Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 34: Problem 36

An electron is moving in the \(+x\) -direction with speed measured at \(50 \mathrm{Mm} / \mathrm{s}\), accurate to \(\pm 10 \% .\) What's the minimum uncertainty in its position?

Short Answer

Expert verified
The minimum uncertainty in the position of the electron is the calculated \(\Delta x\). By accurately inserting the values into the equations, the answer should come out to be approximately \(1.15 \times 10^{-9} m\), subject to the precise values of the constants.

Step by step solution

01

Understand the Given Information

An electron is moving with speed, \(v = 50 \pm 10 \%\) Mm/s. The uncertainty in the speed, \(\Delta v = 10 \% \) of v, so \(\Delta v = 5 \) Mm/s. First the units have to be converted into the standard SI units which are in meters per seconds (m/s). So, \(v = 50 \times 10^6 m/s\) and \(\Delta v = 5 \times 10^6 m/s\).
02

Calculate the Uncertainty in Momentum

Momentum (p) is given by the equation, \(p = mv\), where m is the mass of the electron and v is its speed. The uncertainty in the momentum, \(\Delta p\) can be calculated using the formula: \(\Delta p = m\Delta v\). Substitute m (mass of the electron, \(9.1 \times 10^{-31} kg\)) and \(\Delta v = 5 \times 10^6 m/s\) to calculate \(\Delta p\).
03

Apply Heisenberg's Uncertainty Principle

Heisenberg's Uncertainty Principle states that the product of the uncertainties in the position and momentum of a particle is at least as large as half of Planck's constant (h-bar). Mathematically, this is given by \(\Delta x \Delta p \geq \frac{h}{4\pi}\). To find the minimum uncertainty in position \(\Delta x\), we will rearrange the uncertainty principle equation to \(\Delta x = \frac{h}{4\pi \Delta p}\) and substitute the value of \(\Delta p\) from step 2 and Planck's constant (\(h = 6.626 \times 10^{-34} Js\)) to calculate \(\Delta x\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Is it possible to determine an electron's velocity accurate to \(\pm 1 \mathrm{m} / \mathrm{s}\) while simultaneously finding its position to within \(\pm 1 \mu \mathrm{m} ?\) What about a proton?

Treating the Sun as a \(5800-\mathrm{K}\) blackbody, compare its UV radiance at 200 nm with its visible radiance at its 500 -nm peak wavelength.

(a) Find the highest possible energy for a photon emitted as the electron jumps between two adjacent energy levels in the Bohr hydrogen atom. (b) Which energy levels are involved?

Through what potential difference should you accelerate an electron from rest so its de Broglie wavelength will be the size of a hydrogen atom, about 0.1 nm?

Imagine an atom that, unlike hydrogen, had only three energy levels. If these levels were evenly spaced, how many spectral lines would result? How would their wavelengths compare?
See all solutions

Recommended explanations on Physics Textbooks

Quantum Physics

Read Explanation

Electricity

Read Explanation

Nuclear Physics

Read Explanation

Classical Mechanics

Read Explanation

Solid State Physics

Read Explanation

Geometrical and Physical Optics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free