Bra-ket notation

Bra-ket or bracket (or even bra-c-ket) notation was formulated by Dirac to provide a concise method for performing and describing the linear algebra used throughout the matrix mechanics formulation of quantum mechanics. The notation is in wide use in the field today, and although developed with quantum mechanics in mind it can be employed more generally when working with any vector space. In this notation vectors are represented by kets, such as $|\psi \rangle$, while their corresponding dual vectors are given by bras, $\langle \psi |$. In the context of quantum mechanics the state of a system corresponds to a vector in a Hilbert space, so the state $|\psi \rangle$ is analogous to the wave function $\psi (x)$.

Mathematical description

Let ${\mathcal {H}}$ be a Hilbert space and ${\mathcal {H}}^{*}$ its dual space (which is isomorphic to ${\mathcal {H}}$ if the space is finite-dimensional). Elements of ${\mathcal {H}}$ are then labelled by kets and elements of ${\mathcal {H}}^{*}$ are labelled by bras. Together a bra and a ket can form a Dirac bracket, $\langle \cdot |\cdot \rangle$, which is equal to the inner product between them. The bracket then is a map from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{H}^*\times\mathcal{H}} to a field Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} (in quantum mechanics the field is the complex numbers, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{C}} ).

When the order of the bra and ket is reversed the resulting object is an operator, sometimes called a ket-bra, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\cdot\rangle\langle\cdot|} . This operator is given by the outer product of the ket with the bra, and is a map from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{H}} onto itself since Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left(|\cdot\rangle\langle\cdot|\right)|\cdot\rangle =|\cdot\rangle\langle\cdot|\cdot\rangle=\alpha|\cdot\rangle \in\mathcal{H} } , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha=\langle\cdot|\cdot\rangle\in F} is a scalar. By convention, duplicated vertical bars in an expression are dropped as we have done here (i.e. writing Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle\cdot|\cdot\rangle} instead of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle\cdot||\cdot\rangle} ).

Uses in quantum mechanics

Suppose that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{H}} corresponds to the state space for a quantum system. For example, if the system was a particle in a box then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{H}} would contain every possible state that the particle could occupy. Now let the state of the system be Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\psi\rangle\in\mathcal{H}} , with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\psi\rangle} normalized (that is, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle\psi|\psi\rangle=1} ) and let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{A}} be an operator corresponding to the observable Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} .

Expectation value

The expected result of a measurement of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} is given by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle\hat{A}\rangle=\langle\psi|\hat{A}|\psi\rangle} .

Overlap and probability

The overlap between the state of the system and another state Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\phi\rangle\in\mathcal{H}} is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle\phi|\psi\rangle} , which means that the probability of finding the system in state Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\phi\rangle} is given by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\langle\phi|\psi\rangle|^2} . This can also be seen as the expectation value of the projection operator Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{P}_\phi=|\phi\rangle\langle\phi|} , since this yields Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle\hat{P}_\phi\rangle =\langle\psi|\phi\rangle\langle\phi|\psi\rangle =\langle\psi|\phi\rangle\langle\psi|\phi\rangle^* =|\langle\phi|\psi\rangle|^2 }

Resolution of the identity

If the states Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{|\varphi_i\rangle\}} are the (normalized) eigenstates of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{A}} then the identity operator can be expressed as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I=\sum_i|\varphi_i\rangle\!\langle\varphi_i|} .

This result holds if the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\varphi_i\rangle} are any complete set of orthonormal vectors, which is guaranteed to be the case for the eigenvectors of a Hermitean matrix.