Question

Show that if \(\mathbf{P}\) is a (hermitian) projection operator, so are \(\mathbf{1}-\mathbf{P}\) and U PU for any unitary operator \(\mathbf{U}\).

Short answer

Both \(\mathbf{1}-\mathbf{P}\) and \(\mathbf{U}\mathbf{P}\mathbf{U}\) satisfy the idempotent and Hermitian properties of a projection operator. Therefore, if \(\mathbf{P}\) is a projection operator, so are \(\mathbf{1}-\mathbf{P}\) and \(\mathbf{U}\mathbf{P}\mathbf{U}\) for any unitary operator \(\mathbf{U}\).

Step-by-step solution

Proving \(\mathbf{1}-\mathbf{P}\) is a Projection Operator

First, show that \(\mathbf{1}-\mathbf{P}\) is idempotent. We know that \(\mathbf{P}^2=\mathbf{P}\). Then, \((\mathbf{1}-\mathbf{P})^2=\mathbf{1}^2-\mathbf{1P}-\mathbf{P}\mathbf{1}+\mathbf{P}^2 = \mathbf{1} - 2\mathbf{P} + \mathbf{P} = \mathbf{1} - \mathbf{P}\). Hence, it is idempotent. Secondly, show that \(\mathbf{1}-\mathbf{P}\) is Hermitian. We know that \(\mathbf{P}^{\dagger}=\mathbf{P}\). Then, (\mathbf{1}-\mathbf{P})^{\dagger}=\mathbf{1}^{\dagger}-\mathbf{P}^{\dagger} = \mathbf{1}-\mathbf{P}\). Hence, it is Hermitian.

Proving \(\mathbf{U}\mathbf{P}\mathbf{U}\) is a Projection Operator

First, show that \(\mathbf{U}\mathbf{P}\mathbf{U}\) is idempotent. Start with \((\mathbf{U}\mathbf{P}\mathbf{U})^2\). This simplifies to \(\mathbf{U}\mathbf{P}\mathbf{U}\mathbf{U}\mathbf{P}\mathbf{U}=\mathbf{U}\mathbf{P}\mathbf{1}\mathbf{P}\mathbf{U}=\mathbf{U}\mathbf{P}^2\mathbf{U}=\mathbf{U}\mathbf{P}\mathbf{U}\). Second, prove that \(\mathbf{U}\mathbf{P}\mathbf{U}\) is Hermitian. Start with \((\mathbf{U}\mathbf{P}\mathbf{U})^{\dagger}\) and simplify to \(\mathbf{U}^{\dagger}\mathbf{P}^{\dagger}\mathbf{U}^{\dagger}=\mathbf{U}^{\dagger}\mathbf{P}\mathbf{U}^{\dagger}\) since \(\mathbf{P}\) is Hermitian. Recall the property of unitary operators: \(\mathbf{U}\mathbf{U}^{\dagger}=\mathbf{1}\), then \(\mathbf{U}^{\dagger}=\mathbf{U}\). So \(\mathbf{U}^{\dagger}\mathbf{P}\mathbf{U}^{\dagger}=\mathbf{U}\mathbf{P}\mathbf{U}\). Thus, \(\mathbf{U}\mathbf{P}\mathbf{U}\) is also Hermitian.

Key concepts

Idempotent Operators

Idempotent operators have a special property. When you apply them twice, the result is the same as if you applied them once. Mathematically, this is expressed as an operator \(\mathbf{A}\) being idempotent if \(\mathbf{A}^2 = \mathbf{A}\). This property is crucial in many areas of mathematics and physics.

One of the most common examples of an idempotent operator in linear algebra is a projection operator. Projection operators essentially "project" vectors into a subspace, and doing this operation multiple times doesn't change the result compared to doing it once. For instance, the operator \(\mathbf{P}\) in the problem is such that \(\mathbf{P}^2 = \mathbf{P}\). The step-by-step solution also demonstrated that \(\mathbf{1} - \mathbf{P}\) and \(\mathbf{U}\mathbf{P}\mathbf{U}\), where \(\mathbf{U}\) is unitary, maintain idempotency. This highlights the usefulness of idempotent operators in ensuring consistency across mathematical operations.

Unitary Operators

Unitary operators are vital in quantum mechanics and linear algebra. These operators are characterized by their ability to preserve length or norm of vectors in a complex vector space. A unitary operator \(\mathbf{U}\) satisfies the condition \(\mathbf{U}\mathbf{U}^{\dagger} = \mathbf{U}^{\dagger}\mathbf{U} = \mathbf{1}\), where \(\mathbf{U}^{\dagger}\) represents the conjugate transpose, also known as the Hermitian adjoint of the operator.

The essence of a unitary operator is that it keeps vector norms unchanged, acting like a rotation or reflection in space, without altering the vector's magnitude. In the context of the exercise, using a unitary operator \(\mathbf{U}\) with the projection operator \(\mathbf{P}\) to form \(\mathbf{U}\mathbf{P}\mathbf{U}\) keeps it as a projection operator. Thus, the combination of unitary transformation and projection maintains the necessary properties, showcasing how unitary operators can efficiently transform within a space while maintaining key characteristics of other operators involved.

Hermitian Operators

Hermitian operators are a fundamental concept in quantum mechanics, primarily because they have real eigenvalues, which correspond to observable quantities. A Hermitian operator \(\mathbf{H}\) is its own conjugate transpose, which is expressed mathematically as \(\mathbf{H} = \mathbf{H}^{\dagger}\).

The importance of Hermitian operators lies in their relation to measurable physical properties, like energy in quantum systems. They ensure the results of measurements are real and verifiable. In the exercise, \(\mathbf{P}\) is Hermitian, meaning \(\mathbf{P}^{\dagger} = \mathbf{P}\). It was also shown that both \(\mathbf{1} - \mathbf{P}\) and \(\mathbf{U}\mathbf{P}\mathbf{U}\) maintain this Hermitian property, thereby preserving the universality of Hermitian operators in representing measurable quantities in a transformed state.

Operator Theory

Operator theory is a branch of mathematics focusing on the study of linear operators on function spaces. It blends seamlessly into various scientific fields, including physics and engineering, where operators stand as mathematical tools that act on elements of a space to produce other elements.

Key operator classifications, like idempotent, unitary, and Hermitian, each serve specific roles. Idempotent operators help with repeated processes without altering outcomes. Unitary operators enable rotations and reflections while conserving vector norms, and Hermitian operators link to physical observables due to their real eigenvalues.

The exercise encapsulates these concepts, demonstrating how projection operators, through Hermitian and unitary transformations, retain essential operator properties. Thus, operator theory provides a structured approach to comprehend and analyze the effects and transformations actioned by such operators, offering robust tools to address complex problems in multidimensional spaces.