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 0.120.12 nm and its momentum component along this axis with a standard deviation of 3.0×10−253.0\$$times10^{-25} kg-m/s. $$Use the Heisenberg uncertainty principle to evaluate the validity of this claim.

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

All textbooks

Young & Freedman Calc 14th Edition

Ch 39: Particles Behaving as Waves

Problem 47

Chapter 39, Problem 47

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.

Verified step by step guidance

Step 1: Recall the Heisenberg uncertainty principle, which states that the product of the uncertainties in position (Δx) and momentum (Δp) must satisfy the inequality Δx * Δp ≥ ℏ/2, where ℏ is the reduced Planck's constant (ℏ = 1.0545718 × 10^-34 J·s).

Step 2: Identify the given values in the problem: the standard deviation of the position (Δx) is 0.12 nm (convert this to meters: 0.12 × 10^-9 m), and the standard deviation of the momentum (Δp) is 3.0 × 10^-25 kg·m/s.

Step 3: Calculate the product of the uncertainties Δx * Δp using the given values. Multiply the converted position uncertainty (in meters) by the momentum uncertainty (in kg·m/s).

Step 4: Compare the calculated product Δx * Δp to the minimum value required by the Heisenberg uncertainty principle, ℏ/2. Compute ℏ/2 using ℏ = 1.0545718 × 10^-34 J·s and divide it by 2.

Step 5: Evaluate whether the scientist's claim is valid by checking if the calculated product Δx * Δp meets or exceeds ℏ/2. If it does not, the claim violates the Heisenberg uncertainty principle and is invalid.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

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 of a particle. Mathematically, it is expressed as 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 principle highlights the fundamental limits of measurement in quantum mechanics.

Standard Deviation in Measurement

Standard deviation is a statistical measure that quantifies the amount of variation or dispersion in a set of values. In the context of the Heisenberg Uncertainty Principle, the standard deviations of position (Δx) and momentum (Δp) represent the uncertainties in these measurements. A smaller standard deviation indicates a more precise measurement, but according to the uncertainty principle, this precision in one measurement leads to greater uncertainty in the other.

Planck's Constant

Planck's constant (ħ) is a fundamental constant in quantum mechanics that relates the energy of a photon to its frequency. It has a value of approximately 1.055 x 10^-34 J·s. In the context of the Heisenberg Uncertainty Principle, it sets the scale for the limits of precision in measurements of position and momentum, emphasizing that at quantum scales, these uncertainties are not just theoretical but have real implications for particle behavior.

Recommended video:

Guided course

08:59

Phase Constant of a Wave Function

Related Practice

Textbook Question

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?

1577

views

Textbook Question

(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 $1.0 percent sign$ ?

(b) Repeat part (a) for a proton.

1967

views

Textbook Question

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?

1705

views

Textbook Question

$10.0$ -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?)

1748

views