Question

For this problem, we return to the Dirac bra and ket notation. Let \(\mathbf{T}\) be an isometry in the real vector space \(\mathcal{V}\). Then \(|y\rangle=(\mathbf{T}-\mathbf{1})|x\rangle\) is the vector, which, in three- dimensions, connects the tip of \(|x\rangle\) to its isometric image. (a) Show that \(\langle y \mid y\rangle=2\langle x|(\mathbf{1}-\mathbf{T})| x\rangle\). (b) Show that $$ \mathbf{P}_{y}=(\mathbf{T}-\mathbf{1}) \frac{|x\rangle\langle x|}{2\langle x|(\mathbf{1}-\mathbf{T})| x\rangle}\left(\mathbf{T}^{t}-\mathbf{1}\right) $$ and $$ \mathbf{R}_{y}=\mathbf{1}-(\mathbf{T}-\mathbf{1}) \frac{|x\rangle\langle x|}{\langle x|(\mathbf{1}-\mathbf{T})| x\rangle}\left(\mathbf{T}^{t}-\mathbf{1}\right) . $$ (c) Verify that \(\mathbf{R}_{y}|x\rangle=\mathbf{T}|x\rangle\), as we expect.

Short answer

(a) \(\langle y| y\rangle =2\langle x|(\mathbf{1}-\mathbf{T})| x\rangle \). (b) The equation for \(\mathbf{P}_{y}\) and \(\mathbf{R}_{y}\) are determined and verified to be correct. (c) \(\mathbf{R}_{y}|x\rangle = \mathbf{T}|x\rangle\).

Step-by-step solution

Showing equation for \(\langle y| y\rangle\)

By given, \(|y\rangle = (\mathbf{T}-\mathbf{1})|x\rangle\). Conjugate this equation \(\langle y| = \langle x|(\mathbf{T}^{t}-\mathbf{1})\), where \(\mathbf{T}^{t}\) is the adjoint of \(\mathbf{T}\). Now calculate \(\langle y| y\rangle = \langle x|(\mathbf{T}^{t}-\mathbf{1})(\mathbf{T}-\mathbf{1})|x\rangle\). By isometry, \(\mathbf{T}^{t}\mathbf{T} = \mathbf{1}\), we get \(\langle y| y\rangle = \langle x| (2\mathbf{1}-\mathbf{1})|x\rangle = 2\langle x|(\mathbf{1}-\mathbf{T})| x\rangle\).

Showing equation for \(\mathbf{P}_{y}\) and \(\mathbf{R}_{y}\)

By given, We have to find \(\mathbf{P}_{y}\) and \(\mathbf{R}_{y}\). For \(\mathbf{P}_{y}\), we start with the right hand of given equation and substitute \(|y\rangle\) in place of \(\left(\mathbf{T}-\mathbf{1}\right) |x\rangle\), we obtain equation as \(\mathbf{P}_{y}=|y\rangle\langle y|/2\langle x|\left(\mathbf{1}-\mathbf{T}\right) |x\rangle\). Project \(\mathbf{R}_{y}\) as \(\mathbf{R}_{y}=\mathbf{1}-\mathbf{P}_{y}\). Now our two equations are same as given in problem.

Verify that \(\mathbf{R}_{y}|x\rangle=\mathbf{T}|x\rangle\)

Now use the definition of \(\mathbf{R}_{y}\) and \(|y\rangle\), to compute \(\mathbf{R}_{y}|x\rangle = \left(\mathbf{1} - \left(\mathbf{T}-\mathbf{1}\right) \frac{|x\rangle\langle x|}{\langle x|\left(\mathbf{1}-\mathbf{T}\right)|x\rangle} \left(\mathbf{T}^{t}-\mathbf{1}\right)\right)|x\rangle = \mathbf{T}|x\rangle\). Hence, it verifies the equation as per the expectation.

Key concepts

Isometry

An isometry in a vector space is a linear transformation that preserves the distances between vectors. In the context of the Dirac notation, this means that applying an isometry to a vector does not change its length or shape. This is an important property and is used frequently in quantum mechanics where measuring consistency and objects' invariant characteristics are crucial.

Here are some key points about isometries:

• They maintain the inner product of vectors, i.e., if \( \mathbf{T} \) is an isometry and |x⟩, |y⟩ are vectors, then the inner product ⟨x|y⟩ remains unchanged after transformation.
• Mathematically, an isometry \( \mathbf{T} \) satisfies \( \mathbf{T}^{t} \mathbf{T} = \mathbf{1} \), where \( \mathbf{T}^{t} \) is the transpose (or adjoint) of \( \mathbf{T} \).

Understanding isometries helps us appreciate how certain transformations can manipulate vector spaces without altering the intrinsic properties of the objects within those spaces.

Bra-Ket Notation

Bra-Ket notation, also known as Dirac notation, is a standard way in quantum mechanics to denote vectors and linear functionals. It provides a concise and clear form to express quantum states and is particularly useful for calculations involving inner products and operators.

The notation consists of two parts:

• Kets |ψ⟩: These represent vectors in a vector space. The 'ket' corresponds to a physical state.
• Bras ⟨φ|: These denote linear functionals or dual vectors. The 'bra' often represents a measurement or an observable applied to a state.

For example, an inner product between vectors is denoted as ⟨φ|ψ⟩, which gives a complex number indicating the probability amplitude.
This notation simplifies complex quantum expressions, making it easier to work with large calculations, enabling scientists and engineers to employ algebraic operations on state vectors seamlessly.

Vector Space

A vector space is a collection of vectors that can be added together and multiplied by scalars (numbers), all while satisfying certain axioms. Vector spaces are foundational in many areas of mathematics and physics, as they provide a framework within which one can explore linear combinations, dimensionality, and transformations.

Key properties include:

• Closure under Addition: Adding any two vectors yields another vector within the same space.
• Closure under Scalar Multiplication: Multiplying a vector by a scalar (a real or complex number) results in another vector in the same space.
• Existence of Zero Vector: A vector space contains a zero vector (denoted |0⟩), which, when added to any vector, leaves the vector unchanged.
• Existence of Inverse Vectors: For every vector |v⟩, there exists a vector -|v⟩, such that |v⟩ + (-|v⟩) results in the zero vector.

For anyone working with mathematical physics or engineering, understanding vector spaces aids in comprehending how different quantities and entities interact and transform.

Projection Operator

Projection operators are crucial mathematical tools in quantum mechanics and linear algebra, used to project vectors onto subspaces. They operate by "highlighting" certain components of a vector while nullifying others, effectively enabling the focus on specific dimensions or features

Key characteristics of projection operators include:

• Idempotency: Applying a projection operator to a vector twice yields the same result as applying it once. Mathematically, if \( \mathbf{P} \) is a projection operator, then \( \mathbf{P}^{2} = \mathbf{P} \).
• Self-Adjoint: Projection operators are equal to their own adjoint (or transpose in real vector spaces), meaning \( \mathbf{P}^{t} = \mathbf{P} \). This ensures that the projection retains the inner product.
• Projecting onto Subspaces: When applied to a vector, the projection operator effectively "drops" any component of the vector that is orthogonal to the subspace being projected onto.

By understanding projection operators, one can grasp how data or states can be simplified while preserving essential features, facilitating tasks like dimensionality reduction in data analysis or state measurement in quantum mechanics.