Question

Calculate the uncertainty in the position of (a) an electron moving at a speed of \((3.00 \pm 0.01) \times 10^{5} \mathrm{~m} / \mathrm{s},(\mathbf{b})\) a neutron moving at this same speed. (The masses of an electron and a neutron are given in the table of fundamental constants in the inside cover of the text.) (c) Based on your answers to parts (a) and (b), which can we know with greater precision, the position of the electron or of the neutron?

Short answer

The uncertainty in the position of the electron is \(\Delta x_e \approx 5.78 \times 10^{-9} \mathrm{m}\) and the uncertainty in the position of the neutron is \(\Delta x_n \approx 3.14 \times 10^{-12} \mathrm{m}\). Since the uncertainty in the position of the neutron is less than that of the electron, the position of the neutron can be known with greater precision.

Step-by-step solution

Calculate the uncertainty in momentum of the electron

Using the given uncertainty in speed of the electron, \(\Delta v = 0.01 \times 10^{5} \mathrm{m/s}\), we can calculate the uncertainty in its momentum. From the table of fundamental constants, we can find the mass of the electron \(m_e = 9.109 \times 10^{-31} \mathrm{kg}\). \(\Delta p_e = m_e \Delta v = (9.109 \times 10^{-31} \mathrm{kg})(0.01 \times 10^{5} \mathrm{m/s}) = 9.109 \times 10^{-27} \mathrm{kg \cdot m/s}\)

Calculate the uncertainty in momentum of the neutron

Similarly, using the given uncertainty in speed of the neutron, we can find its uncertainty in momentum. From the table of fundamental constants, we can obtain the mass of the neutron \(m_n = 1.675 \times 10^{-27} \mathrm{kg}\). \(\Delta p_n = m_n \Delta v = (1.675 \times 10^{-27} \mathrm{kg})(0.01 \times 10^{5} \mathrm{m/s}) = 1.675 \times 10^{-23} \mathrm{kg \cdot m/s}\)

Calculate the uncertainty in position of electron using Heisenberg uncertainty principle

Now that we have the uncertainty in momentum for the electron, we can use the Heisenberg uncertainty principle equation to find the uncertainty in position: \(\Delta x_e \Delta p_e \geq \frac{\hbar}{2}\) \(\Delta x_e \geq \frac{\hbar}{2 \Delta p_e}\) where \(\hbar\) is the reduced Planck constant \(= 1.055 \times 10^{-34} \mathrm{Js}\). \(\Delta x_e \geq \frac{1.055 \times 10^{-34} \mathrm{Js}}{2 (9.109 \times 10^{-27} \mathrm{kg \cdot m/s})} \approx 5.78 \times 10^{-9} \mathrm{m}\)

Calculate the uncertainty in position of neutron using Heisenberg uncertainty principle

Similarly, we can find the uncertainty in position for the neutron: \(\Delta x_n \Delta p_n \geq \frac{\hbar}{2}\) \(\Delta x_n \geq \frac{\hbar}{2 \Delta p_n}\) \(\Delta x_n \geq \frac{1.055 \times 10^{-34} \mathrm{Js}}{2 (1.675 \times 10^{-23} \mathrm{kg \cdot m/s})} \approx 3.14 \times 10^{-12} \mathrm{m}\)

Compare the uncertainties in position, and determine which particle's position can be known with greater precision

We have found the uncertainties in position for both the electron and the neutron: \(\Delta x_e \approx 5.78 \times 10^{-9} \mathrm{m}\) \(\Delta x_n \approx 3.14 \times 10^{-12} \mathrm{m}\) Since \(\Delta x_n < \Delta x_e\), the uncertainty in the position of the neutron is less than that of the electron. Therefore, the position of the neutron can be known with greater precision.

Key concepts

Uncertainty in Position

The uncertainty in position is a key concept in quantum mechanics, which arises directly from the Heisenberg uncertainty principle. This principle tells us that there's a limit to how precisely we can know both the position and momentum of a particle simultaneously. The more accurately we know one, the less accurately we can know the other. In practical terms, when we measure the position of a particle with high precision, its momentum becomes less certain, and vice versa.

For example, in the case of the electron moving at a certain speed, the uncertainty in its position is related to the uncertainty in its momentum. By using the Heisenberg uncertainty principle formula \(\Delta x \cdot \Delta p \geq \frac{\hbar}{2}\), scientists can estimate the minimum uncertainty in the electron’s position based on how well the momentum is known.

Momentum Uncertainty

Momentum uncertainty refers to the lack of precision in measuring the momentum of a particle. According to the Heisenberg uncertainty principle, there's always an inherent uncertainty in the measurement of a particle’s momentum if its position is measured precisely. The momentum of a particle, which is the product of its mass and velocity, will have an uncertainty that is obtained from the standard deviation of repeated measurements of its velocity.

This concept is crucial in quantum mechanics as it defines the limit to which we can predict the behavior of particles. It has practical consequences, for example, in designing electron microscopes, where the resolution is ultimately limited by the uncertainty principle.

Electron and Neutron Mass

The mass of subatomic particles like electrons and neutrons is fundamental in calculating the momentum uncertainty and, by extension, the position uncertainty via the Heisenberg uncertainty principle. Electrons are much less massive than neutrons, with typical values taken from fundamental constants: the electron’s mass is around \(9.109 \times 10^{-31}\) kilograms, while the neutron’s mass is about \(1.675 \times 10^{-27}\) kilograms.

The significant difference in mass between electrons and neutrons means they behave differently under similar conditions and explains why, in the given example, the uncertainty in position for a neutron can be pinpointed with greater accuracy as compared to an electron.

Fundamental Constants

Fundamental constants are a set of physical quantities in nature that are universal in scope and remain unchanged across spacetime. They are crucial in the foundation of laws of physics and include constants such as the speed of light in a vacuum \(c\), the gravitational constant \(G\), and the reduced Planck constant \(\hbar\).

The reduced Planck constant, in particular, plays a pivotal role in the Heisenberg uncertainty principle, serving as the bridge between the quantum world and the measurable quantities of momentum and position. It’s a small value, around \(1.055 \times 10^{-34}\) joule-seconds, which indicates why quantum effects are not noticeable in everyday life but become significant at the atomic and subatomic scales.