Question

Show that the product of two unitary operators is always unitary, but the product of two hermitian operators is hermitian if and only if they commute.

Short answer

Under the given definitions and properties, it is shown that the product of two unitary operators is always unitary and the product of two hermitian operators is hermitian if and only if they commute.

Step-by-step solution

Definition and Properties of Unitary Operators

First, it is crucial to understand that in linear algebra, a unitary operator is one which preserves the inner product in a complex Hilbert Space, i.e., the operator \(U\) is unitary if and only if \(U^*U = UU^* = I\), where \(U^*\) is the conjugate transpose of \(U\) and \(I\) is the identity operator.

Proving that Product of Unitary Operators is Unitary

We will consider two unitary operators \(U\) and \(V\). Since they are unitary, we have \(U^*U = UU^* = I\) and \(V^*V = VV^* = I\). Now consider the operator \(W = UV\). We need to prove that \(W\) is unitary, i.e. \(W^*W = WW^* = I\). While checking the necessary condition, we have \(W^*W = (UV)^*(UV) = V^*U^*UV = V^*V = I\). And, for the sufficient condition, we have \(WW^* = UV(UV)^* = UVV^*U^* = UU^* = I\). Hence, \(W\) is unitary. Thus, product of two unitary operators is unitary.

Definition and Properties of Hermitian Operators

In linear algebra, an operator \(H\) is hermitian if it is equal to its own conjugate transpose, i.e., \(H = H^*\).

Proving that Product of Hermitian Operators is Hermitian if They Commute

We will consider two hermitian operators \(A\) and \(B\) which commute, i.e., \(AB = BA\). When these two operators are multiplied together to form an operator \(C\), we get \(C = AB\). To check if \(C\) is hermitian, we need to prove that \(C = C^*\). We get \(C^* = (AB)^* = B^*A^* = BA = AB = C\). Hence, \(C\) is hermitian. Thus, the product of two hermitian operators is hermitian if they commute.

Key concepts

Linear Algebra

Linear algebra is a branch of mathematics that deals with vector spaces, linear mappings between these spaces, and the representation of linear systems in terms of matrices and vectors. It forms the foundation for many areas of mathematics and applied sciences, making it an essential topic to understand.

In the context of quantum mechanics and the exercise, linear algebra provides the language to describe the behavior of quantum states, which are represented as vectors in a complex Hilbert space. Operators, which correspond to physical observables or transformations, act on these vectors. Understanding the properties of these operators, like unitary and hermitian, helps appreciate why their behavior is significant.

Unitary operators, for example, preserve the length of vectors, making them analogous to 'rotations' in quantum systems, while hermitian operators, which correspond to real-world measurements, have real eigenvalues, making them critical in the study of observable quantities like energy or position.

Complex Hilbert Space

A complex Hilbert space is a complete space endowed with an inner product that allows length and angle to be measured. In quantum mechanics, the complex Hilbert space functions as the stage where quantum states reside, and these states evolve according to the unitary operators.

Inner product preservation, which is a property of unitary operators, ensures the probability interpretation of quantum states remains invariant during time evolution. This aspect is crucial since it embodies one of the fundamental principles in quantum mechanics: measurements are probabilistic, and the probabilities do not change unless the system is acted on by some external influence.

The concept of the Hilbert space is deeply tied to linear algebra, since operations on quantum states can be represented as matrices acting on vectors, with unitary and hermitian operators being specific types of matrices that fulfill certain mathematical conditions.

Conjugate Transpose

The conjugate transpose, also known as the Hermitian adjoint, of a matrix is achieved by taking the transpose of the matrix and then taking the complex conjugate of each element. This operation is denoted by the asterisk (\(A^*\) for a matrix A).

In the language of linear algebra, if a matrix is equal to its own conjugate transpose (\(A = A^*\)), it is referred to as hermitian. Hermitian operators represent measurable physical properties and have special significance in quantum mechanics due to their real eigenvalues and orthogonal eigenvectors.

Understanding the conjugate transpose is fundamental when tackling problems involving unitary and hermitian matrices. For instance, unitary operators can be recognized as preserving this inner product structure because the conjugate transpose reverses their action, much like an 'undo' button for their transformation on quantum states.