Question

Show that \((\mathbf{U}+\mathbf{T})(\mathbf{U}-\mathbf{T})=\mathbf{U}^{2}-\mathbf{T}^{2}\) if and only if \([\mathbf{U}, \mathbf{T}]=\mathbf{0} .\)

Short answer

The result \((\mathbf{U}+\mathbf{T})(\mathbf{U}-\mathbf{T})=\mathbf{U}^{2}-\mathbf{T}^{2}\) holds true if and only if the commutator of \(\mathbf{U}\) and \(\mathbf{T}\) equals zero, which indicates that \(\mathbf{U}\) and \(\mathbf{T}\) commute.

Step-by-step solution

Verify the given equation

Start by multiplying out \((\mathbf{U}+\mathbf{T})(\mathbf{U}-\mathbf{T})\). We use the distributive law \( (a + b)(c - d) = ac + bc - ad - bd \) to get \(\mathbf{U}^2 - \mathbf{UT} + \mathbf{TU} - \mathbf{T}^2\).

Simplify the equation

If \([\mathbf{U}, \mathbf{T}]=\mathbf{0}\), then it means that \(\mathbf{UT} - \mathbf{TU} = \mathbf{0}\), which implies \(\mathbf{UT} = \mathbf{TU}\). Therefore, the equation can then be simplified to \(\mathbf{U}^2 - \mathbf{T}^2\).

Conclude

Having shown in steps 1 and 2 that \((\mathbf{U}+\mathbf{T})(\mathbf{U}-\mathbf{T}) = \mathbf{U}^2-\mathbf{T}^2 \) if and only if \([\mathbf{U}, \mathbf{T}]=\mathbf{0}\), we have thereby verified the stated equation.

Key concepts

Distributive Law in Algebra

The distributive law is a fundamental property of algebra that showcases how multiplication interacts with addition or subtraction. It allows us to simplify expressions and solve equations efficiently. When we say a distributive law, such as \((a + b)(c - d) = ac + bc - ad - bd\), we describe how each term in the first parenthesis is multiplied by each term in the second parenthesis.

For example, if we have \((\mathbf{U} + \mathbf{T})(\mathbf{U} - \mathbf{T})\), applying the distributive law means multiplying each matrix in the first set of parenthesis by each matrix in the second. This results in \(\mathbf{U}^2 - \mathbf{UT} + \mathbf{TU} - \mathbf{T}^2\).

The distributive law makes it possible to rearrange and simplify algebraic expressions, whether they involve numbers, variables, or even matrices, as shown in this exercise. It's an essential tool in algebra and other mathematical calculations.

Matrix Multiplication

Matrix multiplication is an operation where two matrices produce a third matrix. Unlike regular multiplication of numbers, matrix multiplication is not just a straightforward operation; it involves the rows of the first matrix and the columns of the second.

For matrices to be multiplied, the number of columns in the first matrix must be equal to the number of rows in the second. When calculating each element of the resulting matrix, you sum the products of the elements from the rows of the first matrix and the corresponding columns of the second matrix.

In our context, when calculating \(\mathbf{U} \cdot \mathbf{T}\) and \(\mathbf{T} \cdot \mathbf{U}\), each element in their resulting matrices comes from these sums of products. However, it's important to note that matrix multiplication is not commutative. This means that \(\mathbf{UT}\) is not necessarily equal to \(\mathbf{TU}\). This is a pivotal aspect in linear algebra and crucial in understanding when matrices commute (i.e., when \(\mathbf{UT = TU}\)), which is central to the solution of our exercise.

Properties of Operators in Quantum Mechanics

In quantum mechanics, operators are mathematical constructs that correspond to physical observables, like momentum and energy. These operators can often be represented by matrices, especially in finite-dimensional spaces.

An essential property among these operators is their commutator. The commutator of two operators \(\mathbf{U}\) and \(\mathbf{T}\) is given by \([\mathbf{U}, \mathbf{T}] = \mathbf{UT} - \mathbf{TU}\). The commutator helps us determine the extent to which two operators are interchangeable.

When \([\mathbf{U}, \mathbf{T}] = 0\), it implies the operators commute, meaning \(\mathbf{UT} = \mathbf{TU}\). This property is significant in quantum mechanics as it indicates that two observables can be simultaneously measured precisely. In our exercise, this commutative condition is crucial to simplifying the equation \((\mathbf{U} + \mathbf{T})(\mathbf{U} - \mathbf{T}) = \mathbf{U}^2 - \mathbf{T}^2\), thus highlighting the importance of understanding these properties when dealing with matrices in quantum mechanics.