TeXipedia

langle

Represents a left angle bracket used for creating mathematical delimiters, particularly in set notation, inner products, and quantum mechanics.

Overview

Essential for mathematical notation where standard parentheses or brackets are insufficient to convey specific meaning. Most commonly paired with \rangle to create angle brackets that serve multiple purposes:

  • Denotes inner products in linear algebra and functional analysis
  • Used in quantum mechanics for bra-ket notation
  • Appears in set builder notation and sequence representations
  • Creates expectation value notation in probability and statistics

Particularly valuable in advanced mathematics and theoretical physics where precise delimiter distinction is crucial.

Examples

Denoting the average or expected value of a quantity A.

⟨A⟩=1N∑i=1Nai\langle A \rangle = \frac{1}{N} \sum_{i=1}^N a_i 
\langle A \rangle = \frac{1}{N} \sum_{i=1}^N a_i

Representing a quantum mechanical expectation value.

⟨ψ∣H∣ψ⟩=E\langle \psi | H | \psi \rangle = E 
\langle \psi | H | \psi \rangle = E

Defining an inner product of vectors x and y.

⟨x,y⟩=∑i=1nxiyi\langle x, y \rangle = \sum_{i=1}^n x_i y_i 
\langle x, y \rangle = \sum_{i=1}^n x_i y_i