Textbook Question
An experiment has four possible outcomes, labeled A to D. The probability of A is PA = 40% and of B is PB = 30%. Outcome C is twice as probable as outcome D. What are the probabilities PC and PD?
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Ch 39: Wave Functions and Uncertainty
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796
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Table of contents
- Ch 01: Concepts of Motion35
- Ch 02: Kinematics in One Dimension75
- Ch 03: Vectors and Coordinate Systems58
- Ch 04: Kinematics in Two Dimensions60
- Ch 05: Force and Motion23
- Ch 06: Dynamics I: Motion Along a Line53
- Ch 07: Newton's Third Law27
- Ch 08: Dynamics II: Motion in a Plane52
- Ch 09: Work and Kinetic Energy56
- Ch 10: Interactions and Potential Energy50
- Ch 11: Impulse and Momentum56
- Ch 12: Rotation of a Rigid Body71
- Ch 13: Newton's Theory of Gravity52
- Ch 14: Fluids and Elasticity44
- Ch 15: Oscillations55
- Ch 16: Traveling Waves37
- Ch 17: Superposition39
- Ch 18: A Macroscopic Description of Matter53
- Ch 19: Work, Heat, and the First Law of Thermodynamics60
- Ch 20: The Micro/Macro Connection53
- Ch 21: Heat Engines and Refrigerators34
- Ch 22: Electric Charges and Forces40
- Ch 23: The Electric Field39
- Ch 24: Gauss' Law32
- Ch 25: The Electric Potential63
- Ch 26: Potential and Field57
- Ch 27: Current and Resistance42
- Ch 28: Fundamentals of Circuits39
- Ch 29: The Magnetic Field47
- Ch 30: Electromagnetic Induction60
- Ch 31: Electromagnetic Fields and Waves32
- Ch 32: AC Circuits63
- Ch 33: Wave Optics32
- Ch 34: Ray Optics57
- Ch 35: Optical Instruments30
- Ch 36: Special Relativity38
- Ch 37: The Foundations of Modern Physics25
- Ch 38: Quantization49
- Ch 39: Wave Functions and Uncertainty31
- Ch 40: One-Dimensional Quantum Mechanics35
- Ch 41: Atomic Physics18
- Ch 42: Nuclear Physics27
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook
Chapter 39, Problem 9
When 5×1012 photons pass through an experimental apparatus, 2.0×109 land in a 0.10-mm-wide strip. What is the probability density at this point?
Verified step by step guidance1
Step 1: Understand the problem. The probability density is defined as the probability per unit area. To calculate it, we need to determine the fraction of photons that land in the specified strip and divide it by the area of the strip.
Step 2: Calculate the fraction of photons that land in the strip. This is given by the ratio of the number of photons landing in the strip to the total number of photons: \( \text{Fraction} = \frac{2.0 \times 10^9}{5.0 \times 10^{12}} \).
Step 3: Determine the area of the strip. The width of the strip is given as \( 0.10 \ \text{mm} \), which needs to be converted to meters: \( 0.10 \ \text{mm} = 0.10 \times 10^{-3} \ \text{m} \). Assume the length of the strip is \( 1 \ \text{m} \) (if not specified, this is a common assumption for such problems). Thus, the area is \( \text{Area} = \text{Width} \times \text{Length} = (0.10 \times 10^{-3}) \times 1 \).
Step 4: Calculate the probability density. The probability density is the fraction of photons divided by the area of the strip: \( \text{Probability Density} = \frac{\text{Fraction}}{\text{Area}} = \frac{\frac{2.0 \times 10^9}{5.0 \times 10^{12}}}{(0.10 \times 10^{-3}) \times 1} \).
Step 5: Simplify the expression to find the probability density. Perform the division and simplify the units to express the final result in terms of \( \text{m}^{-2} \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Photon
A photon is a quantum of electromagnetic radiation, representing the smallest discrete amount of light or other electromagnetic energy. Photons are massless particles that travel at the speed of light and exhibit both wave-like and particle-like properties. Understanding photons is essential for analyzing interactions in experiments involving light and energy transfer.
Probability Density
Probability density refers to the likelihood of finding a particle, such as a photon, in a specific region of space per unit area. It is calculated by dividing the number of particles that land in a given area by the area itself. In this context, it helps quantify how concentrated the photons are in the specified strip, providing insight into their distribution.
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Intro to Density
Area Calculation
Area calculation is fundamental in physics for determining the spatial extent of a region where events occur, such as the landing of photons. In this scenario, the width of the strip (0.10 mm) must be converted to appropriate units (e.g., square meters) to calculate the probability density accurately. This concept is crucial for relating the number of photons to the area they occupy.
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04:05Calculating Work As Area Under F-x Graphs
Related Practice
Textbook Question
FIGURE EX39.13 shows the probability density for an electron that has passed through an experimental apparatus. What is the probability that the electron will land in a 0.010-mm-wide strip at x = 0.000 mm?
Textbook Question
In one experiment, 2000 photons are detected in a 0.10-mm-wide strip where the amplitude of the electromagnetic wave is 10 V/m. How many photons are detected in a nearby 0.10-mm-wide strip where the amplitude is 30 V/m?
Textbook Question
FIGURE EX39.14 is a graph of |ψ(x)|2 for an electron. What is the value of a?
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Textbook Question
FIGURE EX39.12 shows the probability density for an electron that has passed through an experimental apparatus. If 1.0×106 electrons are used, what is the expected number that will land in a 0.010-mm-wide strip at 2.000 mm?