TeXipedia

rangle

Represents the right angle bracket (⟩) used to create mathematical delimiters, commonly paired with \langle for inner products and bra-ket notation.

Overview

Essential in mathematical notation, particularly in quantum mechanics and linear algebra where angle brackets denote specific operations and concepts.

  • Primarily used in bra-ket notation for quantum mechanical states
  • Forms inner product notation when paired with \langle
  • Common in theoretical physics for expectation values
  • Appears in abstract algebra for group theory notation
  • Used to denote averages or means in statistical mechanics

Examples

Denoting the expectation value of a quantum mechanical operator A.

⟨A⟩=∫−∞∞ψ∗Aψdx\langle A \rangle = \int_{-\infty}^{\infty} \psi^* A \psi dx 
\langle A \rangle = \int_{-\infty}^{\infty} \psi^* A \psi dx

Representing a vector inner product in bra-ket notation.

⟨ϕ∣ψ⟩=∫ϕ∗(x)ψ(x)dx\langle \phi | \psi \rangle = \int \phi^*(x) \psi(x) dx 
\langle \phi | \psi \rangle = \int \phi^*(x) \psi(x) dx

Defining an average value in statistical mechanics.

⟨E⟩=1Z∑iEie−βEi\langle E \rangle = \frac{1}{Z} \sum_i E_i e^{-\beta E_i} 
\langle E \rangle = \frac{1}{Z} \sum_i E_i e^{-\beta E_i}