Dirac notation

Why use Dirac notation?

Dirac notation, also called bra-ket notation, is a compact way to express quantum mechanics independent of any particular representation. It separates the abstract state from its coordinate wavefunction.

A ket represents a state vector:

angle$$ A bra is the dual vector: $$langlepsi|$$ The inner product between two states is $$langlephi|psi angle$$ This is a complex number measuring the overlap amplitude between states. ## Kets and wavefunctions The position-space wavefunction is a projection of the state onto a position eigenstate: $$psi(x)=langle x|psi angle$$ Similarly, a momentum-space wavefunction is $$phi(p)=langle p|psi angle$$ The same ket $|psi angle$ can have different representations depending on the basis. ## Outer products and projectors An outer product creates an operator: $$|a anglelangle b|$$ Acting on a state, it gives $$|a anglelangle b|psi angle$$ which means take the overlap with $|b angle$ and produce a vector in the direction $|a angle$. A projector onto normalized state $|a angle$ is $$hat{P}*a=|a anglelangle a|$$ The probability of finding a normalized state $|psi angle$ in state $|a angle$ is $$P(a)=|langle a|psi angle|^2$$ ## Completeness An orthonormal discrete basis satisfies $$langle m|n angle=delta*{mn}$$ and the completeness relation $$sum_n |n anglelangle n|=hat{I}$$ where $hat{I}$ is the identity operator. Inserting the identity lets us expand states: $$|psi angle=hat{I}|psi angle=sum_n |n anglelangle n|psi angle$$ Thus $$|psi angle=sum_n c_n|n angle$$ with $$c_n=langle n|psi angle$$ ## Continuous completeness For the position basis, $$int |x anglelangle x|dx=hat{I}$$ and $$langle x|x' angle=delta(x-x')$$ Then $$|psi angle=int |x anglelangle x|psi angle dx=int psi(x)|x angle dx$$ ## Operators in a basis Matrix elements of an operator are written $$A_{mn}=langle m|hat{A}|n angle$$ This is the connection between abstract operators and matrices. Quantum mechanics can be done with matrices, wavefunctions, or abstract kets; Dirac notation shows they are representations of the same structure. ## Hermitian conjugation Taking the Hermitian conjugate reverses order and conjugates coefficients: $$left(a|psi angle+b|phi angle ight)^*=a^*langlepsi|+b^*langlephi|$$ More precisely, the dagger operation maps kets to bras and operators to adjoints. ## The big idea Dirac notation is a representation-independent language for quantum mechanics. Kets are states, bras are dual states, inner products are amplitudes, outer products are operators, and completeness relations express basis expansion. This notation makes the structure of quantum theory clearer and more flexible.