Schrödinger Picture
States have time dependence. \(|\alpha,t_0;t\rangle_S=|\alpha'\rangle = \hat{U}(t,t_0)|\alpha\rangle = \exp(-i\hat{H}(t-t_0)/\hbar)|\alpha\rangle\). \(A_S(0) = A_S(t)\).
\(\langle b'|A|a'\rangle = \langle Ub|A|Ua\rangle\).
Equation of Motion
\(\hat{H}|\Psi\rangle = i\hbar\frac{\partial}{\partial t}|\Psi\rangle\).
Position Space Hamiltonian
\(i\hbar \frac{\partial}{\partial t}\langle \vec{x}'|\alpha,t_0;t\rangle = \langle \vec{x}'|\hat{H}|\alpha,t_0;t\rangle\), \(i\hbar\frac{\partial}{\partial t}\psi_\alpha(\vec{x}',t) = \langle \vec{x}'|p^2/2m|\alpha,t_0;t\rangle + V(\vec{x}')\psi_\alpha(\vec{x}',t) = \frac{-\hbar^2}{2m}\vec{\nabla}^2\psi_\alpha(\vec{x},t) + V(\vec{x}')\psi_\alpha(\vec{x}',t)\).
\(i\hbar\frac{\partial}{\partial t}\psi_\alpha(\vec{x}',t) = \frac{-\hbar^2}{2m}\vec{\nabla}^2\psi_\alpha(\vec{x},t) + V(\vec{x}')\psi_\alpha(\vec{x}',t)\).
\(-\frac{\hbar^2}{2m}\vec{\nabla}^2+\hat{V}(\vec{x}')=\hat{H}=i\hbar\frac{\partial}{\partial t}\).
Momentum Space Hamiltonian
\(i\hbar\frac{\partial}{\partial t}\Phi(\vec{p}',t) = \frac{\vec{p}'^2}{2m}\Phi(\vec{p}',t) + \langle \vec{p}'|V(i\hbar\vec{\nabla}_p)|\alpha,t_0;t\rangle\).
Example 1
\(\hat{H} = \frac{\hat{P}^2}{2m} + \hat{V}(x) = \frac{\hat{P}^2}{2m}+\frac{1}{\cosh^2\hat{X}}\).
\(-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}u_e(x) + \frac{1}{\cosh^2x}u_E(x) = Eu_E(x)\).
\(\frac{p^2}{2m}\Phi(p)+\frac{1}{\cosh^2(i\hbar\frac{d}{dp})}\Phi_E(p)=E\Phi_E(p)\).
Example 2 (Worksheet)
\(V(x) = fx\), \(\left[\frac{p^2}{2m}+i\hbar f\frac{\partial}{\partial p}\right]\Phi_E(p)=E\Phi_E(p)\).
Example 3 Free Particle
\(\hat{H} = \frac{p^2}{2m}\).
\(\Phi_E(p) = C_1\delta(p-\sqrt{2mE}) + C_2\delta(-p-\sqrt{2mE})\)
\(\Psi_E(x) = C_1\exp(-i(kx+\omega t))\), \(\omega = \frac{E}{\hbar} = \frac{\hbar k^2}{2m}\). \(|v_{ph}|=\frac{\omega(k)}{k}=\mathcal{O}(k)\).
Stationary State
\(|\alpha,t_0\rangle = |E_k\rangle\), \(|\alpha,t_0;t\rangle = \exp(-iE_kt/\hbar)|\alpha,t_0\rangle\). \(\langle \vec{x}'|E_k;t\rangle = \exp(-iE_kt/\hbar)\varphi_k(\vec{x}')\). Plugging this into the Position State Hamiltonian, \(-\frac{\hbar^2}{2m}\vec{\nabla}^2\varphi_k(\vec{x}',t)+V(x')\varphi_k(\vec{x}',t) = i\hbar(-iE_k/\hbar)\varphi_k(\vec{x},t)\). So, \(-\frac{\hbar^2}{2m}u_E(\vec{x}') + V(\vec{x}')u_E(\vec{x}') = E_k u_E(\vec{x}')\).
\(H(\vec{x}')u_E(\vec{x}')=E_ku_E(\vec{x}')\), \(u_E(\vec{x}') = \langle x'|\varphi_k\rangle\).