Generalized Uncertainty Principle

#Physics

Topics

• Heisenberg's Uncertainty Principle

$\displaystyle \sigma_{A}^{2}\sigma_{B}^{2}\geq \left(\frac{1}{2i}{\left\langle{[\hat{A},\hat{B}]}\right\rangle}\right)^{2}$

• Page 138 of Griffiths
• Operators that don't commute have non-zero products of their uncertainties and are called incompatible observables
• Operators that do commute are called compatible observables

Examples

$\displaystyle \Delta x\Delta p\geq \frac{\hbar}{2}$

$\displaystyle \Delta t\Delta E\geq \frac{\hbar}{2}$

• Shown by using $\displaystyle \sigma_{H}\sigma_{q}\geq \frac{\hbar}{2}\left\lvert \frac{ \mathrm{d}{\left\langle{Q}\right\rangle} }{ \mathrm{d}t } \right\rvert$ and $\displaystyle \Delta t\equiv \frac{\sigma_{Q}}{\left\lvert \frac{ \mathrm{d}{\left\langle{Q}\right\rangle} }{ \mathrm{d}t } \right\rvert}$
• $\displaystyle \Delta t$ represents the amount of time it takes for $\displaystyle {\left\langle{Q}\right\rangle}$ to change by one standard deviation
  • Examples (3.5-3.7 in Griffiths):
    • Period of oscillation for superposition of two stationary states
    • Time it takes for a free particle to pass a certain point
    • Lifetime of a particle

$\displaystyle \Delta L_{x}\Delta L_{y}\geq \frac{\hbar}{2}\lvert {\left\langle{L_{z}}\right\rangle}\rvert$