Infinitesimal and Finite Unitary Transformations

Recall

${\hat{U}}^{†} = {\hat{U}}^{- 1}, \hat{U} | ψ ⟩ = | ψ ⟩, {\hat{A}}^{'} = \hat{U} \hat{A} {\hat{U}}^{†}, \hat{A} = {\hat{U}}^{†} {\hat{A}}^{'} \hat{U}$ .

Definition

${\hat{U}}_{ε} (\hat{G}) = I + i ε \hat{G}$

Check

$I = U_{ε} U_{ε}^{†} = (I + i ε \hat{G}) (I - i ε {\hat{G}}^{†}) = I + i ε (\hat{G} - {\hat{G}}^{†}) + O (ε^{2}) = I$ .

Thus, $\hat{G}$ must be Hermitian.

Finite Transformation

${\hat{U}}_{α} (\hat{G}) = {\hat{U}}_{ε} (\hat{G})^{N} = I + N i ε \hat{G} = I + i α \hat{G} = \exp (i α \hat{G})$ .

Applications

Time Translations

Let $\hat{G} = \frac{\hat{H}}{ℏ}$ . Then, ${\hat{U}}_{δ t} (\hat{H}) = I - \frac{i}{ℏ} \hat{H} δ t$ . So,

$\hat{H} | ψ (t) ⟩ = i ℏ \frac{1}{\partial t} | ψ (t) ⟩$
${\hat{U}}_{δ t} | ψ (t) ⟩ = (I - i \hat{H} / ℏ δ t) | ψ (t) ⟩ = | ψ (t) ⟩ - i^{2} δ t \frac{\partial}{\partial t} | ψ (t) ⟩ \approx | ψ (t + δ t) ⟩$ .

Spatial Translations

Let $\hat{G} = \frac{1}{ℏ} \hat{P_{x}}$ . Then,

${\hat{U}}_{δ x} | ψ (x) ⟩ = | ψ (x) ⟩ + \frac{i}{ℏ} (δ x) (- i ℏ \frac{\partial}{\partial x}) = | ψ (x + δ x) ⟩$ .
${\hat{X}}^{'} = {\hat{X}}^{'} + \frac{i}{ℏ} (δ X) [{\hat{P}}_{x}, \hat{X}] \hat{X} + (δ X)$
${\hat{Y}}^{'} = \hat{Y}$

Rotations

$\hat{G} = \frac{\hat{J_{z}}}{ℏ}$ . ${\hat{U}}_{d φ} = I + \frac{i}{ℏ} d φ {\hat{J}}_{z}$