Generalized Uncertainty Principle

Topics

• Heisenberg’s Uncertainty Principle

${\sigma }_A^2{\sigma }_B^2\ge {\left(\frac{1}{2i}⟨[\hat{A},\hat{B}]⟩\right)}^2$

• 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

$\Delta x\Delta p\ge \frac{\hslash }{2}$

$\Delta t\Delta E\ge \frac{\hslash }{2}$

• Shown by using ${\sigma }_H{\sigma }_q\ge \frac{\hslash }{2}\left|\frac{\mathrm{d}⟨Q⟩}{\mathrm{d}t}\right|$ and $\Delta t\equiv \frac{{\sigma }_Q}{\left|\frac{\mathrm{d}⟨Q⟩}{\mathrm{d}t}\right|}$
• $\Delta t$ represents the amount of time it takes for $⟨Q⟩$ 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

$\Delta L_x\Delta L_y\ge \frac{\hslash }{2}|⟨L_z⟩|$