Using the Heisenberg uncertainty principle, calculate Δx for each of the following. a. an electr

Using the Heisenberg uncertainty principle, calculate Δx for...

Using the Heisenberg uncertainty principle, calculate $\Delta x$ for each of the following. a. an electron with $\Delta v=0.100 \mathrm{~m} / \mathrm{s}$ b. a baseball (mass $=145 \mathrm{~g}$ ) with $\Delta v=0.100 \mathrm{~m} / \mathrm{s}$ c. How does the answer in part a compare with the size of a hydrogen atom? d. How does the answer in part b correspond to the size of a baseball?

Using the Heisenberg uncertainty principle, calculate $\Delta x$ for each of the following.
a. an electron with $\Delta v=0.100 \mathrm{~m} / \mathrm{s}$
b. a baseball (mass $=145 \mathrm{~g}$ ) with $\Delta v=0.100 \mathrm{~m} / \mathrm{s}$
c. How does the answer in part a compare with the size of a hydrogen atom?
d. How does the answer in part b correspond to the size of a baseball?

Key Concepts

Measurement Limitations in Quantum Mechanics

In the realm of quantum mechanics, the process of measurement itself imposes limitations on the precision with which certain pairs of properties can be known. These inherent limitations are not due to imperfections in measurement instruments, but are instead a fundamental feature of quantum systems, reflecting the probabilistic nature of quantum states.

Quantum versus Classical Scales

This concept emphasizes the stark contrast between the behavior of microscopic particles and macroscopic objects. While quantum uncertainty plays a crucial role in the behavior of particles like electrons, leading to measurable effects on their position and momentum, classical objects such as baseballs have such large masses that the quantum uncertainties become insignificantly small in everyday measurements.

Planck's Constant

Planck’s constant is a fundamental constant in quantum mechanics that sets the scale at which quantum effects become important. Its extremely small value is the reason why quantum uncertainties, such as those described by the Heisenberg uncertainty principle, are significant only at microscopic scales and are virtually negligible in everyday macroscopic phenomena.

Heisenberg Uncertainty Principle

This principle states that certain pairs of observables, such as position and momentum, cannot be simultaneously measured with arbitrary precision. In essence, the more precisely one of these values is known, the less precisely the other can be determined. This relationship is quantitatively expressed by an inequality involving Planck’s constant, highlighting a fundamental limit imposed by nature.

Momentum Uncertainty

Momentum uncertainty arises due to the intrinsic uncertainty in measuring a particle’s velocity, given its mass. Since momentum is the product of mass and velocity, even a slight imprecision in the velocity measurement leads to a corresponding uncertainty in momentum, which, through the uncertainty principle, directly influences the uncertainty in position.

Transcript

00:01 Heisenberg's uncertainty principle tells us that it's not possible to know exactly the position or the momentum of an object at any given time in space.

00:11 This is expressed as the following equation where there's uncertainty in distance or position as well as an uncertainty in the momentum or mass times velocity.

00:24 And there's always a certain amount of uncertainty that's greater than or equal to a constant value.

00:30 The constant is planck's constant divided by 4 pi.

00:48 So if you're given a certain mass or a certain displacement or a certain speed, you can find what's called the uncertainty using this equation.

00:57 So in our first question, they talk about an electron moving with a certain velocity of 0 .1 meters per second.

01:04 And it asks to find the uncertainty in position.

01:08 So we're solving for delta x.

01:10 So our uncertainty in position is greater than or equal to h.

01:15 Over 4 pi divided by the momentum or mass times velocity.

01:23 The mass of an electron is a constant value equal to 9 .11 times 10 to the negative 31 kilograms.

01:36 Quant's constant is also a constant value, 6 .626 times 10 to the negative 34 joules times second.

01:49 And our velocity is given as 0 .100 meters per second.

01:59 So we can put those values into our equation and solve for the uncertainty in position, which tells us that the uncertainty in position is greater than or equal to 5 .79 times 10 to the negative 4 meters...

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