Hilbert Space Bases

#Physics

$\displaystyle \Psi (x,t)=\braket{ x | \mathcal{S}(t) }$

• The wave function is the $\displaystyle x$ component of $\displaystyle \ket{\mathcal{S}(t)}$ in the basis of position eigenfunctions
• Analogous to $\displaystyle v_{1}=e_{1}\vec{v}$

$\displaystyle \Phi(p,t)=\braket{ p | \mathcal{S}(t) }$

$\displaystyle c_{n}(t)(t)=\braket{ n | \mathcal{S}(t) }$

$\displaystyle \ket{\mathcal{S}(t)}\rightarrow\int \Psi(y,t)\delta(x-y) , \mathrm{d}y=\frac{1}{\sqrt{ 2\pi \hbar }}\int \Phi(p,t)e^{ipx/\hbar} , \mathrm{d}p=\sum c_{n}\psi_{n}(x)e^{-iE_{n}t/\hbar}$

• Different representations of $\displaystyle \ket{\mathcal{S(t)}}$ in different bases

$$

\begin{align}
\ket{S(t)} &= \int \ket{x} \braket{ x | \mathcal{S}(t) } , \mathrm{d}x\equiv \int \Psi(x,t)\ket{x} , \mathrm{d}x \\&=\int \ket{p} \braket{ p | \mathcal{S}(t) } , \mathrm{d}p,\equiv \int \Phi(p,t)\ket{p} , \mathrm{d}p \\&=\sum_{n}\ket{n} \braket{ n | \mathcal{S(t)} },,,\equiv \sum c_{n}(t)\ket{n}
\end{align}
$$

• State vector in position, momentum, and energy bases