2022-05-02

From Canonical Commutation Relation to Heisenberg's Uncertainty Principle

Introducing a real coefficient $\xi$ and an operator $\hat{F}$, where $$ -\infty < \xi < \infty $$ and $$ \hat{F} = \xi \hat{A} + i \hat{B} \quad \hat{F} = \xi \hat{A} - i \hat{B}, $$ so that $$ \begin{aligned} \hat{F}^\dagger \hat{F} &= \xi^2 \hat{A}^2 + \hat{B}^2 + i \xi (\hat{A} \hat{B} - \hat{B} \hat{A}) \\ &= \xi^2 \hat{A} + \hat{B}^2 - \xi \hat{C}. \end{aligned} $$ Let $$ I(\xi) = \braket{ \hat{F}^\dagger \hat{F} } = \int \psi^* \hat{F}^\dagger \hat{F} \psi d {\tau} = \xi^2 \bar{A}^2 + \bar{B}^2 - \xi \bar{C} \geq 0, $$ where $$ \int \psi^* \hat{F}^\dagger \hat{F} \psi d {\tau} = \int \hat{F} \psi ( F \psi )^* d {\tau}. $$ $$ \implies \bar{A}^2 \bar{B}^2 \geq \frac{\bar{C}^2}{4} $$ $$ \begin{align} \hat{A} \rightarrow \hat{A} - \bar{A} \\ \hat{B} \rightarrow \hat{B} - \bar{B} \end{align}$$ $$ \implies \Delta A \cdot \Delta B \geq \frac{|\bar{C}|}{2} $$ $$ \Delta x \Delta p_x \geq \frac{\hbar}{2} $$