Braket Notation

Universal Danker

$$|\Psi ⟩=\left[\begin{array}{c}\Psi (x_1)\\ \Psi (x_2)\\ \Psi (x_3)\\ ⋮\end{array}\right]$$

• Called “ket”
• Basically a regular vector

$$⟨\Psi |=({|\Psi ⟩}^{\top }{)}^{\ast }=\left[\begin{array}{ccccc}\Psi (x_1{)}^{\ast }& \Psi (x_2{)}^{\ast }& \Psi (x_3{)}^{\ast }& \dots & \end{array}\right]$$

• The bra of a wave function is the complex conjugate of the transpose of the ket of the wave function. Transposing and complex conjugating are commutative operations

$⟨\Phi |\Psi ⟩={\varphi }_i^{\ast }{\Psi }_i=\int {\Phi }^{\ast }(x)\Psi (x)\mathrm{d}x$

• Dot Product
• Measures how much two wave functions overlap
• “Bra” can be thought of as a linear operator on ket, or $⟨f|(|g⟩)=\int f^{\ast }g\mathrm{d}x$ when taking the dot product
• The set of all bras is a dual space

$$|\Phi ⟩⟨\Psi |=|\Phi ⟩\otimes ⟨\Psi |=\left[\begin{array}{cccc}{\Phi }_1{\Psi }_1^{\ast }& {\Phi }_1{\Psi }_2^{\ast }& {\Phi }_1{\Psi }_3^{\ast }& \dots \\ {\Phi }_2{\Psi }_1^{\ast }& {\Phi }_2{\Psi }_2^{\ast }& {\Phi }_2{\Psi }_3^{\ast }& \ddots \\ {\Phi }_3{\Psi }_1^{\ast }& {\Phi }_3{\Psi }_2^{\ast }& {\Phi }_3{\Psi }_3^{\ast }& \ddots \\ ⋮& \ddots & \ddots & \ddots \end{array}\right]$$

• Outer product of two wave functions

$|\Psi ⟩⟨\Psi |\Phi ⟩={\text{proj}}_{\Psi }\Phi$

• For normalized wave functions, the outer product of a wave function $\Psi$ with itself enacted on another wave function $\Phi$ gives the projection of $\Phi$ onto $\Psi$

Discrete

$\sum _i|e_n⟩⟨e_n|=I$

• The sum of projection matrices for each dimension $n$ is the identity matrix

$|\Phi ⟩=\sum _i|{\Psi }_i⟩⟨{\Psi }_i|\Phi ⟩$

• Change of basis of $\Phi$ in $\left\{|{\Phi }_i⟩\right\}$ to $\left\{|{\Psi }_i⟩\right\}$ is the sum of projections of $\Phi$ onto ${\Psi }_i$ for finitely many components

Continuous

$\int |e_z⟩⟨e_z|\mathrm{d}x=1$

$|\Phi ⟩=\int |\Psi ⟩⟨\Psi |\Phi ⟩\mathrm{d}x$

• Change of basis for infinitely-many components