Ethan's Notes

    Hilbert Space

    #Math #Physics
    Describes in an infinite-dimensional vector space the wave function of a particle. In the discrete case where a particle can only take three positions, the Hilbert Space would be $\displaystyle \mathbb{C}^{3}$ with each basis vector corresponding to the probability of the particle taking on that position. Also Cauchy Complete
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    Topics

    $\displaystyle \braket{ \Psi_{n} | \Psi_{m} }=\braket{ \Psi_{m} |\Psi_{n} }^{*}$

    • Anti commutative in a way

    $\displaystyle \braket{ \Psi | \Psi }=1 \Leftrightarrow \Psi \text{ is normalized}$

    $\displaystyle \braket{ \Psi_{n} | \Psi_{m} }=0\text{ for }n\neq m\Leftrightarrow \Psi_{n}\text{ and }\Psi_{m}\text{ are orthogonal}$

    $\displaystyle \braket{ \Psi_{n} | \Psi_{m} }=\delta_{nm} \Leftrightarrow {\Psi_{n}}\text{ is orthonormal}$

    • The basis $\displaystyle {\Psi_{n}}$ is orthonormal

    $\displaystyle \Psi(x)=\sum_{n = 1}^{\infty}c_{n}\Psi_{n}(x) \Leftrightarrow {\Psi_{n}}\text{ is complete}$

    $\displaystyle c_{n}=\braket{ \Psi_{n} | \Psi }$