In which of the following are quantum uncertainties significant: measuring simultaneously the speed and location of a baseball, a spitball, or an electron?

Answer 1

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 #pm "0.001 m"# for a #"0.145 kg"# baseball with an average diameter of #"0.072 m"#, then
#("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 #"90 mph"#, or about #"40.2336 m/s"#, clearly, it makes more sense to have an uncertainty of about #"0.0001 m/s"#, which is way larger than #7.273 xx 10^(-31)# #"m/s"#. This makes physical sense, because we can SEE baseballs move.

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 #Deltax = 10^(-16) "m"# (the approx. radius of the electron).

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.

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Answer 2

Quantum uncertainties are important when determining an electron's velocity and position.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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