Quantum Mechanical Translations

Discrete and Continuous Eigenvalues

Discrete

$A | φ_{n} ⟩ = a_{n} | φ_{n} ⟩$

$⟨ φ_{n} | φ_{m} ⟩ = δ_{m}^{n}$

$\sum_{n} | φ_{n} ⟩ ⟨ φ_{n} | = I$

$| ψ ⟩ = \sum_{n} | φ_{n} ⟩ ⟨ φ_{n} | ψ ⟩$ .

$\sum_{n} | ⟨ φ_{n} | ψ ⟩ |^{2} = 1$ .

$⟨ φ | ψ ⟩ = \sum_{n} ⟨ φ | φ_{n} ⟩ ⟨ φ_{n} | ψ ⟩$ .

$⟨ | φ_{n} | A | φ_{m} ⟩ = a_{m} δ_{m}^{n}$ .

Continuous

$B | ξ^{'} ⟩ = b | ξ^{'} ⟩$

$⟨ ξ^{'} | ξ^{″} ⟩ = δ (ξ^{'} - ξ^{″})$

Where the delta function is the Dirac Delta Function.

$\int d ξ^{'} | ξ ⟩ ⟨ ξ^{'} | = I$ .

$| χ ⟩ = \sum_{R} d ξ^{'} | ξ^{'} ⟩ ⟨ ξ^{'} | χ ⟩$ .

$\int_{R} d ξ^{'} | ⟨ ξ^{'} | χ ⟩ |^{2} = 1$ .

$⟨ χ | γ ⟩ = \int_{R} d ξ^{'} χ^{*} (ξ^{'}) γ (ξ^{'})$ .

$⟨ ξ^{″} | B | ξ^{'} ⟩ = b δ (ξ^{″} - ξ^{'})$ .

Position

Example

Gaussian Wave Packets $X$ in one dimension. $X | x^{'} ⟩ = x^{'} | x^{'} ⟩$ . $| ψ ⟩ = \int_{R} d x d x^{'} | x^{'} ⟩ ⟨ x^{'} | ψ ⟩$ .

Practically speaking, $P = | ⟨ x^{'} | ψ ⟩ |^{2} \cdot (δ x)$ . Then, $\int_{R} | ⟨ x^{'} | ψ ⟩ |^{2} d x^{'} = 1$ . $| ψ^{'} ⟩_{a f t e r} = \int_{x^{'} - (δ x) / 2}^{x^{'} + (δ x) / 2} d x | x ⟩ ⟨ x | ψ ⟩$ .

$⟨ x^{'} | ψ ⟩ = ψ (x^{'})$ .

Spatial Translations

${\hat{U}}_{d \vec{x}} = I - i d \vec{x} \cdot \hat{G} = I - i d \vec{x} \cdot \frac{\hat{\vec{p}}}{ℏ}$ , ${\hat{U}}_{d \vec{x}} | {\vec{x}}^{'} ⟩ = | {\vec{x}}^{'} + d \vec{x} ⟩$ .

Successive translations: ${\hat{U}}_{d {\vec{x}}^{'}} {\hat{U}}_{d {\vec{x}}^{″}} = I - \frac{i}{ℏ} \hat{\vec{p}} (d {\vec{x}}^{'} + d {\vec{x}}^{″}) = {\hat{U}}_{d {\vec{x}}^{'} + d {\vec{x}}^{″}}$ .

Momentum Eigenkets from Spatial Translations

${\hat{U}}_{d \vec{x}} | {\vec{p}}^{'} ⟩ = (1 - i / ℏ {\vec{p}}^{'} \cdot d {\vec{x}}^{'}) | {\vec{p}}^{'} ⟩$ .

Inserting Identity

$ψ_{α} (x^{'}) = ⟨ x^{'} | α ⟩$ , $⟨ β | α ⟩ = \int d x^{'} ⟨ β | x^{'} ⟩ ⟨ x^{'} | a ⟩ = \int d x^{'} ψ_{β}^{*} (x^{'}) ψ_{α} (x^{'}) = \int d p^{'} ⟨ β | p^{'} ⟩ ⟨ p^{'} | α ⟩ = \int d p^{'} Ψ_{β}^{*} (p^{'}) Ψ_{α} (p^{'})$

$⟨ β | A | α ⟩ = \iint d x^{'} d x^{″} ψ_{β}^{*} (x^{'}) ⟨ x^{'} | A | x^{″} ⟩ ψ_{α} (x^{″})$ .

Example

$A = {\hat{X}}^{2}$ . Then, $⟨ x^{'} | {\hat{X}}^{2} | x^{″} ⟩ = (x^{″})^{2} δ (x^{″} - x^{'})$ .

$⟨ β | A | α ⟩ = \int d x^{'} ψ_{β}^{*} (x^{'}) ψ_{α} (x^{'}) x^{' 2}$ .

Passive and Active Transformations

Active

$\vec{r} \to {\vec{r}}^{'}$ . The coordinates remain the same. (Our negative sign indicates that this we are using an active translation)

Passive

$\vec{r} \to {\vec{r}}^{'}$ . The coordinate system changes.

Momentum

Operator in Position Basis

$⟨ β | A | α ⟩ = \iint d x^{'} d x^{″} ⟨ β | x^{'} ⟩ ⟨ x^{'} | A | x^{″} ⟩ ⟨ x^{″} | α ⟩ = \iint d x^{'} d x^{″} ψ_{β}^{*} (x^{'}) ψ_{α} (x^{″}) ⟨ x^{'} | A | x^{″} ⟩$

If $A = F (X)$ then this is easy to compute, $⟨ x^{'} | A | x^{″} ⟩ = F (x^{″}) δ (x^{'} - x^{″})$ . So, $⟨ β | A | α ⟩ = \int d x^{'} ψ_{β}^{*} (x^{'}) ψ_{α} (x^{'}) F (x^{'})$ .

If $A = f (\hat{P})$ ? Then, $⟨ x^{'} | f (\hat{P}) | x^{″} ⟩$ . $⟨ p^{'} | A | p^{″} ⟩ = f (p^{″}) δ (p^{'} - p^{″})$ .

Recall

${\hat{U}}_{Δ x} = I - \frac{i}{ℏ} {\hat{P}}_{x} Δ x$ , ${\hat{U}}_{Δ x} | x ⟩ = | x + Δ x ⟩$ .

${\hat{U}}_{Δ x} | α ⟩ = {\hat{U}}_{Δ x} \int d x | x ⟩ ⟨ x | α ⟩ = \int d x | x + Δ x ⟩ ⟨ x | α ⟩ = \int d x^{'} | x^{'} ⟩ ⟨ x^{'} - Δ x | α ⟩ = \int d x^{'} | x^{'} ⟩ [⟨ x^{'} | α - Δ x \frac{\partial}{\partial x^{'}} ⟨ x^{'} | α ⟩] = | α ⟩ - Δ x \int d x^{'} \frac{\partial}{\partial x^{'}} ⟨ x^{'} | α ⟩ = (I - \frac{i}{ℏ} {\hat{P}}_{x} Δ x) | α ⟩$ . So, $\hat{P_{x}} | α ⟩ = - i ℏ \int d x^{'} | x^{'} ⟩ \frac{\partial}{\partial x^{'}} ⟨ x^{'} | α ⟩$ . $⟨ x^{″} | {\hat{P}}_{x} | α ⟩ = - i ℏ \frac{\partial}{\partial x^{″}} ⟨ x^{″} | α ⟩ = - i ℏ \frac{\partial}{\partial x^{″}} ψ_{α} (x^{″})$ .

If $| α ⟩ = | x^{'} ⟩$ then $⟨ x^{″} | {\hat{P}}_{x} | x^{'} ⟩ = - i ℏ \frac{\partial}{\partial x^{″}} δ (x^{″} - x^{'})$ .

Thus, if $A = f (\hat{P})$ then $⟨ x^{'} | f (\hat{P}) | x^{″} ⟩ = \iint d x^{'} d x^{″} ψ_{β}^{*} (x^{'}) f (- i ℏ \frac{\partial}{\partial x^{″}} δ (x^{'} - x^{″})) ψ_{α} (x^{″}) = \int d x^{'} ψ_{β}^{*} (x^{'}) f (- i ℏ \frac{\partial}{\partial x^{'}}) ψ_{α} (x^{'})$

Momentum Space Wavefunction

$\hat{P} | p^{'} ⟩ = p^{'} | p^{'} ⟩$ , $⟨ p^{'} | p^{″} ⟩ = δ (p^{'} - p^{″})$ , $| α ⟩ = \int d p^{'} | p^{'} ⟩ ⟨ p^{'} | α ⟩$ .

${\hat{P}}_{x} ≐ - i ℏ \frac{\partial}{\partial x}$ , $\hat{X} ≐ i ℏ \frac{\partial}{\partial p}$ , $P (p^{'} - Δ p / 2, p^{'} + Δ p / 2) = | ⟨ p^{'} | α ⟩ |^{2} Δ p = | Φ_{α} (p^{'}) |^{2} Δ p$ .

$⟨ x^{″} | \hat{P} | α = - i ℏ \frac{\partial}{\partial x^{″}} ⟨ x^{″} | α ⟩$ .

$| α ⟩ = | p^{'} ⟩$ . Then, $⟨ x^{″} | \hat{P} | α ⟩ = - i ℏ \frac{\partial}{\partial x^{″}} ⟨ x^{″} | p^{'} ⟩ = p^{'} ⟨ x^{″} | p^{'} ⟩$ . So, $p^{'} ⟨ x^{″} | p^{'} ⟩ = - i ℏ \frac{\partial}{\partial x^{″}} ⟨ x^{″} | p^{'} ⟩$ . So, $⟨ x | p ⟩ = C \exp [\frac{i}{ℏ} p x]$ .