Bra–ket notation

Bra–ket notation, also known as Dirac notation, is a standard notation for describing quantum states in the field of quantum mechanics. It was introduced by the physicist Paul Dirac and is widely used in quantum physics and quantum chemistry. The notation is a convenient way to represent complex vectors and linear operators in Hilbert space, which is the mathematical framework for quantum mechanics.

Overview

Bra–ket notation is composed of two main parts: the bra and the ket. These terms are derived from the word "bracket".

Ket Vectors

A ket is a vector in a complex vector space, typically denoted as \(| \psi \rangle\). The symbol \(| \cdot \rangle\) is used to denote a vector, and the letter inside the ket, such as \(\psi\), is a label for the vector. Kets are used to represent quantum states.

For example, a quantum state \(| \psi \rangle\) might represent the state of an electron in a hydrogen atom.

Bra Vectors

A bra is the dual vector to a ket, denoted as \(\langle \phi |\). The bra is the complex conjugate transpose of the ket. In mathematical terms, if \(| \psi \rangle\) is a ket, then \(\langle \psi |\) is the corresponding bra.

The bra \(\langle \phi |\) can be thought of as a row vector, while the ket \(| \psi \rangle\) is a column vector.

Inner Product

The inner product of two states \(| \phi \rangle\) and \(| \psi \rangle\) is written as \(\langle \phi | \psi \rangle\). This is a complex number that represents the "overlap" between the two states. If the states are normalized, the inner product gives the probability amplitude for the transition from state \(| \phi \rangle\) to state \(| \psi \rangle\).

Outer Product

The outer product of a bra \(\langle \phi |\) and a ket \(| \psi \rangle\) is written as \(| \phi \rangle \langle \psi |\). This is an operator that acts on kets to produce other kets. It is used to construct density matrices and projection operators.

Applications

Bra–ket notation is used extensively in quantum mechanics to:

• Describe quantum states and their evolution.
• Calculate probabilities and expectation values.
• Formulate the Schrödinger equation and other fundamental equations of quantum mechanics.
• Analyze systems in quantum computing and quantum information theory.

Also see

• Quantum mechanics
• Hilbert space
• Linear algebra
• Quantum state
• Schrödinger equation
• Quantum computing