Question

Use the defining relation \(\mathbf{L}_{i}=\epsilon_{i j k} x_{j} \mathbf{p}_{k}\) to show that \(x_{j} \mathbf{p}_{k}-x_{k} \mathbf{p}_{j}=\) \(\epsilon_{i j k} \mathbf{L}_{i} .\) In both of these expressions a sum over the repeated indices is understood.

Short answer

By expanding the right-hand side (RHS) of the given expression using the definition of angular momentum and the properties of the Levi-Civita symbol and Kronecker delta, it can be shown to be equivalent to the expression on the left-hand side (LHS), \(x_{j}\mathbf{p}_{k}-x_{k}\mathbf{p}_{j}\), proving the given statement.

Step-by-step solution

Expand the RHS

Begin by expanding the right-hand side (RHS) of the given expression. In accordance with the definition of angular momentum, replace \( \mathbf{L}_{i} \) with \( \epsilon_{ilm} x_{l}\mathbf{p}_{m} \). Now the RHS becomes \( \epsilon_{ijk} \epsilon_{ilm} x_{l} \mathbf{p}_{m} \).

Use the properties of Levi-Civita symbol

Use the property of the Levi-Civita symbol which states that \( \epsilon_{ijk} \epsilon_{ilm} = \delta_{jl} \delta_{km} - \delta_{jm} \delta_{kl} \), where \( \delta_{ij} \) is the Kronecker delta that equals 1 if \( i=j \), and 0 otherwise. Applying this, you get \( (\delta_{jl} \delta_{km} - \delta_{jm} \delta_{kl}) x_{l} \mathbf{p}_{m} \).

Evaluate the expression

Now, evaluate the resulting expression. The Kronecker delta symbols \( \delta_{jl} \) and \( \delta_{km} \) can be used to replace \( l \) with \( j \), and \( m \) with \( k \), respectively. This results in \( x_{j}\mathbf{p}_{k} - x_{k}\mathbf{p}_{j} \), which is equal to the left-hand side (LHS) of the equation, and thus the proof is complete.

Key concepts

Levi-Civita Symbol

The Levi-Civita symbol is a crucial concept in vector calculus and physics. It's typically denoted by \( \epsilon_{ijk} \) and is a three-dimensional antisymmetric tensor. This symbol proves useful in cross products and determinants.

The Levi-Civita symbol has some intriguing properties:

• It is antisymmetric. If you swap two indices, the symbol changes sign, for example \( \epsilon_{ijk} = -\epsilon_{jik} \).
• It equals 1 if \( (i, j, k) \) is an even permutation of (1, 2, 3), and -1 for odd permutations.
• If any two indices are the same, the Levi-Civita symbol is zero, such as \( \epsilon_{iij} = 0 \).

In the context of angular momentum, \( \mathbf{L}_i = \epsilon_{ijk} x_j \mathbf{p}_k \) represents the ith component, demonstrating its usefulness in expressing vector operations succinctly.

Kronecker Delta

The Kronecker delta, denoted as \( \delta_{ij} \), is a mathematical symbol used for indexing and simplifying expressions, particularly in linear algebra and tensor operations. It's a function that serves as an identity for indices.

Here's how the Kronecker delta works:

• It is 1 when the indices match, i.e., \( \delta_{ij} = 1 \) if \( i = j \).
• It becomes 0 when the indices are different, i.e., \( \delta_{ij} = 0 \) if \( i eq j \).

This makes the Kronecker delta incredibly useful for simplifying expressions involving sums over indices. For example, in the proof exercise, it helps replace specific indices to make the equation simpler.

When combined with the Levi-Civita symbol, it helps solve vector identities and simplify expressions like those in physics problems involving angular momentum.

Proof in Physics

A proof in physics often involves demonstrating that one mathematical expression can be transformed into another using logical and mathematical reasoning. This is analogous to proofs in mathematics but within the context of physical laws and theories.

When proving statements like the given angular momentum identity, several steps are involved:

• Begin by clearly writing down the relations or identities you need to prove.
• Use known mathematical tools and symbols, like the Levi-Civita symbol and Kronecker delta, to manipulate expressions.
• Strategically substitute and evaluate expressions to simplify and transform them as needed.
• Check each step carefully to ensure correctness, ensuring the final expressions on both sides equate, thus confirming the proof.

Proofs like this help strengthen understanding by connecting abstract math concepts with tangible physical principles. Mastering these kinds of proofs is essential for developing a deeper comprehension of physics.