## University Physics with Modern Physics (14th Edition)

a. The Heisenberg uncertainty principle limits our knowledge. $$\Delta x \Delta p_x \geq \frac{\hbar}{2}$$ $$\Delta x\;m \Delta v_x \geq \frac{\hbar}{2}$$ We are told that $\ \Delta v_x = (0.010)v_x$ Set the product of uncertainties to its minimum value. $$\Delta x\;m \Delta v_x = \frac{\hbar}{2}$$ $$\Delta x\;m (0.010)v_x = \frac{\hbar}{2}$$ $$v_x =\frac{\hbar}{2m (0.010) \Delta x}=19.3m/s$$ b. Repeat, using the mass of a proton, to find 11 mm/s.