College Physics
11th Edition
ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
Publisher:Raymond A. Serway, Chris Vuille
Chapter1: Units, Trigonometry. And Vectors
Section: Chapter Questions
Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...
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What is the independent wave function for a particle in a box x=0,L
![**Title: Quantum Mechanics Problem: Particle in a Box**
Welcome to the Quantum Mechanics section! Today, we will explore one of the fundamental problems in quantum mechanics: the particle in a box. This exercise involves calculating wave functions and probabilities, essential concepts for understanding quantum systems.
---
**Problem Statement:**
1. **Determine the Time-Independent Wave Function:**
What is the time-independent wave function for a particle in a box [0,L] with an infinite potential?
\[ \psi_n(x) = \ ? \]
2. **Normalize the Wave Function:**
Now calculate the normalization constant by integrating the square of the wave function over the interval [0,L].
\[ \int_{0}^{L} |\psi(x)|^2 \, dx = \ ? \]
3. **Calculate Probability:**
Finally, calculate the probability of finding a particle in a specified region. For instance, determine the probability of finding a particle in the range \( x = 0 \) to \( x = \frac{L}{2} \).
\[ \text{Calculate probability of finding a particle in } \{0 \le x \le \frac{L}{2} \}. \]
---
This problem will guide you through the process of determining wave functions and using them to find probabilities, key techniques in quantum mechanics.
Feel free to explore each step in detail and try solving the integrals. This will bolster your understanding of how quantum states are described mathematically and how these descriptions correlate with physical probabilities.
Happy learning! Remember, the exercise is not just about finding the right answers but also about understanding the underlying principles governing quantum systems.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5886f279-1ae3-47c9-a907-c1e5911c0685%2Fea8e34a1-0d3c-4ec4-b13f-866f1ebc8f3e%2Fl61hilc_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Quantum Mechanics Problem: Particle in a Box**
Welcome to the Quantum Mechanics section! Today, we will explore one of the fundamental problems in quantum mechanics: the particle in a box. This exercise involves calculating wave functions and probabilities, essential concepts for understanding quantum systems.
---
**Problem Statement:**
1. **Determine the Time-Independent Wave Function:**
What is the time-independent wave function for a particle in a box [0,L] with an infinite potential?
\[ \psi_n(x) = \ ? \]
2. **Normalize the Wave Function:**
Now calculate the normalization constant by integrating the square of the wave function over the interval [0,L].
\[ \int_{0}^{L} |\psi(x)|^2 \, dx = \ ? \]
3. **Calculate Probability:**
Finally, calculate the probability of finding a particle in a specified region. For instance, determine the probability of finding a particle in the range \( x = 0 \) to \( x = \frac{L}{2} \).
\[ \text{Calculate probability of finding a particle in } \{0 \le x \le \frac{L}{2} \}. \]
---
This problem will guide you through the process of determining wave functions and using them to find probabilities, key techniques in quantum mechanics.
Feel free to explore each step in detail and try solving the integrals. This will bolster your understanding of how quantum states are described mathematically and how these descriptions correlate with physical probabilities.
Happy learning! Remember, the exercise is not just about finding the right answers but also about understanding the underlying principles governing quantum systems.
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