/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 32 In an experiment with 10,000 ele... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 40: Problem 32

In an experiment with 10,000 electrons, which land symmetrically on both sides of \(x=0,5000\) are detected in the range \(-1.0 \mathrm{cm} \leq x \leq+1.0 \mathrm{cm}, 7500\) are detected in the range \(-2.0 \mathrm{cm} \leq x \leq+2.0 \mathrm{cm},\) and all 10,000 are detected in the range \(-3.0 \mathrm{cm} \leq x \leq+3.0 \mathrm{cm} .\) Draw a graph of a probability density that is consistent with these data. (There may be more than one acceptable answer.)

Short Answer

Expert verified
Given that the total amount of electrons detected was 10,000, it infers that for the three ranges -1.0 to +1.0, -2.0 to +2.0, and -3.0 to +3.0 cm, the corresponding probabilities were 0.5, 0.75, and 1.0 respectively. When plotted on a graph, these points should create a cumulative probability density function. It's symmetrical around x=0, forming a conspicuous 'S' curve. There are other plausible answers, considering that the distribution of probability density within these ranges wasn't explicitly stated.

Step by step solution

01

Understand the distribution of results

Firstly, realize that there are three intervals given, \(-1.0 \mathrm{cm} \leq x \leq+1.0 \mathrm{cm}\), \(-2.0 \mathrm{cm} \leq x \leq+2.0 \mathrm{cm}\), and \(-3.0 \mathrm{cm} \leq x \leq+3.0 \mathrm{cm}\), which contain 5000, 7500 and 10000 electrons respectively. You should maintain the symmetry of the graph on both sides of \(x=0\), as stated in the exercise.
02

Converting numbers into probability

As there are 10,000 electrons in total, each detected electron is equivalent to a probability of 0.0001 (given by 1/10,000). To attain the probability for each range, multiply the number of detected electrons in each range by 0.0001. Therefore, 5000 electrons equates to a probability of 0.5, 7500 to 0.75, and 10000 to 1.0.
03

Drawing the graph

Plot the three probability intervals along the x-axis, ranging from -3.0 cm to +3.0 cm. Mark the vertical line at \(x=0\) to note its symmetry. The y-axis will represent the cumulative probability density, ranging from 0 to 1.0. Plot the three probabilities corresponding to the three ranges along the y-axis, thus, 0.5 at \(x = \pm1.0\), 0.75 at \(x = \pm2.0\), and 1.0 at \(x = \pm3.0\). Next, connect the points to create a line graph鈥攖his represents the cumulative probability distribution. As it's symmetric around x=0, the graph will resemble a 'S' shape.
04

Interpretation of results

The graph obtained corresponds to the probability density function according to the given information. Always note that other possibilities might exist since it is not specified how the probability is distributed in between these provided ranges.

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.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Electron Distribution
Understanding electron distribution is crucial in physics, especially in experiments dealing with particles. In this context, we're focusing on how electrons are detected within specific ranges. The experiment mentions 10,000 electrons distributed symmetrically around the origin (x=0). This symmetry indicates that as many electrons land on the left side as on the right side of the axis.

Key points to remember include:
  • Symmetrical distribution: Equal distribution of electrons on both sides of a central point.
  • Ranges of detection: Electrons detected within specific intervals, showing how spread out they are.
This distribution is essential for interpreting results and understanding the underlying physical phenomena.
Cumulative Probability
Cumulative probability helps us understand the likelihood of an event up to a certain point. In the exercise, it describes how many electrons are detected within various ranges. We convert the number of electrons to probability by dividing by the total number of electrons.

Here's how it works:
  • Probability calculation: Divide the number of electrons detected by the total (10,000) to find the probability.
  • Cumulative aspect: Probabilities add up as you move to a larger range, culminating in 1.0 for the full range.
Cumulative probability gives a comprehensive picture of events occurring up to a certain limit, useful for deriving conclusions from experimental data.
Symmetry in Physics
Symmetry plays a vital role in physics, indicating a balance or uniformity within a system. In this electron experiment, symmetry occurs around the origin, meaning that the electron distribution is even on both sides.

Why is this important?
  • Consistent results: Symmetry ensures that the experimental outcomes are consistent and reliable.
  • Predictability: It allows predictions about how electrons behave under similar conditions.
  • Graphical representation: Symmetrical graphs (like the 'S' shape here) illustrate how values change uniformly around a central point.
In interpreting data, symmetry lets us infer how phenomena are likely to behave under varied scenarios.
Experimental Data Interpretation
Interpreting experimental data involves analyzing results to draw meaningful conclusions. In this case, understanding how electrons distribute helps in visualizing the probability density function.

Steps to take during interpretation:
  • Data conversion: Convert electron counts into probabilities for analysis.
  • Graph plotting: Use graphs to visualize probability distributions and understand relationships.
  • Multiple interpretations: Be aware that data can have more than one interpretation, depending on assumptions.
Data interpretation is both an art and science, requiring careful consideration of available evidence and assumptions.

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

Consider the electron wave function $$\psi(x)=\left\\{\begin{array}{ll}c \sqrt{1-x^{2}} & |x| \leq 1 \mathrm{cm} \\\0 & |x| \geq 1 \mathrm{cm}\end{array}\right.$$ where \(x\) is in \(\mathrm{cm}\). a. Determine the normalization constant \(c\) b. Draw a graph of \(\psi(x)\) over the interval \(-2 \mathrm{cm} \leq x \leq 2 \mathrm{cm} .\) Provide numerical scales on both axes. c. Draw a graph of \(|\psi(x)|^{2}\) over the interval \(-2 \mathrm{cm} \leq x \leq\) 2 cm. Provide numerical scales. d. If \(10^{4}\) electrons are detected, how many will be in the interval \(0.00 \mathrm{cm} \leq x \leq 0.50 \mathrm{cm} ?\)

a. Starting with the expression \(\Delta f \Delta t \approx 1\) for a wave packet, find an expression for the product \(\Delta E \Delta t\) for a photon. b. Interpret your expression. What does it tell you? c. The Bohr model of atomic quantization says that an atom in an excited state can jump to a lower-energy state by emitting a photon. The Bohr model says nothing about how long this process takes. You'll learn in Chapter 42 that the time any particular atom spends in the excited state before emitting a photon is unpredictable, but the average lifetime \Deltat of many atoms can be determined. You can think of \(\Delta t\) as being the uncertainty in your knowledge of how long the atom spends in the excited state. A typical value is \(\Delta t \approx 10\) ns. Consider an atom that emits a photon with a 500 nm wavelength as it jumps down from an excited state. What is the uncertainty in the energy of the photon? Give your answer in eV. d. What is the fractional uncertainty \(\Delta E / E\) in the photon's energy?

\(1.0 \times 10^{10}\) photons pass through an experimental apparatus. How many of them land in a 0.10 -mm-wide strip where the probability density is \(20 \mathrm{m}^{-1} ?\)

A thin solid barrier in the \(x y\) -plane has a \(10-\mu\) m-diameter circular hole. An electron traveling in the z-direction with \(v_{x}=0 \mathrm{m} / \mathrm{s}\) passes through the hole. Afterward, is \(v_{x}\) still zero? If not, within what range is \(v_{x}\) likely to be?

Physicists use laser beams to create an atom trap in which atoms are confined within a spherical region of space with a diameter of about \(1 \mathrm{mm}\). The scientists have been able to cool the atoms in an atom trap to a temperature of approximately \(1 \mathrm{nK},\) which is extremely close to absolute zero, but it would be interesting to know if this temperature is close to any limit set by quantum physics. We can explore this issue with a one-dimensional model of a sodium atom in a 1.0 -mm-long box. a. Estimate the smallest range of speeds you might find for a sodium atom in this box. b. Even if we do our best to bring a group of sodium atoms to rest, individual atoms will have speeds within the range you found in part a. Because there's a distribution of speeds, suppose we estimate that the root-mean-square speed \(v_{\max }\) of the atoms in the trap is half the value you found in part a. Use this \(v_{\operatorname{mat}}\) to estimate the temperature of the atoms when they've been cooled to the limit set by the uncertainty principle.
See all solutions

Recommended explanations on Physics Textbooks

Thermodynamics

Read Explanation

Physics of Motion

Read Explanation

Force

Read Explanation

Wave Optics

Read Explanation

Modern Physics

Read Explanation

Oscillations

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.