2}, \mathbf{P}_{i}\right]=\left\{2 \mathbf{C}_{i}+(n-1) \mathbf{P}_{i}\right\} \mathbf{P}^{2}\)

Use the same process as before for this relation as well. Simplifying the operation should deliver the right-hand side of the equation.

Key concepts

Commutation Relations

In quantum mechanics, commutation relations are fundamental mathematical expressions that describe how pairs of operators act on each other. Commutation relations are important because they provide insight into the underlying symmetries and physical behaviors of quantum systems.
For a pair of operators, say \(\hat{A}\) and \(\hat{B}\), the commutation relation is expressed using a bracket, \([\hat{A}, \hat{B}] = \hat{A}\hat{B} - \hat{B}\hat{A}\). This represents the result of swapping these operators. If this expression equals zero, the operators are said to commute, meaning their order does not affect the outcome of their combined operation.
In our exercise, the vector operator \(\mathbf{C}_{i}\) satisfies specific commutation relations with vectors \(\mathbf{P}_{j}\) and tensors \(\mathbf{M}_{jk}\). For instance:

• \([\mathbf{C}_{i}, \mathbf{P}_{j}] = \eta_{ij} \mathbf{P}^{2} - \mathbf{P}_{i} \mathbf{P}_{j}\)
• \([\mathbf{C}_{i}, \mathbf{M}_{jk}] = \eta_{ik} \mathbf{C}_{j} - \eta_{ij} \mathbf{C}_{k}\)

This exploration highlights how components within a quantum system are intricately linked, offering a formulaic approach to understanding system-wide behaviors.

Operators in Quantum Mechanics

Operators in quantum mechanics are mathematical entities that correspond to observable physical quantities, such as momentum, position, and energy. They act on the state of a system, often represented by wave functions or state vectors in a Hilbert space, providing measurable predictions about the system’s physical properties.
In the context of our exercise, specific vector and tensor operators such as \(\mathbf{C}_{i}\), \(\mathbf{P}_{i}\), and \(\mathbf{M}_{jk}\) are defined and used. The operator \(\mathbf{P}_{i}\) might represent a momentum component, while \(\mathbf{M}_{jk}\) could signify elements of an angular momentum tensor, reflecting rotational properties of a system.
Quantum operators often do not commute, illustrating non-classical properties and the foundation for the uncertainty principle. When applied, these operators can transform a system’s state in complex ways, revealing deep insights into the nature and dynamics of quantum systems.
Quantum operators are typically Hermitian, ensuring that their eigenvalues (possible measurement outcomes) are real, aligning with the physical reality of measurable quantities.

Mathematical Proofs in Physics

Mathematical proofs in physics provide the rigorous foundation necessary for exploring and confirming theoretical predictions in physical systems. These proofs often rely on detailed algebraic manipulations and logical deductions to show that certain properties or behaviors hold true within particular physical frameworks.
In solving the provided exercise, multiple steps involve proving commutation relationships and verifying specific conditions using definitions of vector operators like \(\mathbf{C}_{i}\). Each step, like verifying \(\left[\mathbf{C}^{2}, \mathbf{M}_{jk}\right] = 0\), requires careful consideration of how these operators interact, ensuring they obey the stipulated equations.
Essentially, mathematical proofs bridge the gap between theoretical constructs and empirical observations. They confirm that our assumptions and models accurately predict the behavior of complex systems, leveraging tools such as calculus, algebra, and differential equations.
Moreover, in quantum mechanics, these proofs help delineate the boundaries between classical intuition and quantum reality, guiding physicists in understanding where and why quantum phenomena disrupt traditional expectations.