In which of the following are quantum uncertainties significant: measuring simultaneously the speed and location of a baseball, a spitball, or an electron?
1 Answer
Well, what does quantum mechanics apply to? Small things... VERY small things, i.e. electrons, protons, etc. Things with significant wavelengths and barely any mass.
The Heisenberg Uncertainty Principle can be used to confirm this:
#DeltaxDeltap >= ħ/2#
or
#mDeltaxDeltav >= ħ/2#
#ħ# is on the order of#10^(-34)# #"kg"cdot"m"^2//"s"# .
Actually, it's#ħ = h/(2pi) = 1.0546 xx 10^(-34)# #"kg"cdot"m"^2//"s"# .
THE UNCERTAINTY PRINCIPLE... ON MACROSCOPIC PARTICLES???
If we consider a typical uncertainty of
#("0.145 kg")("0.001 m")Deltav >= 1.0546 xx 10^(-34) "kg"cdot"m"^2//"s"#
And the uncertainty in the velocity would be:
#Deltav >= (1.0546 xx 10^(-34) "kg"cdot"m"^2//"s")/(("0.145 kg")("0.001 m"))#
#>= color(red)(7.273 xx 10^(-31))# #color(red)("m/s")# Well, obviously, we are always sure enough about the velocity of a baseball that this is trivially satisfied...
This is a sign that the Uncertainty Principle does NOT work on macroscopic particles.
If we take the average speed of a baseball to be
THE UNCERTAINTY PRINCIPLE WORKS BEST ON QUANTUM PARTICLES
On the other hand, an electron fares much better when it comes to what we expect. Let's say we were extremely sure about its position, such as
Then the uncertainty in its velocity should be large:
#color(blue)(Deltav) >= (1.0546 xx 10^(-34) "kg"cdot"m"^2//"s")/((9.109 xx 10^(-31) "kg")(10^(-16) "m"))#
#>=# #color(blue)(1.158 xx 10^(12) "m/s")#
This in fact makes sense, because the Uncertainty Principle said, if we are sure about the position, we cannot be that sure about the velocity.
They cannot simultaneously be observed to the same degree of certainty, inasmuch as the particle is quantum-sized.
