Problem#1
A pesky 1.5-mg mosquito is annoying you as you attempt to study physics in your room, which is 5.0 m wide and 2.5 m high. You decide to swat the bothersome insect as it flies toward you, but you need to estimate its speed to make a successful hit. (a) What is the maximum uncertainty in the horizontal position of the mosquito? (b) What limit does the Heisenberg uncertainty principle place on your ability to know the horizontal velocity of this mosquito? Is this limitation a serious impediment to your attempt to swat it?
Answer:
Since we know only that the mosquito is somewhere in the room, there is an uncertainty in its
position. The Heisenberg uncertainty principle tells us that there is an uncertainty in its momentum.
The uncertainty principle is 𝜟x𝜟px ≥ ℏ/2
(a) You know the mosquito is somewhere in the room, so the maximum uncertainty in its
horizontal position is Δx = 5.0 m.
(b) The uncertainty principle gives 𝜟x𝜟px ≥ ℏ/2, and 𝜟px = m𝜟vx since we know the mosquito’s mass. This gives 𝜟x m𝜟vx ≥ ℏ/2, which we can solve for Δvx to get the minimum uncertainty in vx.
𝜟vx = ℏ/2m𝜟x = (1.055 x 10-34 J.s)/[2(1.5 x 10-6 kg)(5.0 m)] = 7.0 x 10-30 m/s, which is hardly a serious impediment!
For something as “large” as a mosquito, the uncertainty principle places a negligible limitation on our ability to measure its speed.
Problem#2
By extremely careful measurement, you determine the x-coordinate of a car’s center of mass with an uncertainty of only The car has a mass of 1200 kg. (a) What is the minimum uncertainty in the x-component of the velocity of the car’s center of mass as prescribed by the Heisenberg uncertainty principle? (b) Does the uncertainty principle impose a practical limit on our ability to make simultaneous measurements of the positions and velocities of ordinary objects like cars, books, and people? Explain.
Answer:
(a) Use 𝜟x𝜟px ≥ ℏ/2, to calculate Δpx and obtain Δvx from this
𝜟vx = ℏ/2m𝜟x = (1.055 x 10-34 J.s)/[2(1200 kg)(1.0 x 10-6 m)] = 4.42 x 10-32 m/s
(b) Even for this very small Δx the minimum Δvx required by the Heisenberg uncertainty
principle is very small. The uncertainty principle does not impose any practical limit on the simultaneous measurements of the positions and velocities of ordinary objects.
Problem#3
A 10.0-g marble is gently placed on a horizontal tabletop that is 1.75 m wide. (a) What is the maximum uncertainty in the horizontal position of the marble? (b) According to the Heisenberg uncertainty principle, what is the minimum uncertainty in the horizontal velocity of the marble? (c) In light of your answer to part (b), what is the longest time the marble could remain on the table? Compare this time to the age of the universe, which is approximately 14 billion years. (Hint: Can you know that the horizontal velocity of the marble is exactly zero?)
Jawab:
Since we know that the marble is somewhere on the table, there is an uncertainty in its position. The Heisenberg uncertainty principle tells us that there is therefore an uncertainty in its momentum.
The uncertainty principle is 𝜟x𝜟px ≥ ℏ/2
(a) Since the marble is somewhere on the table, the maximum uncertainty in its horizontal
position is Δx = 1.75 m.
(b) Following the same procedure as in part (b) of Problem 1, the minimum uncertainty in the
horizontal velocity of the marble is
𝜟vx = ℏ/2m𝜟x = (1.055 x 10-34 J.s)/[2(0,0100 kg)(1,75 x 10-6 m)] = 3.01 x 10-33 m/s
(c) The uncertainty principle tells us that we cannot know that the marble’s horizontal velocity is exactly zero, so the smallest we could measure it to be is 3.01 x 10-33 m/s, from part (b). The longest time it could remain on the table is the time to travel the full width of the table (1.75 m), so
t = x/vx = (1.75 m)/(3.01 x 10-33 m/s) = 5.81 x 1032 s = 1.84 x 1025 years
Since the universe is about 14 x 109 years old, this time is about
(1.84 x 1025 years)/(14 x 109 years) ≈ 1.3 x 1015 times the age of the universe! Don’t hold your breath!
For household objects, the uncertainty principle places a negligible limitation on our ability to measure their speed.
Post a Comment for " The Uncertainty Principle Revisited Problems and Solutions"