6. One electron is trapped in a one-dimensional square well potential with infinitely high sides. Assume that the well extends from x = 0 to x =L. The width of the well is L=306.6pm. Find the energies (E,,E2,E‚,E,) of the four lowest levels in electron volts (eV). a. b. Sketch the probability density functions for the two lowest energy states: P(x), P,(x). Draw an energy level diagram for the for the first four energy levels. С.
6. One electron is trapped in a one-dimensional square well potential with infinitely high sides. Assume that the well extends from x = 0 to x =L. The width of the well is L=306.6pm. Find the energies (E,,E2,E‚,E,) of the four lowest levels in electron volts (eV). a. b. Sketch the probability density functions for the two lowest energy states: P(x), P,(x). Draw an energy level diagram for the for the first four energy levels. С.
Chemistry
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ISBN:9781305957404
Author:Steven S. Zumdahl, Susan A. Zumdahl, Donald J. DeCoste
Publisher:Steven S. Zumdahl, Susan A. Zumdahl, Donald J. DeCoste
Chapter1: Chemical Foundations
Section: Chapter Questions
Problem 1RQ: Define and explain the differences between the following terms. a. law and theory b. theory and...
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Please explain parts c, d, and e. Please include significant figures and units. Thanks for your help!
![**Quantum Mechanics Problem: One-Dimensional Square Well Potential**
In this problem, we consider an electron trapped in a one-dimensional square well potential with infinitely high sides. The width of the well is given as \( L = 306.6 \, \text{pm} \) (picometers). The well extends from \( x = 0 \) to \( x = L \).
### Tasks:
**a. Finding Energy Levels:**
Determine the energies \( E_1, E_2, E_3, E_4 \) of the four lowest levels in electron volts (eV).
**b. Sketching Probability Density Functions:**
Sketch the probability density functions for the two lowest energy states: \( P_1(x) \) and \( P_2(x) \).
**c. Energy Level Diagram:**
Draw an energy level diagram for the first four energy levels.
**d. Determining Wavelengths of Absorbed Photons:**
Find the wavelengths \( \lambda_a \) and \( \lambda_b \) of the two longest wavelength photons that could be absorbed by an electron in the first excited state (the \( n=2 \) state) in this trap. It is assumed that one-dimensional electron traps can absorb and emit photons.
**e. Indicating Transitions on Energy Level Diagram:**
Indicate the transitions involved in part "d." on the energy level diagram.
### Explanation of Related Diagrams and Equations:
1. **Energy Levels:**
The energy levels for a particle in a one-dimensional infinite potential well are given by the equation:
\[ E_n = \frac{n^2 \pi^2 \hbar^2}{2mL^2} \]
where:
- \( n \) is the quantum number (1, 2, 3, ...),
- \( \hbar \) is the reduced Planck's constant,
- \( m \) is the mass of the electron,
- \( L \) is the width of the well.
2. **Probability Density Functions:**
The probability density functions for the lowest energy states in a one-dimensional infinite potential well are given by:
\[ P_n(x) = \left( \sqrt{\frac{2}{L}} \sin \left( \frac{n \pi x}{L} \right) \right)^2 \]
for \( n = 1, 2 \).
3. **Energy Level](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb067f6a6-d6d0-4a81-90c2-26a75bd86046%2F388ec841-8f83-4b3d-8ef3-8d6eaf0d2d98%2Fskemnis_processed.png&w=3840&q=75)
Transcribed Image Text:**Quantum Mechanics Problem: One-Dimensional Square Well Potential**
In this problem, we consider an electron trapped in a one-dimensional square well potential with infinitely high sides. The width of the well is given as \( L = 306.6 \, \text{pm} \) (picometers). The well extends from \( x = 0 \) to \( x = L \).
### Tasks:
**a. Finding Energy Levels:**
Determine the energies \( E_1, E_2, E_3, E_4 \) of the four lowest levels in electron volts (eV).
**b. Sketching Probability Density Functions:**
Sketch the probability density functions for the two lowest energy states: \( P_1(x) \) and \( P_2(x) \).
**c. Energy Level Diagram:**
Draw an energy level diagram for the first four energy levels.
**d. Determining Wavelengths of Absorbed Photons:**
Find the wavelengths \( \lambda_a \) and \( \lambda_b \) of the two longest wavelength photons that could be absorbed by an electron in the first excited state (the \( n=2 \) state) in this trap. It is assumed that one-dimensional electron traps can absorb and emit photons.
**e. Indicating Transitions on Energy Level Diagram:**
Indicate the transitions involved in part "d." on the energy level diagram.
### Explanation of Related Diagrams and Equations:
1. **Energy Levels:**
The energy levels for a particle in a one-dimensional infinite potential well are given by the equation:
\[ E_n = \frac{n^2 \pi^2 \hbar^2}{2mL^2} \]
where:
- \( n \) is the quantum number (1, 2, 3, ...),
- \( \hbar \) is the reduced Planck's constant,
- \( m \) is the mass of the electron,
- \( L \) is the width of the well.
2. **Probability Density Functions:**
The probability density functions for the lowest energy states in a one-dimensional infinite potential well are given by:
\[ P_n(x) = \left( \sqrt{\frac{2}{L}} \sin \left( \frac{n \pi x}{L} \right) \right)^2 \]
for \( n = 1, 2 \).
3. **Energy Level
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