Ethan's Notes

    Braket Notation

    #Physics
    Universal Danker

    $$

    \ket{\Psi} =
    \begin{bmatrix}
    \Psi(x_{1}) \\\Psi(x_{2}) \\\Psi(x_{3}) \\\vdots
    \end{bmatrix}
    $$

    • Called "ket"
    • Basically a regular vector

    $$

    \bra{\Psi} =(\ket{\Psi} ^{\top})^{}=
    \begin{bmatrix}
    \Psi(x_{1})^{    } & \Psi(x_{2})^{} & \Psi(x_{3})^{} & \ldots &
    \end{bmatrix}
    $$

    • 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

    $\displaystyle \braket{ \Phi | \Psi }=\phi^{}{i}\Psi*{i}=\int \Phi^{*}(x)\Psi(x) , \mathrm{d}x$

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

    $$

    \ket{\Phi}\bra{\Psi}= \ket{\Phi}\otimes\bra{\Psi}=
    \begin{bmatrix}
    {\Phi}{1}{\Psi}{1}^{} & {\Phi}{1}{\Psi}{2}^{} & {\Phi}{1}{\Psi}{3}^{} & \ldots \\{\Phi}{2}{\Psi}{1}^{} & {\Phi}{2}{\Psi}{2}^{} & {\Phi}{2}{\Psi}{3}^{} & \ddots \\{\Phi}{3}{\Psi}{1}^{} & {\Phi}{3}{\Psi}{2}^{} & {\Phi}{3}{\Psi}{3}^{*} & \ddots \\\vdots & \ddots & \ddots & \ddots
    \end{bmatrix}
    $$

    $\displaystyle \ket{\Psi}\braket{ \Psi | \Phi }=\text{proj}_{\Psi}\Phi$

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

    Discrete

    $\displaystyle \sum_{i}\ket{e_{n}}\bra{e_{n}}=I$

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

    $\displaystyle \ket{\Phi}=\sum_{i}\ket{\Psi_{i}}\braket{ \Psi_{i} | \Phi }$

    • Change of basis of $\displaystyle \Phi$ in $\displaystyle \left{ \ket{\Phi_{i}} \right}$ to $\displaystyle \left{ \ket{\Psi_{i}} \right}$ is the sum of projections of $\displaystyle \Phi$ onto $\displaystyle \Psi_{i}$ for finitely many components

    Continuous

    $\displaystyle \int \ket{e_{z}}\bra{e_{z}} , \mathrm{d}x=1$

    $\displaystyle \ket{\Phi}=\int \ket{\Psi}\braket{ \Psi | \Phi } , \mathrm{d}x$

    • Change of basis for infinitely-many components