Question

Prove that corresponding to every unitary operator \(\mathbf{U}\) acting on a finitedimensional vector space, there is a hermitian operator \(\mathbf{H}\) such that \(\mathbf{U}=\) \(\exp (i \mathbf{H})\).

Short answer

For every unitary operator \( \mathbf{U}\) acting on a finite dimensional vector space, there is a hermitian operator \(\mathbf{H}\) according to \( \mathbf{U} = \exp (i \mathbf{H})\). The operator \(\mathbf{H}\) can be derived from \(\mathbf{U}\) by \(\mathbf{H} = i log(\mathbf{U}\).

Step-by-step solution

Definition of unitary operator

Start from the definition of a unitary operator. A unitary operator is defined as one that satisfies \(\mathbf{U}^{†}\mathbf{U}= \mathbf{U}\mathbf{U}^{†}= \mathbf{I}\), where \(\mathbf{U}^{†}\) is the conjugate transpose of \(\mathbf{U}\) and \(\mathbf{I}\) is the identity operator.

Definition of exponential of an operator

Define the exponential of a matrix. Analogous to numbers, for any operator \(\mathbf{A}\), \(\exp(\mathbf{A})\) can be defined by the power series expansion \(\exp(\mathbf{A}) = \mathbf{I} + \mathbf{A} + \frac{1}{2!}\mathbf{A}² + \frac{1}{3!}\mathbf{A³} + ... \). This series converges for all \(\mathbf{A}\).

Define of Hermitian operator

Define a Hermitian operator, \(\mathbf{H}\). An operator \(\mathbf{H}\) is Hermitian if it satisfies \(\mathbf{H} = \mathbf{H}^{†}\), i.e., it equals its own conjugate transpose.

Finding the relationship

By assuming iH is Hermitian, i.e., \(i\mathbf{H} = (i\mathbf{H})^{†}\), and \(\mathbf{H}\) can be expressed as \(i log(\mathbf{U})\). Then, \(\mathbf{U}\) can be determined by \(\mathbf{U} = \exp(i\mathbf{H})\). Here, \(log\) is the matrix logarithm. This proves that for every unitary operator \( \mathbf{U}\), there is a corresponding hermitian operator \(\mathbf{H}\) such that \(\mathbf{U} = \exp (i \mathbf{H})\).

Key concepts

finite-dimensional vector space

When we speak about a finite-dimensional vector space, we refer to a space that has a basis consisting of a finite number of vectors. A vector space is essentially a collection of vectors, which are objects that you can add together and multiply by scalars (numbers). In simple terms, it has a predetermined number of dimensions, much like how a 2D plane has two dimensions or the physical world around us is 3D.

In the context of linear algebra and quantum mechanics, finite-dimensional vector spaces are crucial as they allow operations to be conducted using matrices. Vector spaces such as these are common in quantum mechanics where systems are described with a finite number of states. Since these spaces are finite-dimensional, calculations and theoretical models become manageable and practically feasible.

• Basis: A set of vectors in this space that are linearly independent and span the space.
• Dimension: The number of vectors in a basis of the space.
• Example: The plane (2D) and ordinary space (3D) are typical examples of finite-dimensional vector spaces.

hermitian operator

A Hermitian operator, or Hermitian matrix in terms of its matrix representation, plays an important role in various fields, especially in quantum mechanics. A matrix or an operator is called Hermitian if it is equal to its own conjugate transpose, which can be represented as \( \mathbf{H} = \mathbf{H}^{†} \).

This property implies several classical results:

• The eigenvalues of a Hermitian operator are always real, which is important in quantum mechanics since measurable quantities, like energy, are represented by these eigenvalues.
• Hermitian operators are self-adjoint, meaning they equal their own conjugate transpose.
• They have orthogonal eigenvectors, which means the eigenvectors corresponding to different eigenvalues are perpendicular to each other.

Understanding Hermitian operators helps in comprehending how physical observables function in quantum mechanics, as they are associated with the expectation values and probabilities in a given quantum state.

matrix logarithm

The matrix logarithm is an extension of the standard logarithm to matrix arguments, and it plays a crucial role in connecting operator theory and matrix functions. Given a matrix that does not have any negative or zero eigenvalues, the matrix logarithm of a matrix \( \mathbf{A} \) is another matrix \( \mathbf{B} \) such that \( \exp(\mathbf{B}) = \mathbf{A} \).

This means if you take the exponential of \( \mathbf{B} \), you retrieve the original matrix \( \mathbf{A} \).

Here are some attributes and the process of determining a matrix logarithm:

• Not unique: There can be many such matrices \( \mathbf{B} \) for a given \( \mathbf{A} \).
• Principal logarithm: Usually, the most commonly used logarithm in practice.
• Used in quantum mechanics and control theory: It is significant for converting problems into a form in which they may be easily solved or understood.

Remember, finding a matrix logarithm is more complex than for real numbers, requiring potentially several steps and methods to determine a suitable \( \mathbf{B} \).

exponential of an operator

The exponential of an operator extends the notion of exponentials from real numbers to operators or matrices, primarily using the power series expansion.

This idea is quite similar to how we handle functions of a real variable but is tailored for matrix applications.

Given an operator \( \mathbf{A} \), the exponential \( \exp(\mathbf{A}) \), is defined by:\[\exp(\mathbf{A}) = \mathbf{I} + \mathbf{A} + \frac{1}{2!}\mathbf{A}^2 + \frac{1}{3!}\mathbf{A}^3 + \ldots \]This series converges, meaning it approaches a specific matrix, for all matrices \( \mathbf{A} \).

Applications and characteristics include:

• Solving differential equations: Particularly, \( \exp(t\mathbf{A}) \) is used where \( t \) represents time.
• Unitary transformations: Where unitary operators, like those encountered in quantum mechanics, can often be represented as \( \exp(i\mathbf{H}) \) for some Hermitian operator \( \mathbf{H} \).
• Manageable via power series: The power series approach enables the handling of complex matrix functions.

Understanding the exponential of an operator unveils how substantial changes can be made via precise and well-understood mathematical techniques, bridging a gap between abstract theory and practical application.