Schrödinger Picture

States have time dependence. $| α, t_{0}; t ⟩_{S} = | α^{'} ⟩ = \hat{U} (t, t_{0}) | α ⟩ = \exp (- i \hat{H} (t - t_{0}) / ℏ) | α ⟩$ . $A_{S} (0) = A_{S} (t)$ .

$⟨ b^{'} | A | a^{'} ⟩ = ⟨ U b | A | U a ⟩$ .

Equation of Motion

$\hat{H} | Ψ ⟩ = i ℏ \frac{\partial}{\partial t} | Ψ ⟩$ .

Position Space Hamiltonian

$i ℏ \frac{\partial}{\partial t} ⟨ {\vec{x}}^{'} | α, t_{0}; t ⟩ = ⟨ {\vec{x}}^{'} | \hat{H} | α, t_{0}; t ⟩$ , $i ℏ \frac{\partial}{\partial t} ψ_{α} ({\vec{x}}^{'}, t) = ⟨ {\vec{x}}^{'} | p^{2} / 2 m | α, t_{0}; t ⟩ + V ({\vec{x}}^{'}) ψ_{α} ({\vec{x}}^{'}, t) = \frac{- ℏ^{2}}{2 m} {\vec{\nabla}}^{2} ψ_{α} (\vec{x}, t) + V ({\vec{x}}^{'}) ψ_{α} ({\vec{x}}^{'}, t)$ .

$i ℏ \frac{\partial}{\partial t} ψ_{α} ({\vec{x}}^{'}, t) = \frac{- ℏ^{2}}{2 m} {\vec{\nabla}}^{2} ψ_{α} (\vec{x}, t) + V ({\vec{x}}^{'}) ψ_{α} ({\vec{x}}^{'}, t)$ .

$- \frac{ℏ^{2}}{2 m} {\vec{\nabla}}^{2} + \hat{V} ({\vec{x}}^{'}) = \hat{H} = i ℏ \frac{\partial}{\partial t}$ .

Momentum Space Hamiltonian

$i ℏ \frac{\partial}{\partial t} Φ ({\vec{p}}^{'}, t) = \frac{{\vec{p}}^{' 2}}{2 m} Φ ({\vec{p}}^{'}, t) + ⟨ {\vec{p}}^{'} | V (i ℏ {\vec{\nabla}}_{p}) | α, t_{0}; t ⟩$ .

Example 1

$\hat{H} = \frac{{\hat{P}}^{2}}{2 m} + \hat{V} (x) = \frac{{\hat{P}}^{2}}{2 m} + \frac{1}{\cosh^{2} \hat{X}}$ .

$- \frac{ℏ^{2}}{2 m} \frac{d^{2}}{d x^{2}} u_{e} (x) + \frac{1}{\cosh^{2} x} u_{E} (x) = E u_{E} (x)$ .

$\frac{p^{2}}{2 m} Φ (p) + \frac{1}{\cosh^{2} (i ℏ \frac{d}{d p})} Φ_{E} (p) = E Φ_{E} (p)$ .

Example 2 (Worksheet)

$V (x) = f x$ , $[\frac{p^{2}}{2 m} + i ℏ f \frac{\partial}{\partial p}] Φ_{E} (p) = E Φ_{E} (p)$ .

Example 3 Free Particle

$\hat{H} = \frac{p^{2}}{2 m}$ .

$Φ_{E} (p) = C_{1} δ (p - \sqrt{2 m E}) + C_{2} δ (- p - \sqrt{2 m E})$

$Ψ_{E} (x) = C_{1} \exp (- i (k x + ω t))$ , $ω = \frac{E}{ℏ} = \frac{ℏ k^{2}}{2 m}$ . $| v_{p h} | = \frac{ω (k)}{k} = O (k)$ .

Stationary State

$| α, t_{0} ⟩ = | E_{k} ⟩$ , $| α, t_{0}; t ⟩ = \exp (- i E_{k} t / ℏ) | α, t_{0} ⟩$ . $⟨ {\vec{x}}^{'} | E_{k}; t ⟩ = \exp (- i E_{k} t / ℏ) φ_{k} ({\vec{x}}^{'})$ . Plugging this into the Position State Hamiltonian, $- \frac{ℏ^{2}}{2 m} {\vec{\nabla}}^{2} φ_{k} ({\vec{x}}^{'}, t) + V (x^{'}) φ_{k} ({\vec{x}}^{'}, t) = i ℏ (- i E_{k} / ℏ) φ_{k} (\vec{x}, t)$ . So, $- \frac{ℏ^{2}}{2 m} u_{E} ({\vec{x}}^{'}) + V ({\vec{x}}^{'}) u_{E} ({\vec{x}}^{'}) = E_{k} u_{E} ({\vec{x}}^{'})$ .

$H ({\vec{x}}^{'}) u_{E} ({\vec{x}}^{'}) = E_{k} u_{E} ({\vec{x}}^{'})$ , $u_{E} ({\vec{x}}^{'}) = ⟨ x^{'} | φ_{k} ⟩$ .