Hilbert Space Bases

$\Psi (x,t)=⟨x|\mathcal{S}(t)⟩$

• The wave function is the $x$ component of $|\mathcal{S}(t)⟩$ in the basis of position eigenfunctions
• Analogous to $v_1=e_1\vec{v}$

$\Phi (p,t)=⟨p|\mathcal{S}(t)⟩$

$c_n(t)(t)=⟨n|\mathcal{S}(t)⟩$

$|\mathcal{S}(t)⟩\to \int \Psi (y,t)\delta (x-y)\mathrm{d}y=\frac{1}{\sqrt{2\pi \hslash }}\int \Phi (p,t)e^{ipx/\hslash }\mathrm{d}p=\sum c_n{\psi }_n(x)e^{-i{E}_{n}t/\hslash }$

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

$$\begin{array}{rl}|S(t)⟩& =\int |x⟩⟨x|\mathcal{S}(t)⟩\mathrm{d}x\equiv \int \Psi (x,t)|x⟩\mathrm{d}x\\ & =\int |p⟩⟨p|\mathcal{S}(t)⟩\mathrm{d}p\equiv \int \Phi (p,t)|p⟩\mathrm{d}p\\ & =\sum _n|n⟩⟨n|\mathcal{S}(t)⟩\equiv \sum c_n(t)|n⟩\end{array}$$

• State vector in position, momentum, and energy bases