Physlib.QuantumMechanics.DDimensions.Operators.StateObservables.IsEigenvector

Eigenvectors of partial linear maps

Main definitions

`LinearPMap.IsEigenvector`: a nonzero domain vector satisfying `T ψ = μ • ψ`.

3 declarations

definition

$\psi$ is an eigenvector of $T$ with eigenvalue $\mu$

Given a partial linear map $T$ on a complex vector space $H$ and a vector $\psi$ in the domain of $T$ , $\psi$ is an eigenvector of $T$ with eigenvalue $\mu \in \mathbb{C}$ if $T\psi = \mu \psi$ and $\psi$ is non-zero.

theorem

$T \psi = \mu \psi$ for eigenvectors of a partial linear map $T$

#apply_eq

Let $T$ be a partial linear map on a complex vector space $H$ . If $\psi$ is an eigenvector of $T$ with eigenvalue $\mu \in \mathbb{C}$ , then the eigenvalue equation $T\psi = \mu \psi$ holds.

theorem

An eigenvector $\psi$ of a partial linear map satisfies $\psi \neq 0$

#ne_zero

Let $T$ be a partial linear map on a complex vector space $H$ . If $\psi$ is an eigenvector of $T$ with eigenvalue $\mu \in \mathbb{C}$ , then $\psi$ is non-zero, i.e., $\psi \neq 0$ .

Physlib.QuantumMechanics.DDimensions.Operators.StateObservables.IsEigenvector

Eigenvectors of partial linear maps

Main definitions

ψ\psiψ is an eigenvector of TTT with eigenvalue μ\muμ

Tψ=μψT \psi = \mu \psiTψ=μψ for eigenvectors of a partial linear map TTT

An eigenvector ψ\psiψ of a partial linear map satisfies ψ≠0\psi \neq 0ψ=0

$\psi$ is an eigenvector of $T$ with eigenvalue $\mu$

$T \psi = \mu \psi$ for eigenvectors of a partial linear map $T$

An eigenvector $\psi$ of a partial linear map satisfies $\psi \neq 0$