From the Heisenberg uncertainty principle how do you calculate Δx for each of the following: (a) an electron with Δv = 0.340 m/s (b) a baseball (mass = 145 g) with Δv = 0.200 m/s?

The formula for the Heisenberg uncertainty principle is

#color(blue)(|bar(ul(color(white)(a/a) ΔpΔx≥h/(4π) color(white)(a/a)|)))" "#

where

• #Δp# is the uncertainty in the momentum
• #Δx# is the uncertainty in the position
• #h# is Planck's constant (#6.626 × 10^"-34" color(white)(l)"kg·m"^2"s"^"-1"#)

The momentum #p = mv#, and the mass #m# is a constant, so

#Δp =Δ(mv) = mΔv#

The uncertainty principle then becomes

#mΔvΔx≥h/(4π)#

or

#Δx ≥h/(4πmΔv)#

(a) #Δx# for an electron

#Δx ≥h/(4πmΔv) = (6.626 × 10^"-34" color(red)(cancel(color(black)("kg")))·stackrel("m")(color(red)(cancel(color(black)("m"^2))))color(red)(cancel(color(black)("s"^"-1"))))/(4π × 9.109 × 10^"-31" color(red)(cancel(color(black)("kg"))) × 0.340 color(red)(cancel(color(black)("m·s"^"-1")))) = 1.70 × 10^"-4"color(white)(l) "m" = "0.170 mm"#

(b) #Δx# for a baseball

#Δx ≥h/(4πmΔv) = (6.626 × 10^"-34" color(red)(cancel(color(black)("kg")))·stackrel("m")(color(red)(cancel(color(black)("m"^2))))color(red)(cancel(color(black)("s"^"-1"))))/(4π × 0.145 color(red)(cancel(color(black)("kg"))) × 0.200 color(red)(cancel(color(black)("m·s"^"-1")))) = 1.82 × 10^"-33"color(white)(l) "m"#