Problem 46 - General Chemistry: Principles and Modern Applications

Question

Show that the uncertainty principle is not significant when applied to large objects such as automobiles. Assume that \(m\) is precisely known; assign a reasonable value to either the uncertainty in position or the uncertainty in velocity, and estimate a value of the other.

Step-by-Step Solution

Answer

The calculated value for the uncertainty in velocity \(\Delta v\) is extremely small, around the order of \(10^{-34}\) m/s. This value is far too small to have any impact on the measurement or behavior of an automobile. Hence, it can be concluded that the uncertainty principle is not significant when applied to large objects like automobiles.

Step 1: Assign a value for uncertainty

Let's choose an uncertainty in position \(\Delta x\). A suitable value for an automobile might be \(\Delta x = 1\) mm, which represents a very small deviation in the position of a car, considering its size.

Step 2: Identify Known Variables

The mass of an automobile \(m\) can be assumed to be about 1500 kg, a typical mass of a car. Planck's constant \(h = 6.62607015 × 10^{-34}\) m^2 kg / s

Step 3: Calculate uncertainty in velocity

Substitute all the known values into the uncertainty principle formula to solve for \(\Delta v\). We get: \(\Delta v \geq \frac{h}{4\pi m \Delta x}\). After calculation, we will find that \(\Delta v\) is significantly small, implying that for such a large object like a car, the uncertainty principle has negligible impact.

Key Concepts

Macroscopic Objects

When dealing with the world around us, we often focus on macroscopic objects. These are everyday items like cars, books, and people—anything large enough to be seen with the naked eye. Unlike microscopic particles that are governed by quantum mechanics, macroscopic objects behave according to classical physics. For example, when you throw a ball, its trajectory follows predictable laws, allowing you to calculate where it will land. This predictability is due to the negligible effect of quantum uncertainties on large, visible objects. In simpler terms, the larger the mass of an object, the less impact the quantum rules have on it.

• Macroscopic objects include anything visible without a microscope.
• They follow classical, rather than quantum, physics.
• Due to their size, uncertainties in position or velocity are usually too small to notice.

Position Uncertainty

When it comes to quantum mechanics, position uncertainty is a vital concept. This term refers to the inability to determine the exact position of a particle at any given time. According to Heisenberg's Uncertainty Principle, the more precisely you know a particle's position, the less precisely you can know its velocity and vice versa. Now, let's apply this principle to macroscopic objects like a car.
For example, if the position uncertainty (\(\Delta x\)) of a car is as tiny as 1 mm, it may seem insignificant. This small uncertainty makes sense because cars are large compared to microscopic particles. In practical terms, this means knowing the position of your car with an error of just a millimeter, thanks to its size and the limits of human measurement accuracy. This tiny uncertainty indicates why the principle doesn’t affect large objects like cars much in our daily lives.

• Position uncertainty is the inability to know an object's exact location.
• Carrying a minuscule uncertainty for macroscopic objects.
• Makes the uncertainty principle barely noticeable for large objects.

Velocity Uncertainty

Velocity uncertainty refers to how precisely we cannot determine a particle's speed or direction at a specific time. In quantum terms, the more accurately you know an object's position, the less accurately you can know its velocity according to Heisenberg’s Uncertainty Principle. For large objects, like automobiles, this uncertainty becomes less significant. Let's consider a car with a mass of around 1500 kg.
Given the position uncertainty (\(\Delta x\)) of 1 mm, using the uncertainty principle formula \(\Delta v \geq \frac{h}{4\pi m \Delta x}\), we can calculate the velocity uncertainty (\(\Delta v\)). The small magnitude of Planck's constant means that the velocity uncertainty for such a massive object results in an insignificantly small delta in speed. Thus, in reality, you can rely on your car's speedometer for precise speed readings, showing that the effect of the uncertainty principle is negligible for big objects like cars.

• Velocity uncertainty relates to the precision of knowing an object's speed.
• The larger the object, the smaller the impact of velocity uncertainty.
• Planck's constant ensures minimal velocity uncertainty for macroscopic objects.