Ch 39: Particles Behaving as Waves

Young & Freedman Calc - University Physics 14th Edition

Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Not the one you use?Change textbook

Young & Freedman Calc 14th Edition

Ch 39: Particles Behaving as Waves

Problem 45a

Chapter 39, Problem 45a

The uncertainty in the y-component of a proton's position is $2.0 \times 10^{- 12}$ m. What is the minimum uncertainty in a simultaneous measurement of the $y$ -component of the proton's velocity?

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Start by recalling the Heisenberg Uncertainty Principle, which states that the product of the uncertainties in position (Δy) and momentum (Δp_y) is at least as large as the reduced Planck's constant divided by 2: Δy * Δp_y ≥ ℏ / 2. Here, ℏ is the reduced Planck's constant, ℏ = h / (2π), where h = 6.626 × 10^-34 J·s.

Express the momentum uncertainty (Δp_y) in terms of the velocity uncertainty (Δv_y) and the mass of the proton (m): Δp_y = m * Δv_y. The mass of a proton is approximately 1.67 × 10^-27 kg.

Substitute Δp_y = m * Δv_y into the uncertainty relation: Δy * m * Δv_y ≥ ℏ / 2. Rearrange this equation to solve for the velocity uncertainty: Δv_y ≥ ℏ / (2 * m * Δy).

Substitute the given uncertainty in position (Δy = 2.0 × 10^-12 m), the mass of the proton (m = 1.67 × 10^-27 kg), and the reduced Planck's constant (ℏ = 1.055 × 10^-34 J·s) into the equation: Δv_y ≥ (1.055 × 10^-34) / (2 * 1.67 × 10^-27 * 2.0 × 10^-12).

Simplify the expression to find the minimum uncertainty in the y-component of the proton's velocity (Δv_y). Ensure the units are consistent and verify the calculation step-by-step to avoid errors.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Heisenberg Uncertainty Principle

The Heisenberg Uncertainty Principle states that it is impossible to simultaneously know both the exact position and exact momentum (or velocity) of a particle. This principle is fundamental in quantum mechanics and implies that the more precisely one property is measured, the less precisely the other can be controlled or known.

Position and Momentum Relationship

In the context of the uncertainty principle, position and momentum are related through the equation Δx * Δp ≥ ħ/2, where Δx is the uncertainty in position, Δp is the uncertainty in momentum, and ħ is the reduced Planck's constant. This relationship indicates that an increase in the uncertainty of position (Δx) leads to a decrease in the uncertainty of momentum (Δp), which includes velocity.

Planck's Constant

Planck's constant (ħ) is a fundamental constant in quantum mechanics, approximately equal to 1.055 x 10^-34 Js. It plays a crucial role in the Heisenberg Uncertainty Principle, providing the scale at which quantum effects become significant. Understanding this constant is essential for calculating uncertainties in measurements of particles like protons.

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(a) What is the temperature of the hotter star in terms of $T$ ?

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(a) The $x$ -coordinate of an electron is measured with an uncertainty of $0.30$ mm. What is the x-component of the electron's velocity, $v_{x}$ , if the minimum percent uncertainty in a simultaneous measurement of $v_{x}$ is $1.0 percent sign$ ?

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A scientist has devised a new method of isolating individual particles. He claims that this method enables him to detect simultaneously the position of a particle along an axis with a standard deviation of $0.12$ nm and its momentum component along this axis with a standard deviation of $3.0 \times 10^{- 25}$ kg-m/s. Use the Heisenberg uncertainty principle to evaluate the validity of this claim.

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The uncertainty in the y-component of a proton's position is 2.0×10−122.0\(\times\)10^{-12} m. What is the minimum uncertainty in a simultaneous measurement of the yy-component of the proton's velocity?

Verified video answer for a similar problem:

Key Concepts

Heisenberg Uncertainty Principle

Position and Momentum Relationship

Planck's Constant

The uncertainty in the y-component of a proton's position is $2.0 \times 10^{- 12}$ m. What is the minimum uncertainty in a simultaneous measurement of the $y$ -component of the proton's velocity?